WEBVTT

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MAGDALENA TODA: Welcome
to our review of 13.1.

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How many of you didn't
get your exams back?

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I have your exam, and yours.

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And you have to wait.

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I don't have it with me.

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I have it in my office.

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If you have questions
about the score,

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why don't you go ahead and
email me right after class.

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Chapter 13 is a very
physical chapter.

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It has a lot to do with
mechanical engineering,

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with mechanics,
physics, electricity.

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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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Do you remember the definition?

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So we define the work
as a path integral

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along the regular curve.

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And by regular curve-- I'm
sorry if I'm repeating myself,

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but this is part of the
deal-- R is the position

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vector in R3 that is class C1.

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That means differentiable and
derivatives are continuous.

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Plus you are not
allowed to stop.

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

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and not even for a
fraction of a second is he

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or she allowed to
have velocity 0.

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At no point I want
to have velocity 0.

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And that's the position vector.

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And then you have some force
field acting on you-- no,

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acting on the particle
at every moment.

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So you have an F that is
acting at location xy.

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Maybe if you are in space,
let's talk about the xyz,

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where x is a function of
t, y is a functional of t,

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z is a function of t,
which is the same as saying

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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y of t.

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Let me put angular bracket,
although I hate them,

00:02:15.580 --> 00:02:20.140
because you like angular
brackets for vectors.

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F is also a nice function.

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How nice?

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We discussed a
little bit last time.

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It really doesn't
have to be continuous.

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The book assumes it continues.

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It has to be
integrable, so maybe it

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could be piecewise continuous.

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So I had nice enough, was
it continues piecewise.

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And we define the work as
being the path integral over c.

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I keep repeating, because
that's going to be on the final

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as well.

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So all the notions
that are important

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should be given enough
attention in this class.

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Hi.

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So do you guys remember
how we denoted F?

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F was, in general, three
components in our F1, F2, F3.

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They are functions of
the position vector,

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or the position xyz.

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And the position is
a function of time.

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So all in all, after
you do all the work,

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keep in mind that when you
multiply with a dot product,

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the integral will give you what?

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A time integral?

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From a time T0 to a time
T1, you are here at time T0

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and you are here at time T1.

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Maybe your curve is
piecewise, differentiable,

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you don't know what it is.

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But let's assume just a
very nice, smooth arc here.

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Of what?

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Of F1 times what is that?
x prime of t plus F2 times

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y prime of t, plus F3
times z prime of t dt.

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So keep in mind that
Mr. dR is your friend.

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And he was-- what was he?

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Was defined as the
velocity vector multiplied

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by the infinitesimal element dt.

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Say again, the
velocity vector prime

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was a vector in F3 quantified
by the infinitesimal element dt.

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So we reduce this Calc
3 notion path integral

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to a Calc 1 notion, which was a
simple integral from t0 to t1.

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And we've done a
lot of applications.

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What else have we done?

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We've done some
integral of this type

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over another curve, script c.

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I'm repeating mostly for Alex.

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You're caught in the process.

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And there are two or three
people who need an update.

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Maybe I have another
function of g and ds.

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And this is an integral that
in the end will depend on s.

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But s itself depends on t.

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So if I were to re-express
this in terms of d,

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how would I re-express
the whole thing?

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g of s, of t, whatever that
is, then Mr. ds was what?

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STUDENT: s prime of t.

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MAGDALENA TODA: Right.

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So this was the-- that s
prime of t was the speed.

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The speed of the arc of a curve.

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So you have an R of
t and R3, a vector.

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And the speed was,
by definition,

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arc length element was
by definition integral

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from 2t0 to t.

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Of the speed R prime
magnitude d tau.

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I'll have you put tau
because I'm Greek,

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and it's all Greek to me.

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So the tau, some people call
the tau the dummy variable.

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I don't like to call it dumb.

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It's a very smart variable.

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It goes from t0 to t, so what
you have is a function of t.

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This guy is speed.

00:06:32.900 --> 00:06:38.880
So when you do that
here, ds becomes speed,

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R prime of t times dt.

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This was your old friend ds.

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And let me put it on top
of this guy with speed.

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Because he was so
important to you,

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you cannot forget about him.

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So that was review of--
reviewing of 13.1 and 13.2

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There were some things in
13.3 that I pointed out

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to you are important.

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13.3 was independence of path.

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Everybody write, magic-- no.

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Magic section.

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No, have to be serious.

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So that's independence of path
of certain type of integrals,

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of some integrals.

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And an integral like
that, a path integral

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is independent of path.

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When would such an animal--
look at this pink animal,

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inside-- when would
this not depend

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on the path you are taking
between two given points?

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So I can move on another
arc and another arc

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and another regular arc, and
all sorts of regular arcs.

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It doesn't matter
which path I'm taking--

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STUDENT: If that
force is conservative.

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MAGDALENA TODA: If the
force is conservative.

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Excellent, Alex.

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And what did it mean for a
force to be conservative?

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How many of you
know-- it's no shame.

00:08:15.880 --> 00:08:17.350
Just raise hands.

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If you forgot what it is,
don't raise your hand.

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But if you remember what it
means for a force F force

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field-- may the force be
with you-- be conservative,

00:08:27.340 --> 00:08:30.157
then what do you do?

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Say F is conservative
by definition.

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When, if and only, F
there is a so-called--

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STUDENT: Scalar.

00:08:50.750 --> 00:08:51.916
MAGDALENA TODA: --potential.

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Scalar potential, thank you.

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I'll fix that.

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A scalar potential function f.

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Instead of there is, I
didn't want to put this.

00:09:09.850 --> 00:09:11.650
Because a few
people told me they

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got scared about
the symbolistics.

00:09:14.570 --> 00:09:17.146
This means "there exists."

00:09:17.146 --> 00:09:22.380
OK, smooth potential
such that-- at least

00:09:22.380 --> 00:09:26.800
is differential [INAUDIBLE]
1 such that the nabla of f--

00:09:26.800 --> 00:09:28.240
what the heck is that?

00:09:28.240 --> 00:09:32.180
The gradient of this little
f will be the given F.

00:09:32.180 --> 00:09:38.500
And we saw all sorts of wizards
here, like, Harry Potter,

00:09:38.500 --> 00:09:42.930
[INAUDIBLE] well,
there are many,

00:09:42.930 --> 00:09:47.670
Alex, Erin, many,
many-- Matthew.

00:09:47.670 --> 00:09:49.540
So what did they do?

00:09:49.540 --> 00:09:50.961
They guessed the
scalar potential.

00:09:50.961 --> 00:09:53.370
I had to stop because
there are 10 of them.

00:09:53.370 --> 00:09:55.940
It's a whole school
of Harry Potter.

00:09:55.940 --> 00:09:58.760
How do they find the little f?

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Through witchcraft.

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No.

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Normally you should--

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STUDENT: I've actually
done it through witchcraft.

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Tell you that?

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MAGDALENA TODA: You did.

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I think you can do it
through witchcraft.

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But practically everybody
has the ability to guess.

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Why do we have the ability
to guess and check?

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Because our brain does
the integration for you.

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Whether you tell your
brain to stop or not,

00:10:24.160 --> 00:10:27.250
when your brain, for example,
sees is kind of function--

00:10:27.250 --> 00:10:30.340
and now I'm gonna
test your magic skills

00:10:30.340 --> 00:10:32.040
on a little harder one.

00:10:32.040 --> 00:10:36.040
I didn't want to do an
R2 value vector function.

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Let me go to R3.

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But I know that you have
your witchcraft handy.

00:10:42.070 --> 00:10:47.650
So let's say somebody
gave you a force field

00:10:47.650 --> 00:10:55.620
that is yz i plus xzj plus xyk.

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And you're going to jump and
say this is a piece of cake.

00:10:58.900 --> 00:11:03.680
I can see the scalar potential
and just wave my magic wand,

00:11:03.680 --> 00:11:06.950
and I get it.

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STUDENT: [INAUDIBLE]

00:11:08.099 --> 00:11:09.390
MAGDALENA TODA: Oh my god, yes.

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Guys, you saw it fast.

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OK, I should be proud of you.

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And I am proud of you.

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I've had made classes
where the students couldn't

00:11:18.610 --> 00:11:21.930
see any of the scalar
potentials that I gave them,

00:11:21.930 --> 00:11:23.930
that I asked them to guess.

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How did you deal with it?

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You integrate this
with respect to F?

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In the back of
your mind you did.

00:11:29.927 --> 00:11:31.510
And then you guessed
one, and then you

00:11:31.510 --> 00:11:34.120
said, OK so should be xyz.

00:11:34.120 --> 00:11:36.790
Does it verify my
other two conditions?

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And you say, oh yeah, it does.

00:11:38.460 --> 00:11:42.550
Because of I prime with respect
to y, I have exactly xz.

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If I prime with respect to c I
have exactly xy, so I got it.

00:11:46.510 --> 00:11:49.560
And even if somebody
said xyz plus 7,

00:11:49.560 --> 00:11:51.160
they would still be right.

00:11:51.160 --> 00:11:56.060
In the end you can have
any xyz plus a constant.

00:11:56.060 --> 00:11:58.260
In general it's not
so easy to guess.

00:11:58.260 --> 00:12:02.020
But there are lots of examples
of conservative forces where

00:12:02.020 --> 00:12:06.590
you simply cannot see the scalar
potential or cannot deduce it

00:12:06.590 --> 00:12:09.850
like in a few seconds.

00:12:09.850 --> 00:12:12.970
Expect something easy,
though, like that,

00:12:12.970 --> 00:12:14.945
something that you can see.

00:12:14.945 --> 00:12:15.820
Let's see an example.

00:12:15.820 --> 00:12:19.260
Assume this is your force field
acting on a particle that's

00:12:19.260 --> 00:12:23.510
moving on a curving space.

00:12:23.510 --> 00:12:29.130
And it's stubborn and it
decides to move on a helix,

00:12:29.130 --> 00:12:32.560
because it's a-- I don't
know what kind of particle

00:12:32.560 --> 00:12:34.696
would move on a
helix, but suppose

00:12:34.696 --> 00:12:39.490
a lot of particles, just a
little train or a drunken bug

00:12:39.490 --> 00:12:40.690
or something.

00:12:40.690 --> 00:12:45.070
And you were moving
on another helix.

00:12:45.070 --> 00:12:52.170
Now suppose that helix will
be R of t equals cosine t

00:12:52.170 --> 00:13:01.110
sine t and t where you
have t as 0 to start with.

00:13:01.110 --> 00:13:02.352
What do I have at 0?

00:13:02.352 --> 00:13:05.640
The point 1, 0, 0.

00:13:05.640 --> 00:13:09.120
That's the point,
let's call it A.

00:13:09.120 --> 00:13:12.870
And let's call this B. I
don't know what I want to do.

00:13:12.870 --> 00:13:16.580
I'll just do a
complete rotation,

00:13:16.580 --> 00:13:18.510
just to make my life easier.

00:13:18.510 --> 00:13:23.120
And this is B. And that
will be A at t equals 0

00:13:23.120 --> 00:13:25.160
and B equals 2 pi.

00:13:25.160 --> 00:13:32.840


00:13:32.840 --> 00:13:35.352
So what will this be at B?

00:13:35.352 --> 00:13:37.120
STUDENT: 1, 0, 2 pi.

00:13:37.120 --> 00:13:40.430
MAGDALENA TODA: 1, 0, and 2 pi.

00:13:40.430 --> 00:13:44.320
So you perform a complete
rotation and come back.

00:13:44.320 --> 00:13:49.750
Now, if your force is
conservative, you are lucky.

00:13:49.750 --> 00:13:53.100
Because you know the theorem
that says in that case

00:13:53.100 --> 00:13:57.820
the work integral will
be independent of path.

00:13:57.820 --> 00:14:03.910
And due to the theorem in-- what
section was that again-- 13.3,

00:14:03.910 --> 00:14:06.890
independence of path,
you know that this

00:14:06.890 --> 00:14:11.720
is going to be-- let me rewrite
it one more time with gradient

00:14:11.720 --> 00:14:16.150
of f instead of big F.

00:14:16.150 --> 00:14:19.615
And this will become what,
f of the q-- not the q.

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,

00:14:24.065 --> 00:14:24.565
right?

00:14:24.565 --> 00:14:28.050


00:14:28.050 --> 00:14:29.030
What does this mean?

00:14:29.030 --> 00:14:33.410
You have to measure
the-- to evaluate

00:14:33.410 --> 00:14:39.440
the coordinates of
this function xyz

00:14:39.440 --> 00:14:50.080
where t equals 2 pi minus
xyz where t equals what?

00:14:50.080 --> 00:14:52.404
0.

00:14:52.404 --> 00:14:56.830
And now I have to be
careful, because I

00:14:56.830 --> 00:14:58.200
have to evaluate them.

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.

00:15:06.632 --> 00:15:08.100
In the end it doesn't matter.

00:15:08.100 --> 00:15:12.530
I can get 0-- I
can get 0 for this

00:15:12.530 --> 00:15:14.700
and get 0 for that as well.

00:15:14.700 --> 00:15:20.600
So when this is 2 pi I get
x equals 1, y equals 0,

00:15:20.600 --> 00:15:23.210
and t equals 2 pi.

00:15:23.210 --> 00:15:26.650
So in the end, both products
are 0 and I got a 0.

00:15:26.650 --> 00:15:31.890
So although the [INAUDIBLE]
works very hard-- I mean,

00:15:31.890 --> 00:15:36.450
works hard in our perception
to get from a point

00:15:36.450 --> 00:15:38.730
to another-- the work is 0.

00:15:38.730 --> 00:15:39.380
Why?

00:15:39.380 --> 00:15:42.470
Because it's a vector
value thing inside.

00:15:42.470 --> 00:15:46.560
And there are some
annihilations going on.

00:15:46.560 --> 00:15:50.473
So that reminds me
of another example.

00:15:50.473 --> 00:15:52.828
So we are done
with this example.

00:15:52.828 --> 00:15:55.654
Let's go back to our washer.

00:15:55.654 --> 00:15:57.560
I was just doing
laundry last night

00:15:57.560 --> 00:16:01.360
and I was thinking of
the washer example.

00:16:01.360 --> 00:16:04.960
And I thought of a small
variation of the washer

00:16:04.960 --> 00:16:09.430
example, just assuming that
I would give you a pop quiz.

00:16:09.430 --> 00:16:12.120
And I'm not giving you
a pop quiz right now.

00:16:12.120 --> 00:16:14.940
But if I gave you
a pop quiz now,

00:16:14.940 --> 00:16:20.850
I would ask you example
two, the washer.

00:16:20.850 --> 00:16:24.412


00:16:24.412 --> 00:16:27.770
It is performing
a circular motion,

00:16:27.770 --> 00:16:30.000
and I want to know
the work performed

00:16:30.000 --> 00:16:36.320
by the centrifugal force
between various points.

00:16:36.320 --> 00:16:48.180
So have the circular motion,
the centrifugal force.

00:16:48.180 --> 00:16:50.892
This is the
centrifugal, I'm sorry.

00:16:50.892 --> 00:16:54.065
I'll take the centrifugal force.

00:16:54.065 --> 00:16:56.480
And that was last
time we discussed

00:16:56.480 --> 00:17:04.960
that, that was extending
the radius of the initial--

00:17:04.960 --> 00:17:07.220
the vector value position.

00:17:07.220 --> 00:17:11.560
So you have that in
every point, xi plus yj.

00:17:11.560 --> 00:17:14.670
And you want F to
be able xi plus yj.

00:17:14.670 --> 00:17:20.368
But it points outside
from the point

00:17:20.368 --> 00:17:22.965
on the circular trajectory.

00:17:22.965 --> 00:17:26.040


00:17:26.040 --> 00:17:31.350
And I asked you, find
out what you performed

00:17:31.350 --> 00:17:38.686
by F in one full rotation.

00:17:38.686 --> 00:17:43.760


00:17:43.760 --> 00:17:48.975
We gave the equation of motion,
being cosine t y sine t,

00:17:48.975 --> 00:17:51.560
if you remember from last time.

00:17:51.560 --> 00:17:57.390
And then W2, let's
say, is performed by F

00:17:57.390 --> 00:18:01.910
from t equals 0 to t equals pi.

00:18:01.910 --> 00:18:03.496
I want that as well.

00:18:03.496 --> 00:18:12.570
And W2 performed by F from
t-- that makes t0 to t

00:18:12.570 --> 00:18:20.030
equals pi-- t equals 0
to t equals pi over 4.

00:18:20.030 --> 00:18:21.830
These are all very
easy questions,

00:18:21.830 --> 00:18:24.970
and you should be able to
answer them in no time.

00:18:24.970 --> 00:18:28.430
Now, let me tell you something.

00:18:28.430 --> 00:18:29.710
We are in plane, not in space.

00:18:29.710 --> 00:18:30.790
But it doesn't matter.

00:18:30.790 --> 00:18:35.440
It's like the third quadrant
would be 0, piece of cake.

00:18:35.440 --> 00:18:39.997
Your eye should be so
well-trained that when

00:18:39.997 --> 00:18:41.580
you look at the force
field like that,

00:18:41.580 --> 00:18:44.159
and people talk about what
you should ask yourself,

00:18:44.159 --> 00:18:44.950
is it conservative?

00:18:44.950 --> 00:18:48.390


00:18:48.390 --> 00:18:51.010
And it is conservative.

00:18:51.010 --> 00:18:53.830
And that means little f is what?

00:18:53.830 --> 00:18:56.900


00:18:56.900 --> 00:18:59.700
Nitish said that yesterday.

00:18:59.700 --> 00:19:00.860
Why did you go there?

00:19:00.860 --> 00:19:02.630
You want to sleep today?

00:19:02.630 --> 00:19:05.510
I'm just teasing you.

00:19:05.510 --> 00:19:08.470
I got so comfortable with
you sitting in the front row.

00:19:08.470 --> 00:19:10.070
STUDENT: I took his spot.

00:19:10.070 --> 00:19:12.071
STUDENT: She doesn't like
you sitting over here.

00:19:12.071 --> 00:19:13.070
MAGDALENA TODA: It's OK.

00:19:13.070 --> 00:19:13.920
It's fine.

00:19:13.920 --> 00:19:16.710
I still give him credit
for what he said last time.

00:19:16.710 --> 00:19:19.570
So do you guys remember,
he gave us this answer?

00:19:19.570 --> 00:19:23.070
x squared plus y squared over
2, and he found the scalar

00:19:23.070 --> 00:19:26.980
potential through witchcraft
in about a second and a half?

00:19:26.980 --> 00:19:28.190
OK.

00:19:28.190 --> 00:19:31.430
We are gonna conclude something.

00:19:31.430 --> 00:19:36.420
Do you remember that I found the
answer by find the explanation?

00:19:36.420 --> 00:19:39.160
I got W to be 0.

00:19:39.160 --> 00:19:45.340
But if I were to find another
explanation why the work would

00:19:45.340 --> 00:19:50.180
be 0 in this case, it
would have been 0 anyway

00:19:50.180 --> 00:19:53.220
for any force field.

00:19:53.220 --> 00:19:58.117
Even if I took the F
to be something else.

00:19:58.117 --> 00:20:03.320
Assume that F would be
G. Really wild, crazy,

00:20:03.320 --> 00:20:06.970
but still differentiable
vector value function.

00:20:06.970 --> 00:20:08.700
G differential.

00:20:08.700 --> 00:20:15.050
Would the work that we
want be the same for G?

00:20:15.050 --> 00:20:15.727
STUDENT: Yeah.

00:20:15.727 --> 00:20:16.560
MAGDALENA TODA: Why?

00:20:16.560 --> 00:20:18.310
STUDENT: Because of
displacement scenario.

00:20:18.310 --> 00:20:20.680
MAGDALENA TODA: Since
it's conservative,

00:20:20.680 --> 00:20:23.150
you have a closed loop.

00:20:23.150 --> 00:20:25.800
So the closed loop
will say, thick F

00:20:25.800 --> 00:20:30.340
at that terminal point minus
thick F at the initial point.

00:20:30.340 --> 00:20:33.620
But if a loop motion,
your terminal point

00:20:33.620 --> 00:20:35.290
is the initial point.

00:20:35.290 --> 00:20:36.040
Duh.

00:20:36.040 --> 00:20:40.902
So you have the
same point, the P

00:20:40.902 --> 00:20:44.490
equals qe if it's
a closed curve.

00:20:44.490 --> 00:20:48.294
So for a closed curve--
we also call that a loop.

00:20:48.294 --> 00:20:50.210
With a basketball, it
would have been too easy

00:20:50.210 --> 00:20:54.920
and you would have gotten a
dollar for free like that.

00:20:54.920 --> 00:20:57.930
So any closed curve
is called a loop.

00:20:57.930 --> 00:21:01.610
If your force field is
conservative-- attention,

00:21:01.610 --> 00:21:05.390
you might have examples
like that in the exams--

00:21:05.390 --> 00:21:08.990
then it doesn't matter
who little f is,

00:21:08.990 --> 00:21:12.510
if p equals q you get 0 anyway.

00:21:12.510 --> 00:21:15.750
But the reason why I
said you would get 0

00:21:15.750 --> 00:21:20.950
on the example of last time
was a slightly different one.

00:21:20.950 --> 00:21:24.367
What does the engineer
say to himself?

00:21:24.367 --> 00:21:25.700
STUDENT: Force is perpendicular.

00:21:25.700 --> 00:21:26.360
MAGDALENA TODA: Yeah.

00:21:26.360 --> 00:21:27.000
Very good.

00:21:27.000 --> 00:21:28.990
Whenever the force
is perpendicular

00:21:28.990 --> 00:21:33.180
to the trajectory, I'm going
to get 0 for the force.

00:21:33.180 --> 00:21:36.710
Because at every
moment the dot product

00:21:36.710 --> 00:21:41.160
between the force and the
displacement direction,

00:21:41.160 --> 00:21:45.190
which would be like dR, the
tangent to the displacement,

00:21:45.190 --> 00:21:46.750
would be [INAUDIBLE].

00:21:46.750 --> 00:21:50.190
And cosine of [INAUDIBLE] is 0.

00:21:50.190 --> 00:21:50.770
Duh.

00:21:50.770 --> 00:21:52.810
So that's another reason.

00:21:52.810 --> 00:21:59.822
Reason of last time
was F perpendicular

00:21:59.822 --> 00:22:05.330
to the R prime
direction, R prime

00:22:05.330 --> 00:22:11.030
being the velocity-- look,
when I'm moving in a circle,

00:22:11.030 --> 00:22:13.030
this is the force.

00:22:13.030 --> 00:22:14.520
And I'm moving.

00:22:14.520 --> 00:22:17.660
This is my velocity, is
the tangent to the circle.

00:22:17.660 --> 00:22:22.460
And the velocity and the normal
are always perpendicular,

00:22:22.460 --> 00:22:23.110
at every point.

00:22:23.110 --> 00:22:24.000
That's why I have 0.

00:22:24.000 --> 00:22:26.600


00:22:26.600 --> 00:22:32.100
So note that even if I
didn't take a close look,

00:22:32.100 --> 00:22:36.020
why would the answer
be from 0 to pi?

00:22:36.020 --> 00:22:38.670
Still?

00:22:38.670 --> 00:22:42.685
0 because of that.

00:22:42.685 --> 00:22:43.640
0.

00:22:43.640 --> 00:22:47.000
How about from 0 to pi over 4?

00:22:47.000 --> 00:22:49.890
Still 0.

00:22:49.890 --> 00:22:52.170
And of course if somebody
would not believe them,

00:22:52.170 --> 00:22:54.860
if somebody would not
understand the theory,

00:22:54.860 --> 00:22:57.605
they would do the work and
they would get to the answer

00:22:57.605 --> 00:23:01.046
and say, oh my
god, yeah, I got 0.

00:23:01.046 --> 00:23:02.660
All right?

00:23:02.660 --> 00:23:03.670
OK.

00:23:03.670 --> 00:23:09.796
Now, what if somebody--
and I want to spray this.

00:23:09.796 --> 00:23:11.730
Can I go ahead and
erase the board

00:23:11.730 --> 00:23:15.160
and move onto example
three or whatever?

00:23:15.160 --> 00:23:16.420
Yes?

00:23:16.420 --> 00:23:17.210
OK.

00:23:17.210 --> 00:23:19.081
All right.

00:23:19.081 --> 00:23:21.520
STUDENT: Could you say
non-conservative force?

00:23:21.520 --> 00:23:23.570
MAGDALENA TODA: Yeah,
that's what I-- exactly.

00:23:23.570 --> 00:23:26.371
You are a mind reader.

00:23:26.371 --> 00:23:31.838
You are gonna guess my mind.

00:23:31.838 --> 00:23:44.263


00:23:44.263 --> 00:23:46.800
And I'm going to
pick a nasty one.

00:23:46.800 --> 00:23:49.845
And since I'm doing
review anyway,

00:23:49.845 --> 00:23:50.970
you may have one like that.

00:23:50.970 --> 00:23:55.250
And you may have both one that
involves a conservative force

00:23:55.250 --> 00:24:00.110
field and one that does not
involve a conservative force

00:24:00.110 --> 00:24:00.690
field.

00:24:00.690 --> 00:24:07.450
And we can ask you, find us the
work belong to different path.

00:24:07.450 --> 00:24:11.690
And I've done this
type of example before.

00:24:11.690 --> 00:24:15.450
Let's take F of
x and y in plane.

00:24:15.450 --> 00:24:29.050
In our two I take xyi
plus x squared y of j.

00:24:29.050 --> 00:24:34.650
And the problem would
involve my favorite picture,

00:24:34.650 --> 00:24:39.000
y equals x squared and y
equals x, our two paths.

00:24:39.000 --> 00:24:40.520
One is the straight path.

00:24:40.520 --> 00:24:43.700
One is the [INAUDIBLE] path.

00:24:43.700 --> 00:24:46.280
They go from 0,
0 to 1, 1 anyway.

00:24:46.280 --> 00:24:49.220


00:24:49.220 --> 00:24:58.250
And I'm asking you to
find W1 along path one

00:24:58.250 --> 00:25:01.250
and W2 along path two.

00:25:01.250 --> 00:25:06.280
And of course,
example three, if this

00:25:06.280 --> 00:25:10.270
were conservative
you would say, oh,

00:25:10.270 --> 00:25:11.895
it doesn't matter
what path I'm taking,

00:25:11.895 --> 00:25:14.930
I'm still getting
the same answer.

00:25:14.930 --> 00:25:17.499
But is this conservative?

00:25:17.499 --> 00:25:18.425
STUDENT: No.

00:25:18.425 --> 00:25:20.177
Because you said it wasn't.

00:25:20.177 --> 00:25:21.260
MAGDALENA TODA: Very good.

00:25:21.260 --> 00:25:22.870
So how do you know?

00:25:22.870 --> 00:25:26.220
That's one test
when you are in two.

00:25:26.220 --> 00:25:30.708
There is the magic test that
says-- let's say this is M,

00:25:30.708 --> 00:25:36.700
and let's say this is N. You
would have to check if M sub--

00:25:36.700 --> 00:25:37.271
STUDENT: y.

00:25:37.271 --> 00:25:38.020
MAGDALENA TODA: y.

00:25:38.020 --> 00:25:38.519
Very good.

00:25:38.519 --> 00:25:39.640
I'm proud of you.

00:25:39.640 --> 00:25:42.260
You're ready for
3350, by the way.

00:25:42.260 --> 00:25:44.030
Is equal to N sub x.

00:25:44.030 --> 00:25:46.600
M sub y is x.

00:25:46.600 --> 00:25:49.180
N sub x is 2xy.

00:25:49.180 --> 00:25:51.530
They are not equal.

00:25:51.530 --> 00:25:55.240
So that's me crying that I have
to do the work twice and get--

00:25:55.240 --> 00:25:57.634
probably I'll get two
different examples.

00:25:57.634 --> 00:26:00.970


00:26:00.970 --> 00:26:03.510
If you read the book--
I'm afraid to ask

00:26:03.510 --> 00:26:09.320
how many of you opened the
book at section 13.2, 13.3.

00:26:09.320 --> 00:26:12.000
But did you read
it, any of them?

00:26:12.000 --> 00:26:13.000
STUDENT: Nitish read it.

00:26:13.000 --> 00:26:15.400
MAGDALENA TODA: Oh, good.

00:26:15.400 --> 00:26:20.140
There is another criteria for
a force to be conservative.

00:26:20.140 --> 00:26:22.120
If you are, it's piece of cake.

00:26:22.120 --> 00:26:23.296
You do that, right?

00:26:23.296 --> 00:26:24.337
MAGDALENA TODA: Yes, sir?

00:26:24.337 --> 00:26:25.620
STUDENT: Curl has frequency 0.

00:26:25.620 --> 00:26:27.494
MAGDALENA TODA: The curl
criteria, excellent.

00:26:27.494 --> 00:26:29.000
The curl has to be zero.

00:26:29.000 --> 00:26:38.977
So if F in R 3 is
conservative, then you'll

00:26:38.977 --> 00:26:40.060
get different order curve.

00:26:40.060 --> 00:26:42.052
Curl F is 0.

00:26:42.052 --> 00:26:44.640
Now let's check what
the heck was curl.

00:26:44.640 --> 00:26:47.610
You see, mathematics
is not a bunch

00:26:47.610 --> 00:26:51.910
of these joint discussions
like other sciences.

00:26:51.910 --> 00:26:55.340
In mathematics, if you don't
know a section or you skipped

00:26:55.340 --> 00:26:58.450
it, you are sick, you
have a date that day,

00:26:58.450 --> 00:27:02.700
you didn't study, then it's
all over because you cannot

00:27:02.700 --> 00:27:06.650
understand how to work out the
problems and materials if you

00:27:06.650 --> 00:27:08.000
skip the section.

00:27:08.000 --> 00:27:12.020
Curl was the one
where we learned

00:27:12.020 --> 00:27:16.150
that we used the determinant.

00:27:16.150 --> 00:27:17.300
That's the easiest story.

00:27:17.300 --> 00:27:20.740
It came with a t-shirt,
but that t-shirt really

00:27:20.740 --> 00:27:25.800
doesn't help because
it's easier to,

00:27:25.800 --> 00:27:28.270
instead of memorizing
the formula,

00:27:28.270 --> 00:27:31.440
you set out the determinant.

00:27:31.440 --> 00:27:33.865
So you have the operator
derivative with respect

00:27:33.865 --> 00:27:40.760
to x, y z followed by what?

00:27:40.760 --> 00:27:43.100
F1, F2, F3.

00:27:43.100 --> 00:27:46.945
Now in your case, I'm
asking you if you did it

00:27:46.945 --> 00:27:51.680
for this F, what is
the third component?

00:27:51.680 --> 00:27:52.510
STUDENT: The 0.

00:27:52.510 --> 00:27:54.800
MAGDALENA TODA: The
0, so this guy is 0.

00:27:54.800 --> 00:27:59.740
This guy is X squared
Y, and this guy is xy.

00:27:59.740 --> 00:28:01.690
And it should be
a piece of cake,

00:28:01.690 --> 00:28:03.990
but I want to do
it one more time.

00:28:03.990 --> 00:28:08.520
I times the minor derivative
of 0 with respect to y

00:28:08.520 --> 00:28:11.540
is 0 minus derivative
of x squared

00:28:11.540 --> 00:28:15.410
y respect to 0, all
right, plus j minus

00:28:15.410 --> 00:28:17.860
j because I'm alternating.

00:28:17.860 --> 00:28:19.965
You've known enough
in your algebra

00:28:19.965 --> 00:28:22.840
to know why I'm expanding
along the first row.

00:28:22.840 --> 00:28:25.700
I have a minus, all
right, then the x

00:28:25.700 --> 00:28:33.600
of 0, 0 derivative of xy
respect to the 0 plus k times

00:28:33.600 --> 00:28:37.550
the minor corresponding
to k derivative 2xy.

00:28:37.550 --> 00:28:45.050


00:28:45.050 --> 00:28:46.325
Oh, and the derivative--

00:28:46.325 --> 00:28:49.062


00:28:49.062 --> 00:28:52.504
STUDENT: Yeah, this
is the n equals 0.

00:28:52.504 --> 00:28:54.170
MAGDALENA TODA: Oh,
yeah, that's the one

00:28:54.170 --> 00:28:58.700
where it's not a because
that's not conservative.

00:28:58.700 --> 00:28:59.830
So what do you get.

00:28:59.830 --> 00:29:04.950
You get 2xy minus x, right?

00:29:04.950 --> 00:29:07.100
But I don't know how to
write it better than that.

00:29:07.100 --> 00:29:08.100
Well, it doesn't matter.

00:29:08.100 --> 00:29:09.510
Leave it like that.

00:29:09.510 --> 00:29:18.080
So this would be 0 if it only
if x would be 0, but otherwise y

00:29:18.080 --> 00:29:18.860
was 1/2.

00:29:18.860 --> 00:29:22.850
But in general, it
is not a 0, good.

00:29:22.850 --> 00:29:30.470
So F is not
conservative, and then we

00:29:30.470 --> 00:29:32.400
can say goodbye
to the whole thing

00:29:32.400 --> 00:29:39.680
here and move on to
computing the works.

00:29:39.680 --> 00:29:42.020
What is the only
way we can do that?

00:29:42.020 --> 00:29:46.491
By parameterizing
the first path,

00:29:46.491 --> 00:29:48.926
but I didn't say which
one is the first path.

00:29:48.926 --> 00:29:52.578
This is the first path, so
x of t equals t, and y of t

00:29:52.578 --> 00:29:55.257
equals t is your
parameterization.

00:29:55.257 --> 00:30:03.910
The simplest one, and then
W1 will be integral of-- I'm

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.

00:30:10.135 --> 00:30:14.540
Times x prime of
t plus x squared

00:30:14.540 --> 00:30:21.550
y times y prime of t dt where--

00:30:21.550 --> 00:30:31.372
STUDENT: Isn't that just
xy dx y-- never mind.

00:30:31.372 --> 00:30:33.565
MAGDALENA TODA: This is F2.

00:30:33.565 --> 00:30:36.560
And this is x prime,
and this is y prime

00:30:36.560 --> 00:30:40.394
because this thing is
just-- I have no idea.

00:30:40.394 --> 00:30:41.852
STUDENT: Right,
but what I'm asking

00:30:41.852 --> 00:30:46.719
is that not the same as just
F1 dx because we're going

00:30:46.719 --> 00:30:49.630
to do a chain rule anyway.

00:30:49.630 --> 00:30:53.750
MAGDALENA TODA: If I put
the x, I cannot put this.

00:30:53.750 --> 00:30:57.440
OK, this times that is dx.

00:30:57.440 --> 00:30:59.900
This guy times this guy is dx.

00:30:59.900 --> 00:31:01.400
STUDENT: But then
you can't use your

00:31:01.400 --> 00:31:03.980
MAGDALENA TODA: Then I
cannot use the t's then.

00:31:03.980 --> 00:31:06.486
STUDENT: All right, there we go.

00:31:06.486 --> 00:31:11.160
MAGDALENA TODA: All right, so
I have integral from 0 to 1 t,

00:31:11.160 --> 00:31:15.390
t times 1 t squared.

00:31:15.390 --> 00:31:18.260
If I make a mistake, that would
be a silly algebra mistake

00:31:18.260 --> 00:31:18.760
[INAUDIBLE].

00:31:18.760 --> 00:31:21.595


00:31:21.595 --> 00:31:23.980
All right, class.

00:31:23.980 --> 00:31:33.540
t cubed times 1dt,
how much is this?

00:31:33.540 --> 00:31:38.002
t cubed over 3 plus t
to the fourth over 4.

00:31:38.002 --> 00:31:39.252
STUDENT: It's just 2-- oh, no.

00:31:39.252 --> 00:31:45.100


00:31:45.100 --> 00:31:48.120
MAGDALENA TODA: Very good.

00:31:48.120 --> 00:31:52.500
Do not expect that we kill you
with computations on the exams,

00:31:52.500 --> 00:31:55.010
but that's not what we want.

00:31:55.010 --> 00:31:58.250
We want to test if you have
the basic understanding of what

00:31:58.250 --> 00:32:03.267
this is all about, not to
kill you with, OK, that.

00:32:03.267 --> 00:32:05.350
I'm not going to say that
in front of the cameras,

00:32:05.350 --> 00:32:06.805
but everybody knows that.

00:32:06.805 --> 00:32:08.270
There are professors
who would like

00:32:08.270 --> 00:32:09.690
to kill you with computations.

00:32:09.690 --> 00:32:12.180
Now, we're living in
a different world.

00:32:12.180 --> 00:32:15.590
If I gave you a long
polynomial sausage here

00:32:15.590 --> 00:32:17.660
and I ask you to
work with it, that

00:32:17.660 --> 00:32:21.090
doesn't mean that I'm smart
because MATLAB can do it.

00:32:21.090 --> 00:32:25.100
Mathematica, you get some
very nice simplifications

00:32:25.100 --> 00:32:28.500
over there, so I'm
just trying to see

00:32:28.500 --> 00:32:35.730
if rather than being able
to compute with no error,

00:32:35.730 --> 00:32:40.670
you are having the basic
understanding of the concept.

00:32:40.670 --> 00:32:44.520
And the rest can been done
by the mathematical software,

00:32:44.520 --> 00:32:49.650
which, nowadays, most
mathematicians are using.

00:32:49.650 --> 00:32:52.780
If you asked me 15
years ago, I think

00:32:52.780 --> 00:32:57.660
I knew colleagues at all the
ranks in academia who would not

00:32:57.660 --> 00:33:01.594
touch Mathematica or
MATLAB or Maple say

00:33:01.594 --> 00:33:04.810
that's like tool from
evil or something,

00:33:04.810 --> 00:33:08.072
but now everybody uses.

00:33:08.072 --> 00:33:10.760
Engineers use mostly
MATLAB as I told you.

00:33:10.760 --> 00:33:15.910
Mathematicians use both
MATLAB and Mathematica.

00:33:15.910 --> 00:33:19.450
Some of them use Maple,
especially the ones who

00:33:19.450 --> 00:33:23.002
have demos for K-12
level teachers,

00:33:23.002 --> 00:33:26.135
but MATLAB is a wonderful
tool, very pretty powerful

00:33:26.135 --> 00:33:27.583
in many ways.

00:33:27.583 --> 00:33:30.830
If you are doing any kind
of linear algebra project--

00:33:30.830 --> 00:33:33.760
I noticed three or four of you
are taking linear algebra-- you

00:33:33.760 --> 00:33:39.070
can always rely on MATLAB being
the best of all of the above.

00:33:39.070 --> 00:33:40.230
OK, W2.

00:33:40.230 --> 00:33:42.970


00:33:42.970 --> 00:33:49.830
For W2, I have a parabola, and
it's, again, a piece of cake.

00:33:49.830 --> 00:33:54.545
X prime will be 1,
y prime will be 2t.

00:33:54.545 --> 00:33:56.485
When I write down
the whole thing,

00:33:56.485 --> 00:33:58.667
I have to pay a little
bit of attention

00:33:58.667 --> 00:34:02.480
when I substitute
especially when I'm

00:34:02.480 --> 00:34:04.595
taking an exam under pressure.

00:34:04.595 --> 00:34:08.860


00:34:08.860 --> 00:34:13.909
x squared is t
squared, y is t squared

00:34:13.909 --> 00:34:17.270
times y prime, which is 2t.

00:34:17.270 --> 00:34:19.570
So now this is x prime.

00:34:19.570 --> 00:34:21.139
This is y prime.

00:34:21.139 --> 00:34:24.704
Let me change colors.

00:34:24.704 --> 00:34:26.924
All politicians change colors.

00:34:26.924 --> 00:34:29.300
But I'm not a
politician, but I'm

00:34:29.300 --> 00:34:34.030
thinking it's useful for you
to see who everybody was.

00:34:34.030 --> 00:34:38.690
This is the F1 in terms of t.

00:34:38.690 --> 00:34:46.214
That's the idea of what that
is, and this is F2 in terms of t

00:34:46.214 --> 00:34:48.121
as well.

00:34:48.121 --> 00:34:49.810
Oh, my God, another answer?

00:34:49.810 --> 00:34:53.540
Absolutely, I'm going to
get an another answer.

00:34:53.540 --> 00:34:57.276
Is it obviously to everybody
I'm going to get another answer?

00:34:57.276 --> 00:34:58.080
STUDENT: Yeah.

00:34:58.080 --> 00:35:01.495
MAGDALENA TODA: So I don't
have to put the t's here,

00:35:01.495 --> 00:35:03.790
but I thought it was
sort of neat to see

00:35:03.790 --> 00:35:05.950
that t goes from 0 to 1.

00:35:05.950 --> 00:35:08.760
And what do I get?

00:35:08.760 --> 00:35:16.355
This whole lot of them is t
cubed plus 2 t to the fifth.

00:35:16.355 --> 00:35:19.040


00:35:19.040 --> 00:35:26.002
So when I do the integration,
I get t to the 4 over 4 plus--

00:35:26.002 --> 00:35:27.335
shut up, Magdalena, get people--

00:35:27.335 --> 00:35:29.745


00:35:29.745 --> 00:35:30.620
STUDENT: [INAUDIBLE].

00:35:30.620 --> 00:35:32.040
MAGDALENA TODA: Very good.

00:35:32.040 --> 00:35:36.010
Yeah, he's done
the simplification.

00:35:36.010 --> 00:35:37.570
STUDENT: You get
the same values.

00:35:37.570 --> 00:35:40.330


00:35:40.330 --> 00:35:45.340
Plug in 1, you get 7/12 again.

00:35:45.340 --> 00:35:48.190
MAGDALENA TODA: So I'm
asking you-- OK, what was it?

00:35:48.190 --> 00:35:56.830
Solve 0, 1-- so I'm
asking why do you

00:35:56.830 --> 00:36:00.888
think we get the same value?

00:36:00.888 --> 00:36:03.310
Because the force
is not conservative,

00:36:03.310 --> 00:36:06.870
and I went on another path.

00:36:06.870 --> 00:36:10.075
I went on one path, and
I went on another path.

00:36:10.075 --> 00:36:15.656
And look, obviously my
expression was different.

00:36:15.656 --> 00:36:18.870
It's like one of those
math games or UIL games.

00:36:18.870 --> 00:36:20.970
And look at the algebra.

00:36:20.970 --> 00:36:23.521
The polynomials are different.

00:36:23.521 --> 00:36:25.960
What was my luck here?

00:36:25.960 --> 00:36:27.143
I took 1.

00:36:27.143 --> 00:36:27.726
STUDENT: Yeah.

00:36:27.726 --> 00:36:29.490
MAGDALENA TODA: I
could have taken 2.

00:36:29.490 --> 00:36:35.990
So if instead of 1, I would
have taken another number,

00:36:35.990 --> 00:36:38.330
then the higher the power,
the bigger the number

00:36:38.330 --> 00:36:39.490
would have been.

00:36:39.490 --> 00:36:40.407
I could have taken 2--

00:36:40.407 --> 00:36:42.114
STUDENT: You could
have taken negative 1,

00:36:42.114 --> 00:36:44.240
and you still wouldn't
have got the same answer.

00:36:44.240 --> 00:36:48.940
MAGDALENA TODA: Yeah, there
are many reasons why that is.

00:36:48.940 --> 00:36:53.900
But anyway, know that when you
take 1, 1 to every power is 1.

00:36:53.900 --> 00:36:55.470
And yeah, you were lucky.

00:36:55.470 --> 00:36:58.430
But in general, keep in
mind that if the force is

00:36:58.430 --> 00:37:01.500
conservative, in
general, in most examples

00:37:01.500 --> 00:37:04.510
you're not going to get the
same answer for the work

00:37:04.510 --> 00:37:10.680
because it does depend on
the path you want to take.

00:37:10.680 --> 00:37:18.210
I think I have reviewed quite
everything that I wanted.

00:37:18.210 --> 00:37:27.330


00:37:27.330 --> 00:37:29.870
So I should be ready
to move forward.

00:37:29.870 --> 00:37:32.630


00:37:32.630 --> 00:37:42.326
So I'm saying we are done
with sections 13.1, 13.2,

00:37:42.326 --> 00:37:48.920
and 13.3, which was my
favorite because it's not

00:37:48.920 --> 00:37:50.970
just the integral of
the path that I like,

00:37:50.970 --> 00:37:54.600
but it's the so-called
fundamental theorem of calculus

00:37:54.600 --> 00:38:04.870
3, which says, fundamental
theorem of the path integral

00:38:04.870 --> 00:38:12.030
saying that you have f of the
endpoint minus f of the origin,

00:38:12.030 --> 00:38:14.430
where little f is
that scalar potential

00:38:14.430 --> 00:38:17.310
as the linear function
was concerned.

00:38:17.310 --> 00:38:24.380
I'm going to call it the
fundamental theorem of path

00:38:24.380 --> 00:38:26.060
integral.

00:38:26.060 --> 00:38:29.230
Last time I told you the
fundamental theorem of calculus

00:38:29.230 --> 00:38:31.800
is Federal Trade Commission.

00:38:31.800 --> 00:38:34.620
We refer to that in Calc 1.

00:38:34.620 --> 00:38:39.080
But this one is the fundamental
theorem of path integral.

00:38:39.080 --> 00:38:42.815
Remember it because at
least one problem out of 15

00:38:42.815 --> 00:38:44.648
or something on the
final, and there are not

00:38:44.648 --> 00:38:45.564
going to be very many.

00:38:45.564 --> 00:38:48.815
It's going to ask you to
know that result. This is

00:38:48.815 --> 00:38:51.950
an important theorem.

00:38:51.950 --> 00:38:55.970
And another important theorem
that is starting right now

00:38:55.970 --> 00:38:57.810
is Green's theorem.

00:38:57.810 --> 00:39:02.690
Green's theorem is
a magic result. I

00:39:02.690 --> 00:39:04.905
have a t-shirt with it.

00:39:04.905 --> 00:39:06.410
I didn't bring it today.

00:39:06.410 --> 00:39:08.290
Maybe I'm going to bring
it next time First,

00:39:08.290 --> 00:39:12.376
I want you to see
the result, and then

00:39:12.376 --> 00:39:15.230
I'll bring the t-shirt
to the exam, so OK.

00:39:15.230 --> 00:39:18.080


00:39:18.080 --> 00:39:24.870
Assume that you have a
soup called Jordan curve.

00:39:24.870 --> 00:39:27.960


00:39:27.960 --> 00:39:32.380
You see, mathematicians don't
follow mathematical objects

00:39:32.380 --> 00:39:34.370
by their names.

00:39:34.370 --> 00:39:37.370
We are crazy people, but
we don't have a big ego.

00:39:37.370 --> 00:39:41.890
We would not say a theorem
of myself or whatever.

00:39:41.890 --> 00:39:45.340
We never give our names to that.

00:39:45.340 --> 00:39:50.930
But all through calculus you
saw all sorts of results.

00:39:50.930 --> 00:39:57.090
Like you see the Jordan
curve is a terminology,

00:39:57.090 --> 00:40:00.200
but then you see
everywhere the Linus rule.

00:40:00.200 --> 00:40:02.360
Did Linus get to
call it his own rule?

00:40:02.360 --> 00:40:06.320
No, but Euler's
number, these are

00:40:06.320 --> 00:40:08.515
things that were
discovered, and in honor

00:40:08.515 --> 00:40:11.640
of that particular
mathematician,

00:40:11.640 --> 00:40:13.290
we call them names.

00:40:13.290 --> 00:40:15.593
We call them the name
of the mathematician.

00:40:15.593 --> 00:40:18.430


00:40:18.430 --> 00:40:22.560
Out of curiosity for
0.5 extra credit points,

00:40:22.560 --> 00:40:25.140
find out who Jordan was.

00:40:25.140 --> 00:40:32.810
Jordan curve is a closed
curve that, in general,

00:40:32.810 --> 00:40:34.595
could be piecewise continuous.

00:40:34.595 --> 00:40:40.610


00:40:40.610 --> 00:40:43.230
So you have a closed
loop over here.

00:40:43.230 --> 00:40:50.170
So in general, I could
have something like that

00:40:50.170 --> 00:40:54.102
that does not enclose.

00:40:54.102 --> 00:40:56.542
That encloses a
domain without holes.

00:40:56.542 --> 00:41:04.350


00:41:04.350 --> 00:41:07.540
Holes are functions
of the same thing.

00:41:07.540 --> 00:41:10.240
STUDENT: So doesn't it
need to be continuous?

00:41:10.240 --> 00:41:12.440
MAGDALENA TODA:
No, I said it is.

00:41:12.440 --> 00:41:13.775
STUDENT: You said, piecewise.

00:41:13.775 --> 00:41:15.170
MAGDALENA TODA: Ah, piecewise.

00:41:15.170 --> 00:41:16.448
This is piecewise.

00:41:16.448 --> 00:41:17.762
STUDENT: Oh, so it's piecewise.

00:41:17.762 --> 00:41:18.262
OK.

00:41:18.262 --> 00:41:20.430
MAGDALENA TODA: So you
have a bunch of arcs.

00:41:20.430 --> 00:41:23.050
Finitely many, let's
say, in your case.

00:41:23.050 --> 00:41:26.230
Finitely many arcs,
they have corners,

00:41:26.230 --> 00:41:29.740
but you can see define the
integral along such a path.

00:41:29.740 --> 00:41:33.676


00:41:33.676 --> 00:41:37.530
Oh, and also for another
0.5 extra credit,

00:41:37.530 --> 00:41:40.140
find out who Mr.
Green was because he

00:41:40.140 --> 00:41:43.490
has several theorems that
are through mathematics

00:41:43.490 --> 00:41:46.360
and free mechanics and
variation calculus.

00:41:46.360 --> 00:41:50.860
There are several identities
that are called Greens.

00:41:50.860 --> 00:41:52.420
There is this famous
Green's theorem,

00:41:52.420 --> 00:41:54.890
but there are Green's
first identity,

00:41:54.890 --> 00:41:57.844
Green's second identity,
and all sorts of things.

00:41:57.844 --> 00:42:01.825
And find out who Mr.
Green was, and as a total,

00:42:01.825 --> 00:42:03.826
you have 1 point extra credit.

00:42:03.826 --> 00:42:08.870
And you can turn in a regular
essay like a two-page thing.

00:42:08.870 --> 00:42:12.602
You want biography of these
mathematicians if you want,

00:42:12.602 --> 00:42:15.500
just a few paragraphs.

00:42:15.500 --> 00:42:19.210
So what does Green's theorem do?

00:42:19.210 --> 00:42:26.200
Green's theorem is
a remarkable result

00:42:26.200 --> 00:42:31.083
which links the path integral
to the double integral.

00:42:31.083 --> 00:42:38.268
It's a remarkable
and famous result.

00:42:38.268 --> 00:42:48.330
And that links the path
integral on the closed

00:42:48.330 --> 00:43:07.262
curve to a double integral
over the domain enclosed.

00:43:07.262 --> 00:43:09.761
I can see the domain
inside, but you

00:43:09.761 --> 00:43:15.160
have to understand it's
enclosed by the curve.

00:43:15.160 --> 00:43:20.740


00:43:20.740 --> 00:43:24.444
All right, and assume
that you have--

00:43:24.444 --> 00:43:35.970
M and N are C1 functions of
x and y, what does it mean?

00:43:35.970 --> 00:43:37.620
M is a function of xy.

00:43:37.620 --> 00:43:40.300
N is a function of xy in plane.

00:43:40.300 --> 00:43:43.360
Both of them are differentiable
with continuous derivative.

00:43:43.360 --> 00:43:46.692


00:43:46.692 --> 00:43:47.830
They are differentiable.

00:43:47.830 --> 00:43:49.515
You can take the
partial derivatives,

00:43:49.515 --> 00:43:51.840
and all the partial
derivatives are continuous.

00:43:51.840 --> 00:43:55.496
That's what we mean
by being C1 functions.

00:43:55.496 --> 00:43:58.610
And there the magic
happens, so let me show you

00:43:58.610 --> 00:44:02.320
where the magic happens.

00:44:02.320 --> 00:44:06.360
This in the box,
the path integral

00:44:06.360 --> 00:44:20.637
over c of M dx plus Ndy is
equal to the double integral

00:44:20.637 --> 00:44:22.390
over the domain enclosed.

00:44:22.390 --> 00:44:24.250
OK, this is the c.

00:44:24.250 --> 00:44:27.160
On the boundary you
go counterclockwise

00:44:27.160 --> 00:44:29.415
like any respectable
mathematician

00:44:29.415 --> 00:44:33.880
would go in a trigonometric
sense, just counterclockwise.

00:44:33.880 --> 00:44:36.776
And this is the domain
being closed by c.

00:44:36.776 --> 00:44:40.060


00:44:40.060 --> 00:44:44.260
And you put here the
integral, which is magic.

00:44:44.260 --> 00:44:46.240
This is easy to
remember for you.

00:44:46.240 --> 00:44:48.170
This is not easy to
remember unless I

00:44:48.170 --> 00:44:49.980
take the t-shirt to
the exam, and you

00:44:49.980 --> 00:44:52.068
cheat by looking at my t-shirt.

00:44:52.068 --> 00:44:54.589
No, by the time of the
exam, I promised you

00:44:54.589 --> 00:44:59.310
you are going to have at least
one week, seven days or more,

00:44:59.310 --> 00:45:03.310
10-day period in which
we will study samples,

00:45:03.310 --> 00:45:05.692
various samples of old finals.

00:45:05.692 --> 00:45:08.320
I'm going to go ahead and
send you some by email.

00:45:08.320 --> 00:45:11.440
Do you mind?

00:45:11.440 --> 00:45:13.676
In the next week
after this week, we

00:45:13.676 --> 00:45:15.400
are going to start reviewing.

00:45:15.400 --> 00:45:20.240
And by dA I mean dxdy, the
usual area limit in Cartesian

00:45:20.240 --> 00:45:23.710
coordinates the way you
are used to it the most.

00:45:23.710 --> 00:45:27.320


00:45:27.320 --> 00:45:29.720
And then, Alex is looking
at it and said, well,

00:45:29.720 --> 00:45:32.380
then I tell her that
the most elegant way

00:45:32.380 --> 00:45:34.690
is to put it with dxdy.

00:45:34.690 --> 00:45:38.480
This is what we call a
one form in mathematics.

00:45:38.480 --> 00:45:39.750
What is a one form.

00:45:39.750 --> 00:45:43.945
It is a linear combination of
this infinitesimal elements

00:45:43.945 --> 00:45:47.490
dxdy in plane with some
scalar functions of x

00:45:47.490 --> 00:45:49.491
and y in front of her.

00:45:49.491 --> 00:45:50.750
OK, so what do we do?

00:45:50.750 --> 00:45:52.505
We integrate the one form.

00:45:52.505 --> 00:45:57.460
The book doesn't talk about one
forms because the is actually

00:45:57.460 --> 00:46:00.727
written for the average
student, the average freshman

00:46:00.727 --> 00:46:04.920
or the average
sophomore, but I think

00:46:04.920 --> 00:46:08.250
we have an exposure to
the notion of one form,

00:46:08.250 --> 00:46:10.770
so I can get a little bit
more elegant and more rigorous

00:46:10.770 --> 00:46:12.290
in my speech.

00:46:12.290 --> 00:46:15.752
If you are a graduate
student, you most likely

00:46:15.752 --> 00:46:18.140
would know this is a one form.

00:46:18.140 --> 00:46:22.140
That's actually the
definition of a one form.

00:46:22.140 --> 00:46:23.475
And you'll say, what is this?

00:46:23.475 --> 00:46:27.284
This is actually two
form, but you are

00:46:27.284 --> 00:46:28.450
going to say, wait a minute.

00:46:28.450 --> 00:46:30.610
I have a scalar
function, whatever

00:46:30.610 --> 00:46:34.430
that is, from the integration
in front of the dxdy

00:46:34.430 --> 00:46:39.740
you want but you never said
that dxdy is a two form.

00:46:39.740 --> 00:46:44.990
Actually, I did, and I
didn't call it a two form.

00:46:44.990 --> 00:46:47.280
Do you remember that
I introduced to you

00:46:47.280 --> 00:46:50.434
some magic wedge product?

00:46:50.434 --> 00:46:53.640


00:46:53.640 --> 00:46:57.750
And we said, this is
a tiny displacement.

00:46:57.750 --> 00:46:59.750
Dx infinitesimal is small.

00:46:59.750 --> 00:47:02.360
Imagine how much the
video we'll there

00:47:02.360 --> 00:47:04.960
is an infinitesimal
displacement dx

00:47:04.960 --> 00:47:07.570
and an infinitesimal
displacement dy,

00:47:07.570 --> 00:47:10.700
and you have some
sort of a sign area.

00:47:10.700 --> 00:47:15.130
So we said, we don't
just take dxdy,

00:47:15.130 --> 00:47:19.070
but we take a product
between dxdy with a wedge,

00:47:19.070 --> 00:47:21.700
meaning that if I
change the order,

00:47:21.700 --> 00:47:24.140
I'm going to have minus dy here.

00:47:24.140 --> 00:47:29.140
This is typical exterior
derivative theory-- exterior

00:47:29.140 --> 00:47:31.456
derivative theory.

00:47:31.456 --> 00:47:34.940
And it's a theory that
starts more or less

00:47:34.940 --> 00:47:36.450
at the graduate level.

00:47:36.450 --> 00:47:39.510
And many people get their
master's degree in math

00:47:39.510 --> 00:47:43.452
and never get to see it, and
I pity them, but this life.

00:47:43.452 --> 00:47:47.470
On the other hand, when
you have dx, which dx--

00:47:47.470 --> 00:47:50.546
the area between dx and dx is 0.

00:47:50.546 --> 00:47:53.400
So we're all very happy
I get rid of those.

00:47:53.400 --> 00:47:55.850
When I have the sign
between the displacement,

00:47:55.850 --> 00:47:57.570
dy and itself is 0.

00:47:57.570 --> 00:47:59.990
So these are the
basic properties

00:47:59.990 --> 00:48:05.420
that we started
about the sign area.

00:48:05.420 --> 00:48:07.720
I want to show you what happens.

00:48:07.720 --> 00:48:16.360
I'm going to-- yeah,
I'm going to erase here.

00:48:16.360 --> 00:48:22.690


00:48:22.690 --> 00:48:25.856
I'm going to show
you later I'm going

00:48:25.856 --> 00:48:32.700
to prove this theorem to you
later using these tricks that I

00:48:32.700 --> 00:48:35.096
just showed you here.

00:48:35.096 --> 00:48:53.656
I will provide proof
to this formula, OK?

00:48:53.656 --> 00:48:57.500
And let's take a look at
that, and we say, well,

00:48:57.500 --> 00:49:00.010
can I memorize that by
the time of the final?

00:49:00.010 --> 00:49:01.660
Yes, you can.

00:49:01.660 --> 00:49:13.070
What is beautiful about
this, it can actually

00:49:13.070 --> 00:49:18.510
help you solve problems that you
didn't think would be possible.

00:49:18.510 --> 00:49:20.920
For example, example
1, and I say,

00:49:20.920 --> 00:49:26.400
that would be one of
the most basic ones.

00:49:26.400 --> 00:49:38.710
Find the geometric meaning of
the integral over a c where

00:49:38.710 --> 00:49:39.890
c is a closed loop.

00:49:39.890 --> 00:49:41.920
OK, c is a loop.

00:49:41.920 --> 00:49:47.388
Piecewise define Jordan
curve-- Jordan curve.

00:49:47.388 --> 00:49:49.645
And I integrate out
of something weird.

00:49:49.645 --> 00:49:51.144
And you say, oh, my God.

00:49:51.144 --> 00:49:51.950
Look at her.

00:49:51.950 --> 00:49:59.360
She picked some weird function
where the path from the dx

00:49:59.360 --> 00:50:05.970
is M, and the path in front of
dy is N, the M and N functions.

00:50:05.970 --> 00:50:07.780
Why would pick like that?

00:50:07.780 --> 00:50:11.260
You wouldn't know yet, but
if you apply Green's theorem,

00:50:11.260 --> 00:50:14.040
assuming you believe
it's true, you

00:50:14.040 --> 00:50:18.271
have double integral over the
domain enclosed by this loop.

00:50:18.271 --> 00:50:24.820
The loop is enclosing
this domain of what?

00:50:24.820 --> 00:50:32.080
Now, I'm trying to shut up,
and I'm want you to talk.

00:50:32.080 --> 00:50:35.542
What am I going to
write over here?

00:50:35.542 --> 00:50:36.940
STUDENT: 1 plus 1.

00:50:36.940 --> 00:50:40.600
MAGDALENA TODA: 1 plus
1, how fun is that?

00:50:40.600 --> 00:50:46.130
Y minus 1, 1 plus 1 equals
2 last time I checked,

00:50:46.130 --> 00:50:49.370
and this is dA.

00:50:49.370 --> 00:50:52.900
And what do you think
this animal would be?

00:50:52.900 --> 00:50:56.090
The cast of 2 always can escape.

00:50:56.090 --> 00:51:00.566
So if we don't want
it, just kick it out.

00:51:00.566 --> 00:51:04.480
What is the remaining
double integral for d of DA?

00:51:04.480 --> 00:51:07.310
You have seen this guy all
through the Calculus 3 course.

00:51:07.310 --> 00:51:09.850
You're tired of it.

00:51:09.850 --> 00:51:13.780
You said, I cannot wait for
this semester to be over

00:51:13.780 --> 00:51:19.060
because this is the double
integral of 1dA over d.

00:51:19.060 --> 00:51:21.940
What in the world is that?

00:51:21.940 --> 00:51:24.364
That is the--

00:51:24.364 --> 00:51:25.330
STUDENT: --area.

00:51:25.330 --> 00:51:26.700
MAGDALENA TODA: Area, very good.

00:51:26.700 --> 00:51:31.150
This is the area of the
domain d inside the curve.

00:51:31.150 --> 00:51:34.980
The shaded area is this.

00:51:34.980 --> 00:51:39.060
So you have discovered
something beautiful

00:51:39.060 --> 00:51:46.530
that the area of the domain
enclosed by a Jordan curve

00:51:46.530 --> 00:51:51.040
equals 1/2 because you pull
the two out in front here,

00:51:51.040 --> 00:51:56.320
it's going to be 1/2 of the path
integrals over the boundary.

00:51:56.320 --> 00:51:58.590
This is called
boundary of a domain.

00:51:58.590 --> 00:52:00.380
c is the boundary of the domain.

00:52:00.380 --> 00:52:04.990


00:52:04.990 --> 00:52:06.940
Some mathematicians--
I don't know

00:52:06.940 --> 00:52:10.950
how far you want to go with your
education, but in a few years

00:52:10.950 --> 00:52:13.300
you might become
graduate students.

00:52:13.300 --> 00:52:18.700
And even some engineers use this
notation boundary of d, del d.

00:52:18.700 --> 00:52:22.230
That means the boundaries,
the frontier of a domain.

00:52:22.230 --> 00:52:24.430
The fence of a ranch.

00:52:24.430 --> 00:52:27.050
That is the del d, but
don't tell the rancher

00:52:27.050 --> 00:52:30.598
because he will take his gun
out and shoot you thinking

00:52:30.598 --> 00:52:33.860
that you are off the hook or
you are after something weird.

00:52:33.860 --> 00:52:38.340
So that's the boundary
of the domain.

00:52:38.340 --> 00:52:42.444
And then you have
minus ydx plus xdy.

00:52:42.444 --> 00:52:46.210


00:52:46.210 --> 00:52:48.210
MAGDALENA TODA: We discover
something beautiful.

00:52:48.210 --> 00:52:50.210
Something important.

00:52:50.210 --> 00:52:52.760
And now I'm asking,
with this exercise--

00:52:52.760 --> 00:52:59.480
one which I could even--
I could even call a lemma.

00:52:59.480 --> 00:53:03.964
Lemma is not quite a
theorem, because it's based--

00:53:03.964 --> 00:53:05.130
could be based on a theorem.

00:53:05.130 --> 00:53:09.460
It's a little result that can
be proved in just a few lines

00:53:09.460 --> 00:53:12.480
without being something
sophisticated based

00:53:12.480 --> 00:53:15.800
on something you
knew from before.

00:53:15.800 --> 00:53:19.950
So this is called a lemma.

00:53:19.950 --> 00:53:26.250
When you have a sophisticated
area to compute--

00:53:26.250 --> 00:53:30.020
or even can you prove-- if you
believe in Green's theorem,

00:53:30.020 --> 00:53:33.410
can you prove that the
area inside the circle

00:53:33.410 --> 00:53:34.780
is pi r squared?

00:53:34.780 --> 00:53:39.966
Can you prove that the
area inside of an ellipse

00:53:39.966 --> 00:53:41.490
is-- I don't know what.

00:53:41.490 --> 00:53:44.370
Do you know the area
inside of an ellipse?

00:53:44.370 --> 00:53:47.561
Nobody taught me in school.

00:53:47.561 --> 00:53:50.800
I don't know why
it's so beautiful.

00:53:50.800 --> 00:53:55.150
I learned what an ellipse
was in eleventh grade

00:53:55.150 --> 00:53:59.190
in high school and again
a review as a freshman

00:53:59.190 --> 00:54:01.310
analytic geometry.

00:54:01.310 --> 00:54:03.070
So we've seen conics again--

00:54:03.070 --> 00:54:04.945
STUDENT: I think we did
conics in 10th grade.

00:54:04.945 --> 00:54:06.684
We might have seen it.

00:54:06.684 --> 00:54:08.100
MAGDALENA TODA:
But nobody told me

00:54:08.100 --> 00:54:10.170
like-- I give you an ellipse.

00:54:10.170 --> 00:54:12.040
Compute the area inside.

00:54:12.040 --> 00:54:13.170
I had no idea.

00:54:13.170 --> 00:54:15.420
And I didn't know
the formula until I

00:54:15.420 --> 00:54:17.820
became an assistant professor.

00:54:17.820 --> 00:54:19.490
I was already in my thirties.

00:54:19.490 --> 00:54:23.970
That's a shame to see that
thing for the first time OK.

00:54:23.970 --> 00:54:27.900
So let's see if we believe
this lemma, and the Green's

00:54:27.900 --> 00:54:28.736
theorem of course.

00:54:28.736 --> 00:54:31.624
But let's apply the
lemma, primarily

00:54:31.624 --> 00:54:33.492
from the Green's theorem.

00:54:33.492 --> 00:54:36.600
Can we actually prove
that the area of the disk

00:54:36.600 --> 00:54:40.890
is pi r squared and the
area of the ellipse--

00:54:40.890 --> 00:54:43.330
inside the ellipse
will be god knows what.

00:54:43.330 --> 00:54:47.180
And we will discover
that by ourselves.

00:54:47.180 --> 00:54:49.385
I think that's the
beauty of mathematics.

00:54:49.385 --> 00:54:53.630
Because every now and then
even if you discover things

00:54:53.630 --> 00:54:56.280
that people have known
for hundreds of years,

00:54:56.280 --> 00:54:58.080
it still gives you
the satisfaction

00:54:58.080 --> 00:55:01.840
that you did something by
yourself-- all on yourself.

00:55:01.840 --> 00:55:06.350
Like, when you feel
build a helicopter or you

00:55:06.350 --> 00:55:07.730
build a table.

00:55:07.730 --> 00:55:09.898
There are many more
beautiful tables

00:55:09.898 --> 00:55:12.050
that were built before
you, but still it's

00:55:12.050 --> 00:55:14.980
a lot of satisfaction that
you do all by yourself.

00:55:14.980 --> 00:55:16.850
It's the same with mathematics.

00:55:16.850 --> 00:55:23.386
So let's see what we can
do now for the first time.

00:55:23.386 --> 00:55:24.590
Not for the first time.

00:55:24.590 --> 00:55:28.030
We do it in other ways.

00:55:28.030 --> 00:55:37.608
Can you prove using the lemma
or Green's theorem-- which

00:55:37.608 --> 00:55:43.125
is the same thing-- either
one-- that the area of the disk

00:55:43.125 --> 00:55:47.850
of radius r-- this is the r.

00:55:47.850 --> 00:55:52.330
so this the radius
r is pi r squared.

00:55:52.330 --> 00:55:56.290


00:55:56.290 --> 00:55:57.530
I hope so.

00:55:57.530 --> 00:55:59.700
And the answer is, I hope so.

00:55:59.700 --> 00:56:00.660
And that's all.

00:56:00.660 --> 00:56:03.540


00:56:03.540 --> 00:56:09.460
Area of the disk of radius r.

00:56:09.460 --> 00:56:10.240
Oh my god.

00:56:10.240 --> 00:56:12.620
So you go, well.

00:56:12.620 --> 00:56:19.420
If I knew the parameterization
of that boundary C,

00:56:19.420 --> 00:56:20.600
it would be a piece of cake.

00:56:20.600 --> 00:56:25.640
Because I would just-- I know
how to do a path integral now.

00:56:25.640 --> 00:56:27.620
I've learned in the
previous sections,

00:56:27.620 --> 00:56:30.460
so this should be easy.

00:56:30.460 --> 00:56:32.690
Can we do that?

00:56:32.690 --> 00:56:33.330
So let's see.

00:56:33.330 --> 00:56:36.360


00:56:36.360 --> 00:56:38.290
Without computing
the double integral,

00:56:38.290 --> 00:56:41.270
because I can always do
that with polar coordinates.

00:56:41.270 --> 00:56:42.880
And we are going to do that.

00:56:42.880 --> 00:56:47.640


00:56:47.640 --> 00:56:49.544
Let's do that as
well, as practice.

00:56:49.544 --> 00:56:53.801
Because so you
review for the exam.

00:56:53.801 --> 00:56:57.100


00:56:57.100 --> 00:57:00.940
But another way to
do it would be what?

00:57:00.940 --> 00:57:06.640
1/2 integral over the circle.

00:57:06.640 --> 00:57:13.900
And how do I parametrize a
circle of fixed radius r?

00:57:13.900 --> 00:57:14.830
Who tells me?

00:57:14.830 --> 00:57:18.020
x of t will be--
that was Chapter 10.

00:57:18.020 --> 00:57:20.872
Everything is a
circle in mathematics.

00:57:20.872 --> 00:57:21.800
STUDENT: r cosine t.

00:57:21.800 --> 00:57:22.925
MAGDALENA TODA: r cosine t.

00:57:22.925 --> 00:57:24.210
Excellent.

00:57:24.210 --> 00:57:25.786
y of t is?

00:57:25.786 --> 00:57:26.640
STUDENT: r sine t.

00:57:26.640 --> 00:57:29.420
MAGDALENA TODA: r sine t.

00:57:29.420 --> 00:57:32.360
So, finally I'm going to
go ahead and use this one.

00:57:32.360 --> 00:57:37.065
And I'm going to say, well,
minus y to be plugged in.

00:57:37.065 --> 00:57:39.800


00:57:39.800 --> 00:57:43.000
This is minus y.

00:57:43.000 --> 00:57:44.580
Multiply by dx.

00:57:44.580 --> 00:57:47.550
Well, you say, wait a
minute. dx with respect.

00:57:47.550 --> 00:57:48.750
What is dx?

00:57:48.750 --> 00:57:52.410
dx is just x prime dt.

00:57:52.410 --> 00:57:54.690
Dy is just y prime dt.

00:57:54.690 --> 00:57:56.370
And t goes out.

00:57:56.370 --> 00:57:57.520
It's banished.

00:57:57.520 --> 00:57:59.620
No, he's the most important guy.

00:57:59.620 --> 00:58:02.970
So t goes from something
to something else.

00:58:02.970 --> 00:58:05.260
We will see that later.

00:58:05.260 --> 00:58:06.970
What is x prime dt?

00:58:06.970 --> 00:58:12.850
X prime is minus r sine
theta-- sine t, Magdalena.

00:58:12.850 --> 00:58:15.740
Minus r sine t.

00:58:15.740 --> 00:58:18.260
That was x prime.

00:58:18.260 --> 00:58:19.430
Change the color.

00:58:19.430 --> 00:58:23.240
Give people some
variation in their life.

00:58:23.240 --> 00:58:32.056
Plus r cosine t,
because this x--

00:58:32.056 --> 00:58:32.930
STUDENT: [INAUDIBLE].

00:58:32.930 --> 00:58:39.310


00:58:39.310 --> 00:58:46.440
MAGDALENA TODA: --times
the y, which is r cosine t.

00:58:46.440 --> 00:58:49.150
So it suddenly became beautiful.

00:58:49.150 --> 00:58:52.480
It looks-- first it looks ugly,
but now it became beautiful.

00:58:52.480 --> 00:58:52.980
Why?

00:58:52.980 --> 00:58:54.365
How come it became beautiful?

00:58:54.365 --> 00:58:56.740
STUDENT: Because you got sine
squared plus cosine square.

00:58:56.740 --> 00:58:58.281
MAGDALENA TODA:
Because I got a plus.

00:58:58.281 --> 00:59:01.850
If you pay attention, plus sine
squared plus cosine squared.

00:59:01.850 --> 00:59:04.970
So I have, what is sine
squared plus cosine squared?

00:59:04.970 --> 00:59:07.804
I heard that our
students in trig--

00:59:07.804 --> 00:59:11.859
Poly told me-- who still don't
know that this is the most

00:59:11.859 --> 00:59:13.650
important thing you
learn in trigonometry--

00:59:13.650 --> 00:59:15.150
is Pythagorean theorem.

00:59:15.150 --> 00:59:15.860
Right?

00:59:15.860 --> 00:59:24.236
So you have 1/2 integral

00:59:24.236 --> 00:59:27.075
STUDENT: r squared--

00:59:27.075 --> 00:59:28.450
MAGDALENA TODA:
r-- no, I'm lazy.

00:59:28.450 --> 00:59:30.920
I'm going slow-- r.

00:59:30.920 --> 00:59:32.730
dt.

00:59:32.730 --> 00:59:34.916
T from what to what?

00:59:34.916 --> 00:59:36.780
From 0 times 0.

00:59:36.780 --> 00:59:39.880
I'm starting whatever
I want, actually.

00:59:39.880 --> 00:59:44.214
I go counterclockwise
I'm into pi.

00:59:44.214 --> 00:59:45.986
STUDENT: Why is
that not r squared?

00:59:45.986 --> 00:59:47.808
It should be r squared.

00:59:47.808 --> 00:59:49.176
MAGDALENA TODA: I'm sorry, guys.

00:59:49.176 --> 00:59:50.090
I'm sorry.

00:59:50.090 --> 00:59:53.550
I don't know what
I am-- r squared.

00:59:53.550 --> 00:59:57.060
1/2 r squared times 2 pi.

00:59:57.060 --> 01:00:00.740


01:00:00.740 --> 01:00:03.410
So we have pi r squared.

01:00:03.410 --> 01:00:06.870
And if you did not
tell me it's r squared,

01:00:06.870 --> 01:00:09.710
we wouldn't have
gotten the answer.

01:00:09.710 --> 01:00:10.210
That's good.

01:00:10.210 --> 01:00:14.130


01:00:14.130 --> 01:00:16.257
What's the other way to do it?

01:00:16.257 --> 01:00:18.530
If a problem on
the final would ask

01:00:18.530 --> 01:00:22.172
you prove in two different
ways that the rubber

01:00:22.172 --> 01:00:25.130
disk is pi r squared using
Calc 3, or whatever--

01:00:25.130 --> 01:00:26.130
STUDENT: Would require--

01:00:26.130 --> 01:00:28.350
MAGDALENA TODA: The
double integral, right?

01:00:28.350 --> 01:00:28.850
Right?

01:00:28.850 --> 01:00:31.141
STUDENT: Could have done
Cartesian coordinates as well.

01:00:31.141 --> 01:00:33.105
If that counts as a second way.

01:00:33.105 --> 01:00:33.980
MAGDALENA TODA: Yeah.

01:00:33.980 --> 01:00:34.830
You can-- OK.

01:00:34.830 --> 01:00:36.360
What could this be?

01:00:36.360 --> 01:00:37.140
Oh my god.

01:00:37.140 --> 01:00:42.280
This would be minus 1
to 1 minus square root

01:00:42.280 --> 01:00:45.966
1 minus x squared to square
root 1 minus x squared.

01:00:45.966 --> 01:00:46.849
Am i right guys?

01:00:46.849 --> 01:00:47.390
STUDENT: Yep.

01:00:47.390 --> 01:00:48.700
MAGDALENA TODA: 1 dy dx.

01:00:48.700 --> 01:00:51.165
Of course it's a pain.

01:00:51.165 --> 01:00:53.706
STUDENT: You could double that
and set the bottoms both equal

01:00:53.706 --> 01:00:55.074
to 0.

01:00:55.074 --> 01:00:55.990
MAGDALENA TODA: Right.

01:00:55.990 --> 01:01:01.420
So we can do by symmetry--

01:01:01.420 --> 01:01:02.720
STUDENT: Yeah.

01:01:02.720 --> 01:01:05.060
MAGDALENA TODA: I'm--
shall I erase or leave it.

01:01:05.060 --> 01:01:07.790
Are you understand
what Alex is saying?

01:01:07.790 --> 01:01:12.192
This is 2i is the integral
that you will get.

01:01:12.192 --> 01:01:13.650
STUDENT: Just write
it next to it--

01:01:13.650 --> 01:01:14.990
MAGDALENA TODA: I tell
you four times, you

01:01:14.990 --> 01:01:16.490
see, Alex, because you have--

01:01:16.490 --> 01:01:16.820
STUDENT: Oh, yeah.

01:01:16.820 --> 01:01:19.069
MAGDALENA TODA: --symmetry
with respect to the x-axis,

01:01:19.069 --> 01:01:21.420
and symmetry with
respect to y-axis.

01:01:21.420 --> 01:01:26.725
And you can take 0
to 1 and 0 to that.

01:01:26.725 --> 01:01:29.210
And you have x from 0 to 1.

01:01:29.210 --> 01:01:33.740
You have y from 0 to stop.

01:01:33.740 --> 01:01:35.440
Square root of 1 minus x square.

01:01:35.440 --> 01:01:37.465
Like the strips.

01:01:37.465 --> 01:01:41.600
And you have 4
times that A1, which

01:01:41.600 --> 01:01:44.925
would be the area of
the first quadratic.

01:01:44.925 --> 01:01:46.470
You can do that, too.

01:01:46.470 --> 01:01:46.970
It's easier.

01:01:46.970 --> 01:01:50.100
But the best way to do that is
not in Cartesian coordinates.

01:01:50.100 --> 01:01:52.770
The best way is to do
it in polar coordinates.

01:01:52.770 --> 01:01:56.570
Always remember
your Jacobian is r.

01:01:56.570 --> 01:02:00.892
So if you have
Jacobian r-- erase.

01:02:00.892 --> 01:02:03.420
Let's put r here again.

01:02:03.420 --> 01:02:08.270
And then dr d theta.

01:02:08.270 --> 01:02:10.280
But now you say, wait
a minute, Magdalena.

01:02:10.280 --> 01:02:11.780
You said r is fixed.

01:02:11.780 --> 01:02:12.439
Yes.

01:02:12.439 --> 01:02:13.980
And that's why I
need to learn Greek,

01:02:13.980 --> 01:02:15.762
because it's all Greek to me.

01:02:15.762 --> 01:02:18.860
Instead of r I put
rho as a variable.

01:02:18.860 --> 01:02:23.560
And I say, rho is
between 0 and r.

01:02:23.560 --> 01:02:25.300
r is fixed.

01:02:25.300 --> 01:02:27.200
That's my [INAUDIBLE].

01:02:27.200 --> 01:02:32.130
Big r is not usually written
as a variable from 0 to some.

01:02:32.130 --> 01:02:33.485
I cannot use that.

01:02:33.485 --> 01:02:37.260
So I have to us a Greek letter,
whether I like it or not.

01:02:37.260 --> 01:02:39.580
And theta is from 0 to 2 pi.

01:02:39.580 --> 01:02:41.750
And I still get the same thing.

01:02:41.750 --> 01:02:47.050
I get r-- rho squared
over 2 between 0 and r.

01:02:47.050 --> 01:02:48.610
And I have 2 pi.

01:02:48.610 --> 01:02:53.350
And in the end that means
pi r squared, and I'm back.

01:02:53.350 --> 01:02:56.260
And you say, wait,
this is Example 4.

01:02:56.260 --> 01:02:57.462
Whatever example.

01:02:57.462 --> 01:02:59.180
Is it Example 4, 5?

01:02:59.180 --> 01:03:01.090
You say, this is
a piece of cake.

01:03:01.090 --> 01:03:05.610
I have two methods showing
me that area of the disk

01:03:05.610 --> 01:03:06.965
is so pi r squared.

01:03:06.965 --> 01:03:08.204
It's so trivial.

01:03:08.204 --> 01:03:12.140
Yeah, then let's move
on and do the ellipse.

01:03:12.140 --> 01:03:14.740
Or we could have been
smart and done the ellipse

01:03:14.740 --> 01:03:16.710
from the beginning.

01:03:16.710 --> 01:03:18.600
And then the circular
disk would have

01:03:18.600 --> 01:03:23.010
been just a trivial, particular
example of the ellipse.

01:03:23.010 --> 01:03:25.010
But let's do the ellipse
with this magic formula

01:03:25.010 --> 01:03:26.740
that I just taught you.

01:03:26.740 --> 01:03:29.830


01:03:29.830 --> 01:03:34.250
In the finals-- I'm going to
send you a bunch of finals.

01:03:34.250 --> 01:03:36.630
You're going to be
amused, because you're

01:03:36.630 --> 01:03:38.460
going to look at
them and you say,

01:03:38.460 --> 01:03:41.650
regardless of the year and
semester when the final was

01:03:41.650 --> 01:03:44.220
given for Calc 3,
there was always

01:03:44.220 --> 01:03:49.040
one of the problems at the
end using direct application

01:03:49.040 --> 01:03:50.960
of Green's theorem.

01:03:50.960 --> 01:03:53.290
So Green's theorem
is an obsession,

01:03:53.290 --> 01:03:55.214
and not only at Tech.

01:03:55.214 --> 01:03:58.780
I was looking UT Austin,
A&M, other schools--

01:03:58.780 --> 01:04:05.990
California Berkley-- all the
Calc 3 courses on the final

01:04:05.990 --> 01:04:10.620
have at least one application--
direct application

01:04:10.620 --> 01:04:12.360
applying principal.

01:04:12.360 --> 01:04:12.860
OK.

01:04:12.860 --> 01:04:17.250


01:04:17.250 --> 01:04:18.960
So what did I say?

01:04:18.960 --> 01:04:21.715
I said that we have
to draw an ellipse.

01:04:21.715 --> 01:04:25.426
How do we draw an ellipse
without making it up?

01:04:25.426 --> 01:04:26.845
That's the question.

01:04:26.845 --> 01:04:28.737
STUDENT: Draw a circle.

01:04:28.737 --> 01:04:30.156
MAGDALENA TODA: Draw a circle.

01:04:30.156 --> 01:04:32.060
Good answer.

01:04:32.060 --> 01:04:33.580
OK.

01:04:33.580 --> 01:04:35.060
All right.

01:04:35.060 --> 01:04:40.010
And guys this
started really bad.

01:04:40.010 --> 01:04:43.433
So I'm doing what I can.

01:04:43.433 --> 01:04:46.391


01:04:46.391 --> 01:04:49.349
I should have tried
more coffee today,

01:04:49.349 --> 01:04:52.460
because I'm getting
insecure and very shaky.

01:04:52.460 --> 01:04:52.960
OK.

01:04:52.960 --> 01:04:58.100
So I have the ellipse
in standard form

01:04:58.100 --> 01:05:01.620
of center O, x squared over
x squared plus y squared

01:05:01.620 --> 01:05:05.300
over B squared equals 1.

01:05:05.300 --> 01:05:07.890
And now you are going to
me who is A and who is B?

01:05:07.890 --> 01:05:08.920
What are they called?

01:05:08.920 --> 01:05:10.314
Semi--

01:05:10.314 --> 01:05:11.105
STUDENT: Semiotics.

01:05:11.105 --> 01:05:12.188
MAGDALENA TODA: Semiotics.

01:05:12.188 --> 01:05:15.480
A and B. Good.

01:05:15.480 --> 01:05:18.900
Find the area.

01:05:18.900 --> 01:05:21.810
I don't like-- OK.

01:05:21.810 --> 01:05:27.080
Let's put B inside, and let's
put C outside the boundary.

01:05:27.080 --> 01:05:42.690
So area of the ellipse domain
D will be-- by the lemma-- 1/2

01:05:42.690 --> 01:05:46.010
integral over C.

01:05:46.010 --> 01:05:47.250
This is C. Is not f.

01:05:47.250 --> 01:05:48.200
Don't confuse it.

01:05:48.200 --> 01:05:50.616
It is my beautiful
script C. I've

01:05:50.616 --> 01:05:52.420
tried to use it many times.

01:05:52.420 --> 01:05:55.240
Going to be minus y dx plus xdy.

01:05:55.240 --> 01:05:58.380


01:05:58.380 --> 01:05:59.490
Again, why was that?

01:05:59.490 --> 01:06:04.530
Because we said this
is M and this is N,

01:06:04.530 --> 01:06:09.110
and Green's theorem will give
you double integral of N sub x

01:06:09.110 --> 01:06:10.570
minus M sub y.

01:06:10.570 --> 01:06:13.910
So you have 1 minus
minus 1, which is 2.

01:06:13.910 --> 01:06:15.910
And 2 knocked that out.

01:06:15.910 --> 01:06:16.410
OK.

01:06:16.410 --> 01:06:19.090
That's how we prove it.

01:06:19.090 --> 01:06:19.720
OK.

01:06:19.720 --> 01:06:24.210
Problem is that I do not the
parametrization of the ellipse.

01:06:24.210 --> 01:06:28.220
And if somebody doesn't help me,
I'm going to be in big trouble.

01:06:28.220 --> 01:06:32.620


01:06:32.620 --> 01:06:34.420
And I'll start
cursing and I'm not

01:06:34.420 --> 01:06:37.070
allowed to curse in
front of the classroom.

01:06:37.070 --> 01:06:40.760
But you can help me on
that, because this reminds

01:06:40.760 --> 01:06:46.040
you of a famous Greek identity.

01:06:46.040 --> 01:06:48.520
The fundamental trig identity.

01:06:48.520 --> 01:06:51.610
If this would be cosine
squared of theta,

01:06:51.610 --> 01:06:55.260
and this would be sine squared
of theta, as two animals,

01:06:55.260 --> 01:06:56.710
their sum would be 1.

01:06:56.710 --> 01:07:00.630
And whenever you have sums
of sum squared thingies,

01:07:00.630 --> 01:07:03.834
then you have to think trig.

01:07:03.834 --> 01:07:06.710
So, what would be
good as a parameter?

01:07:06.710 --> 01:07:07.300
OK.

01:07:07.300 --> 01:07:10.240
What would be good
as a parametrization

01:07:10.240 --> 01:07:12.214
to make this come true?

01:07:12.214 --> 01:07:15.012
STUDENT: You have the cosine
of theta would equal x over x.

01:07:15.012 --> 01:07:15.970
MAGDALENA TODA: Uh-huh.

01:07:15.970 --> 01:07:18.126
So then x would be A times--

01:07:18.126 --> 01:07:19.410
STUDENT: The cosine of theta.

01:07:19.410 --> 01:07:21.590
MAGDALENA TODA:
Do you like theta?

01:07:21.590 --> 01:07:23.690
You don't, because
you're not Greek.

01:07:23.690 --> 01:07:25.050
That's the problem.

01:07:25.050 --> 01:07:26.840
If you were Greek,
you would like it.

01:07:26.840 --> 01:07:29.260
We had a colleague who
is not here anymore.

01:07:29.260 --> 01:07:30.790
Greek from Cypress.

01:07:30.790 --> 01:07:38.250
And he could claim that the
most important-- most important

01:07:38.250 --> 01:07:40.230
alphabet is the
Greek one, and that's

01:07:40.230 --> 01:07:44.210
why the mathematicians
adopted it.

01:07:44.210 --> 01:07:45.150
OK?

01:07:45.150 --> 01:07:47.150
B sine t.

01:07:47.150 --> 01:07:48.270
How do you check?

01:07:48.270 --> 01:07:49.290
You always think, OK.

01:07:49.290 --> 01:07:51.460
This over that is cosine.

01:07:51.460 --> 01:07:53.490
This over this is sine.

01:07:53.490 --> 01:07:54.340
I square them.

01:07:54.340 --> 01:07:56.210
I get exactly that
and I get a 1.

01:07:56.210 --> 01:07:56.710
Good.

01:07:56.710 --> 01:07:57.670
I'm in good shape.

01:07:57.670 --> 01:08:01.380
I know that this
implicit equation--

01:08:01.380 --> 01:08:04.710
this is an implicit
equation-- happens if and only

01:08:04.710 --> 01:08:11.080
if I have this system of
the parametrization with t

01:08:11.080 --> 01:08:17.000
between-- anything I want,
including the basic 0 to 2

01:08:17.000 --> 01:08:18.420
pi interval.

01:08:18.420 --> 01:08:22.380
And then if I were to move
all around for time real t

01:08:22.380 --> 01:08:26.274
I would wind around that the
circle infinitely many times.

01:08:26.274 --> 01:08:29.270
Between time equals
minus infinity--

01:08:29.270 --> 01:08:33.060
that nobody remembers-- and
time equals plus infinity--

01:08:33.060 --> 01:08:36.180
that nobody will
ever get to know.

01:08:36.180 --> 01:08:38.550
So those are the values of it.

01:08:38.550 --> 01:08:41.366
All the real values, actually.

01:08:41.366 --> 01:08:44.960
I only needed from 0 to 2
pi to wind one time around.

01:08:44.960 --> 01:08:46.660
And this is the idea.

01:08:46.660 --> 01:08:48.514
I wind one time around.

01:08:48.514 --> 01:08:51.149
Now people-- you're going
to see mathematicians

01:08:51.149 --> 01:08:52.740
are not the greatest people.

01:08:52.740 --> 01:09:01.254
I've seen engineers and
physicists use a lot this sign.

01:09:01.254 --> 01:09:02.420
Do you know what this means?

01:09:02.420 --> 01:09:04.420
STUDENT: It means
one full revolution.

01:09:04.420 --> 01:09:06.550
MAGDALENA TODA: It
means a full revolution.

01:09:06.550 --> 01:09:10.410
You're going to have
a loop-- loops, that's

01:09:10.410 --> 01:09:11.240
whatever you want.

01:09:11.240 --> 01:09:13.380
Here and goes counterclockwise.

01:09:13.380 --> 01:09:15.720
And they put this
little sign showing

01:09:15.720 --> 01:09:21.790
I'm going counterclockwise on
a closed curved, or a loop.

01:09:21.790 --> 01:09:22.509
All right.

01:09:22.509 --> 01:09:24.439
Don't think they are crazy.

01:09:24.439 --> 01:09:27.479
This was used in lots
of scientific papers

01:09:27.479 --> 01:09:30.810
in math, physics, and
engineering, and so on.

01:09:30.810 --> 01:09:31.310
OK.

01:09:31.310 --> 01:09:34.850


01:09:34.850 --> 01:09:36.660
Let's do it then.

01:09:36.660 --> 01:09:38.279
Can we do it by ourselves?

01:09:38.279 --> 01:09:39.210
I think so.

01:09:39.210 --> 01:09:39.810
That's see.

01:09:39.810 --> 01:09:42.370
1/2 is 1.

01:09:42.370 --> 01:09:45.310
And I don't like
the pink marker.

01:09:45.310 --> 01:09:47.270
Integral log.

01:09:47.270 --> 01:09:51.930
Time from 0 to 2 pi
should be measured.

01:09:51.930 --> 01:09:55.740
y minus B sine t.

01:09:55.740 --> 01:10:01.730


01:10:01.730 --> 01:10:04.345
dx-- what tells me that?

01:10:04.345 --> 01:10:06.810
STUDENT: B minus--

01:10:06.810 --> 01:10:07.223


01:10:07.223 --> 01:10:08.306
MAGDALENA TODA: Very good.

01:10:08.306 --> 01:10:09.800
Minus A sine t.

01:10:09.800 --> 01:10:10.796
How hard is that?

01:10:10.796 --> 01:10:16.274
It's a piece of cake Plus x--

01:10:16.274 --> 01:10:18.179
STUDENT: A cosine.

01:10:18.179 --> 01:10:19.262
MAGDALENA TODA: Very good.

01:10:19.262 --> 01:10:21.760
A cosine t.

01:10:21.760 --> 01:10:23.090
TImes--

01:10:23.090 --> 01:10:25.630
STUDENT: B cosine t.

01:10:25.630 --> 01:10:28.790
MAGDALENA TODA: --B cosine t.

01:10:28.790 --> 01:10:30.130
And dt.

01:10:30.130 --> 01:10:32.750
And this thing-- look at it.

01:10:32.750 --> 01:10:33.460
It's huge.

01:10:33.460 --> 01:10:35.700
It looks huge, but it's
so beautiful, because--

01:10:35.700 --> 01:10:36.569
STUDENT: AB.

01:10:36.569 --> 01:10:37.360
MAGDALENA TODA: AB.

01:10:37.360 --> 01:10:38.805
Why is it AB?

01:10:38.805 --> 01:10:44.030
It's AB because sine squared
plus cosine squared inside

01:10:44.030 --> 01:10:46.180
becomes 1.

01:10:46.180 --> 01:10:49.990
And I have plus AB,
plus AB, AB out.

01:10:49.990 --> 01:10:52.020
Kick out the AB.

01:10:52.020 --> 01:10:57.090
Kick out the A and
the B and you get

01:10:57.090 --> 01:11:01.750
something beautiful-- sine
squared t plus cosine squared

01:11:01.750 --> 01:11:03.420
t is your old friend.

01:11:03.420 --> 01:11:04.840
And he says, I'm 1.

01:11:04.840 --> 01:11:08.049
Look how beautiful
life is for you.

01:11:08.049 --> 01:11:09.498
Finally, we proved it.

01:11:09.498 --> 01:11:10.947
What did we prove?

01:11:10.947 --> 01:11:11.913
We are almost there.

01:11:11.913 --> 01:11:12.879
We got a 1/2.

01:11:12.879 --> 01:11:15.780


01:11:15.780 --> 01:11:18.600
A constant value kick out, AB.

01:11:18.600 --> 01:11:21.504


01:11:21.504 --> 01:11:22.472
STUDENT: Times 2 pi.

01:11:22.472 --> 01:11:23.597
MAGDALENA TODA: Times 2 pi.

01:11:23.597 --> 01:11:26.840


01:11:26.840 --> 01:11:28.010
Good.

01:11:28.010 --> 01:11:30.110
2 goes away.

01:11:30.110 --> 01:11:33.300
And we got a magic thing that
nobody taught us in school,

01:11:33.300 --> 01:11:34.730
because they were mean.

01:11:34.730 --> 01:11:37.360
They really didn't want
us to learn too much.

01:11:37.360 --> 01:11:38.760
That's the thingy.

01:11:38.760 --> 01:11:40.540
AB pi.

01:11:40.540 --> 01:11:45.220
AB pi is what we were
hoping for, because, look.

01:11:45.220 --> 01:11:47.820
I mean it's almost
too good to be true.

01:11:47.820 --> 01:11:53.610
Well, it's a disk of radius
r, A and B are equal.

01:11:53.610 --> 01:11:55.870
And they are the
radius of the disk.

01:11:55.870 --> 01:11:58.730
And that's why we
have pi r squared

01:11:58.730 --> 01:12:00.972
as a particular
example of the disk

01:12:00.972 --> 01:12:04.924
of the area of this ellipse.

01:12:04.924 --> 01:12:07.641
When I saw it the first
time, I was like, well,

01:12:07.641 --> 01:12:12.500
I'm glad that I lived to be
30 or something to learn this.

01:12:12.500 --> 01:12:17.510
Because nobody had shown it
to me in K-12 or in college.

01:12:17.510 --> 01:12:22.534
And I was a completing-- I was
a PhD and I didn't know it.

01:12:22.534 --> 01:12:25.378
And then I said, oh,
that's why-- pi AB.

01:12:25.378 --> 01:12:26.800
Yes, OK.

01:12:26.800 --> 01:12:27.770
All right.

01:12:27.770 --> 01:12:31.870
So it's so easy to
understand once you-- well.

01:12:31.870 --> 01:12:33.493
Once you learn the section.

01:12:33.493 --> 01:12:34.909
If you don't learn
the section you

01:12:34.909 --> 01:12:38.970
will not be able to understand.

01:12:38.970 --> 01:12:39.470
OK.

01:12:39.470 --> 01:12:39.970
All right.

01:12:39.970 --> 01:12:42.300
I'm going to go
ahead and erase this.

01:12:42.300 --> 01:12:44.950
And I'll show you
an example that

01:12:44.950 --> 01:12:49.500
was popping up like an obsession
with the numbers changed

01:12:49.500 --> 01:12:53.190
in most of the final exams
that happen in the last three

01:12:53.190 --> 01:12:58.900
years, regardless of
who wrote the exam.

01:12:58.900 --> 01:13:04.590
Because this problem really
matches the learning outcomes,

01:13:04.590 --> 01:13:08.530
oh, just about any university--
any good university

01:13:08.530 --> 01:13:10.690
around the world.

01:13:10.690 --> 01:13:12.240
So you'll say, wow.

01:13:12.240 --> 01:13:13.110
It's so easy.

01:13:13.110 --> 01:13:16.754
I could not believe it
that-- how easy it is.

01:13:16.754 --> 01:13:25.664
But once you see it, you
will-- you'll say, wow.

01:13:25.664 --> 01:13:26.660
It's easy.

01:13:26.660 --> 01:13:34.626


01:13:34.626 --> 01:13:35.126
OK.

01:13:35.126 --> 01:13:41.102


01:13:41.102 --> 01:13:44.120
[CHATTER]

01:13:44.120 --> 01:13:46.034
Let's try this one.

01:13:46.034 --> 01:13:48.519
You have a circle.

01:13:48.519 --> 01:13:57.770
and the circle will be
a circle radius r given

01:13:57.770 --> 01:14:03.590
and origin 0 of 4, 9, 0, and 0.

01:14:03.590 --> 01:14:08.520


01:14:08.520 --> 01:14:17.290
And I'm going to
write-- I'm going

01:14:17.290 --> 01:14:19.970
to give you-- first I'm going
to give you a very simple one.

01:14:19.970 --> 01:14:31.171


01:14:31.171 --> 01:14:37.989
Compute in the
simplest possible way.

01:14:37.989 --> 01:14:41.570
If you don't want to
parametrize the circle--

01:14:41.570 --> 01:14:43.400
you can always
parametrize the circle.

01:14:43.400 --> 01:14:44.302
Right?

01:14:44.302 --> 01:14:45.420
But you don't want to.

01:14:45.420 --> 01:14:49.270
You want to do it the
fastest possible way

01:14:49.270 --> 01:14:51.440
without parameterizing
the circle.

01:14:51.440 --> 01:14:53.880
Without writing down
what I'm writing down.

01:14:53.880 --> 01:14:55.110
You are in a hurry.

01:14:55.110 --> 01:14:58.980
You have 20-- 15 minutes
left of your final.

01:14:58.980 --> 01:15:00.700
And you're looking
at me and say, I

01:15:00.700 --> 01:15:02.220
hope I get an A in this final.

01:15:02.220 --> 01:15:05.887
So what do you have to
remember when you look at that?

01:15:05.887 --> 01:15:09.660


01:15:09.660 --> 01:15:14.770
M and M. M and M.
No, M and N. OK.

01:15:14.770 --> 01:15:18.690
And you have to remember
that you are over a circle

01:15:18.690 --> 01:15:20.150
so you have a closed loop.

01:15:20.150 --> 01:15:21.780
And that's a Jordan curve.

01:15:21.780 --> 01:15:24.130
That's enclosing a disk.

01:15:24.130 --> 01:15:28.070
So you have a relationship
between the path

01:15:28.070 --> 01:15:34.650
integral along the C and the
area along the D-- over D.

01:15:34.650 --> 01:15:36.350
Which is of what?

01:15:36.350 --> 01:15:38.720
Is N sub x minus M sub y.

01:15:38.720 --> 01:15:41.300
So let me write it
in this form, which

01:15:41.300 --> 01:15:46.120
is the same thing my students
mostly prefer to write it as.

01:15:46.120 --> 01:15:48.710
N sub x minus M sub y.

01:15:48.710 --> 01:15:51.790
The t-shirt I have
has it written

01:15:51.790 --> 01:15:56.960
like that, because it was
bought from nerdytshirt.com

01:15:56.960 --> 01:16:01.220
And it was especially
created to impress nerds.

01:16:01.220 --> 01:16:04.300
And of course if you
look at the del notation

01:16:04.300 --> 01:16:07.380
that gives you that kind
of snobbish attitude

01:16:07.380 --> 01:16:11.670
that you aren't a scientist.

01:16:11.670 --> 01:16:12.220
OK.

01:16:12.220 --> 01:16:16.332
So what is this
going to be then?

01:16:16.332 --> 01:16:19.390
Double integral over d.

01:16:19.390 --> 01:16:22.390
And sub x is up
here so it gave 5.

01:16:22.390 --> 01:16:24.750
And sub y is a piece of cake.

01:16:24.750 --> 01:16:36.530
3 dx dy equals 2 out times
the area of the disk, which

01:16:36.530 --> 01:16:38.466
is something you know.

01:16:38.466 --> 01:16:40.886
And I'm not going to ask you
to prove that all over again.

01:16:40.886 --> 01:16:42.760
So you have to say 2.

01:16:42.760 --> 01:16:46.586
I know the area of the
disk-- pi r squared.

01:16:46.586 --> 01:16:48.002
And that's the answer.

01:16:48.002 --> 01:16:49.418
And you leave the room.

01:16:49.418 --> 01:16:50.362
And that's it.

01:16:50.362 --> 01:16:52.260
It's almost too
easy to believe it,

01:16:52.260 --> 01:16:58.280
but it was always there in
the simplest possible way.

01:16:58.280 --> 01:17:02.610
And now I'm wondering, if I
were to give you something hard,

01:17:02.610 --> 01:17:08.040
because-- you know my theory
that when you practice

01:17:08.040 --> 01:17:11.810
at something in
the classroom you

01:17:11.810 --> 01:17:16.650
have to be working on harder
things in the classroom

01:17:16.650 --> 01:17:19.310
to do better in the exam.

01:17:19.310 --> 01:17:22.990
So let me cook up
something ugly for you.

01:17:22.990 --> 01:17:25.830
The same kind of disk.

01:17:25.830 --> 01:17:28.250
And I'm changing the functions.

01:17:28.250 --> 01:17:33.055
And I'll make it
more complicated.

01:17:33.055 --> 01:17:36.140


01:17:36.140 --> 01:17:40.478
Let's see how you
perform on this one.

01:17:40.478 --> 01:17:47.240


01:17:47.240 --> 01:17:49.710
We avoided that one,
probably, on finals

01:17:49.710 --> 01:17:52.478
because I think the
majority of students

01:17:52.478 --> 01:17:58.060
wouldn't have understood what
theorem they needed to apply.

01:17:58.060 --> 01:17:59.690
It looks a little bit scary.

01:17:59.690 --> 01:18:01.910
But let's say that I've
given you the hint,

01:18:01.910 --> 01:18:05.003
apply Greens theorem
on the same path

01:18:05.003 --> 01:18:10.380
integral, which is a circle
of origin 0 and radius r.

01:18:10.380 --> 01:18:14.410
I now draw counterclockwise.

01:18:14.410 --> 01:18:18.490
You apply Green's theorem and
you say, I know how to do this,

01:18:18.490 --> 01:18:21.130
because now I know the theorem.

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.

01:18:27.270 --> 01:18:30.870
It said P and Q. Do you
want to put P and Q?

01:18:30.870 --> 01:18:31.750
I put P and Q.

01:18:31.750 --> 01:18:34.740
So I can-- I can have this
like it is on my t-shirt.

01:18:34.740 --> 01:18:39.230
So this is going
to be P sub x-- no.

01:18:39.230 --> 01:18:39.730
Q sub x.

01:18:39.730 --> 01:18:40.730
Sorry.

01:18:40.730 --> 01:18:44.560
M and N. So the second
one with respect to x.

01:18:44.560 --> 01:18:48.560
The one that sticks to the y
is prime root respect to x.

01:18:48.560 --> 01:18:54.344
The one that sticks to dx is
prime root with respect to y.

01:18:54.344 --> 01:18:56.885
And I think one
time-- the one time

01:18:56.885 --> 01:19:01.140
when that my friend and
colleague wrote that,

01:19:01.140 --> 01:19:02.680
he did it differently.

01:19:02.680 --> 01:19:05.595
He wrote something
like, just-- I'll

01:19:05.595 --> 01:19:09.750
put-- I don't remember what.

01:19:09.750 --> 01:19:10.800
He put this one.

01:19:10.800 --> 01:19:13.506


01:19:13.506 --> 01:19:17.470
Then the student
was used to dx/dy

01:19:17.470 --> 01:19:19.250
and got completely confused.

01:19:19.250 --> 01:19:25.625
So pay attention to
what you are saying.

01:19:25.625 --> 01:19:29.400
Most of us write it
in x and y first.

01:19:29.400 --> 01:19:32.960
And we can see that the
derivative with respect

01:19:32.960 --> 01:19:38.870
to x of q, because that is
the one next to be the y.

01:19:38.870 --> 01:19:42.446
When he gave it to me
like that, he messed up

01:19:42.446 --> 01:19:44.760
everybody's notations.

01:19:44.760 --> 01:19:46.000
No.

01:19:46.000 --> 01:19:47.190
Good students steal data.

01:19:47.190 --> 01:19:49.770
So you guys have to
put it in standard form

01:19:49.770 --> 01:19:52.825
and pay attention to
what you are doing.

01:19:52.825 --> 01:19:53.770
All right.

01:19:53.770 --> 01:19:57.140
So that one form can
be swapped by people

01:19:57.140 --> 01:19:58.620
who try to play games.

01:19:58.620 --> 01:20:02.570


01:20:02.570 --> 01:20:08.312
Now in this one-- So you
have q sub x minus b sub y.

01:20:08.312 --> 01:20:15.990
You have 3x squared minus
minus, or just plus, 3y squared.

01:20:15.990 --> 01:20:16.490
Good.

01:20:16.490 --> 01:20:17.460
Wonderful.

01:20:17.460 --> 01:20:20.370
Am I happy, do you
think I'm happy?

01:20:20.370 --> 01:20:22.740
Why would I be so happy?

01:20:22.740 --> 01:20:25.790
Why is this a happy thing?

01:20:25.790 --> 01:20:27.740
I could have had
something more wild.

01:20:27.740 --> 01:20:28.270
I don't.

01:20:28.270 --> 01:20:30.210
I'm happy I don't.

01:20:30.210 --> 01:20:31.760
Why am I so happy?

01:20:31.760 --> 01:20:34.360
Let's see.

01:20:34.360 --> 01:20:39.850
3 out over the disk.

01:20:39.850 --> 01:20:41.710
Is this ringing a bell?

01:20:41.710 --> 01:20:48.850


01:20:48.850 --> 01:20:49.880
Yeah.

01:20:49.880 --> 01:20:53.150
It's r squared if I
do this in former.

01:20:53.150 --> 01:20:58.640
So if I do this in former,
its going to be rdr, d theta.

01:20:58.640 --> 01:21:01.340
So life is not as
hard as you believe.

01:21:01.340 --> 01:21:04.035
It can look like
a harder problem,

01:21:04.035 --> 01:21:06.250
but in reality, it's not really.

01:21:06.250 --> 01:21:11.970
So I have 3 times-- now, I
have r squared, I have r cubed.

01:21:11.970 --> 01:21:16.780
r cubed dr d theta, r between.

01:21:16.780 --> 01:21:20.560


01:21:20.560 --> 01:21:30.710
r was between 0 and big
R. Theta will always

01:21:30.710 --> 01:21:34.150
be between 0 and 2 pi.

01:21:34.150 --> 01:21:42.316
So, I want you, without
me to compute the answer

01:21:42.316 --> 01:21:44.454
and tell me what you got.

01:21:44.454 --> 01:21:46.839
STUDENT: Just say it?

01:21:46.839 --> 01:21:48.270
MAGDALENA TODA: Yep.

01:21:48.270 --> 01:21:52.580
STUDENT: 3/2, pi
r to the fourth.

01:21:52.580 --> 01:21:54.350
MAGDALENA TODA: So
how did you do that?

01:21:54.350 --> 01:21:56.610
You said, r to the
4 over 4, coming

01:21:56.610 --> 01:22:00.470
from integration times the 2 pi,
coming from integration times

01:22:00.470 --> 01:22:01.966
3.

01:22:01.966 --> 01:22:05.312
Are you guys with me?

01:22:05.312 --> 01:22:08.690
Is everybody with me on this?

01:22:08.690 --> 01:22:12.130
OK so, we will simplify
the answer, we'll do that.

01:22:12.130 --> 01:22:16.696
What regard is the
radius of the disk?

01:22:16.696 --> 01:22:18.320
STUDENT: How did he
solve that integral

01:22:18.320 --> 01:22:20.008
without switching the poles?

01:22:20.008 --> 01:22:26.456


01:22:26.456 --> 01:22:28.936
MAGDALENA TODA: It would
have been a killer.

01:22:28.936 --> 01:22:30.920
Let me write it out.

01:22:30.920 --> 01:22:32.408
[LAUGHTER]

01:22:32.408 --> 01:22:34.910
Because you want to
write it out, of course.

01:22:34.910 --> 01:22:40.550
OK, 3 integral, integral
x squared plus y

01:22:40.550 --> 01:22:45.450
squared, dy/dx, just to make
my life a little bit funnier,

01:22:45.450 --> 01:22:50.280
and then y between minus
square root-- you're

01:22:50.280 --> 01:22:52.060
looking for trouble, huh?

01:22:52.060 --> 01:22:59.580
Y squared minus x squared to
r squared minus s squared.

01:22:59.580 --> 01:23:01.250
And again, you could
do what you just

01:23:01.250 --> 01:23:04.536
said, split into four integrals
over four different domains,

01:23:04.536 --> 01:23:07.440
or two up and down.

01:23:07.440 --> 01:23:11.950
And minus r and are
you guys with me?

01:23:11.950 --> 01:23:14.890
And then, when you go
and integrate that,

01:23:14.890 --> 01:23:22.960
you integrate with respect
to y-- [INAUDIBLE].

01:23:22.960 --> 01:23:25.300
Well he's right,
so you can get x

01:23:25.300 --> 01:23:27.835
squared y plus y cubed over 3.

01:23:27.835 --> 01:23:30.770


01:23:30.770 --> 01:23:34.240
Between those points, minus 12.

01:23:34.240 --> 01:23:36.120
And from that moment,
that would just

01:23:36.120 --> 01:23:39.563
leave it and go for a walk.

01:23:39.563 --> 01:23:43.260
I will not have the
patience to do this.

01:23:43.260 --> 01:23:44.800
Just a second, Matthew.

01:23:44.800 --> 01:23:46.680
For this kind of
stuff, of course

01:23:46.680 --> 01:23:50.410
I could put this in Maple.

01:23:50.410 --> 01:23:54.150
You know Maple has these
little interactive fields,

01:23:54.150 --> 01:23:55.940
like little squares?

01:23:55.940 --> 01:23:58.940
And you go inside there
and add your endpoints.

01:23:58.940 --> 01:24:02.740
And even if it looks very ugly,
Maple will spit you the answer.

01:24:02.740 --> 01:24:05.668
If you know your
syntax and do it right,

01:24:05.668 --> 01:24:07.620
even if you don't
switch to polar

01:24:07.620 --> 01:24:10.060
coordinates or put
it in Cartesian.

01:24:10.060 --> 01:24:12.980
Give it the right data, and
it's going to spit the answer.

01:24:12.980 --> 01:24:13.942
Yes, Matthew?

01:24:13.942 --> 01:24:16.106
STUDENT: I was
out of the room, I

01:24:16.106 --> 01:24:19.242
was wondering why
it's now y cubed.

01:24:19.242 --> 01:24:21.450
MAGDALENA TODA: Because if
you integrate with respect

01:24:21.450 --> 01:24:24.140
to y first--

01:24:24.140 --> 01:24:27.460
STUDENT: Because when I
walked out, it was negative y.

01:24:27.460 --> 01:24:29.084
MAGDALENA TODA: If
I didn't put minus.

01:24:29.084 --> 01:24:30.250
STUDENT: It's a new problem.

01:24:30.250 --> 01:24:32.120
That's what he's confused about.

01:24:32.120 --> 01:24:34.700
He walked out of the room
during the previous problem

01:24:34.700 --> 01:24:36.420
and came back after this one.

01:24:36.420 --> 01:24:37.709
And now he's confused.

01:24:37.709 --> 01:24:40.000
MAGDALENA TODA: You don't
care about what I just asked?

01:24:40.000 --> 01:24:40.380
STUDENT: Oh.

01:24:40.380 --> 01:24:40.880
No.

01:24:40.880 --> 01:24:44.125


01:24:44.125 --> 01:24:45.942
I like the polar coordinates.

01:24:45.942 --> 01:24:47.650
MAGDALENA TODA: Let
me ask you a question

01:24:47.650 --> 01:24:50.100
before I talk any further.

01:24:50.100 --> 01:24:53.250
I was about to put a plus here.

01:24:53.250 --> 01:24:56.460
What would have been the problem
if I had put a plus here?

01:24:56.460 --> 01:24:59.810


01:24:59.810 --> 01:25:02.560
If I worked this out,
I would have gotten

01:25:02.560 --> 01:25:05.930
x squared minus y squared.

01:25:05.930 --> 01:25:08.300
Would that have been
the end of the world?

01:25:08.300 --> 01:25:10.040
No.

01:25:10.040 --> 01:25:16.156
But it would have complicated
my life a little bit more.

01:25:16.156 --> 01:25:20.990
Let's do that one as well.

01:25:20.990 --> 01:25:22.422
STUDENT: I was
just curious of how

01:25:22.422 --> 01:25:25.247
you do any of these problems
when you can't switch to polar.

01:25:25.247 --> 01:25:27.830
MAGDALENA TODA: Right, let's see
what-- because Actually, even

01:25:27.830 --> 01:25:32.440
in this case, life is not so
hard, not as hard as you think.

01:25:32.440 --> 01:25:34.750
The persistence in that matters.

01:25:34.750 --> 01:25:37.620
You never give up on a
problem that freaks you out.

01:25:37.620 --> 01:25:41.240
That's the definition
of a mathematician.

01:25:41.240 --> 01:25:48.380
3x squared minus 3y
squared over dx/dy.

01:25:48.380 --> 01:25:50.270
Do it slowly because
I'm not in a hurry.

01:25:50.270 --> 01:25:55.670
We are almost done with 13.4.

01:25:55.670 --> 01:25:57.407
This is OK, right?

01:25:57.407 --> 01:25:59.355
Just the minus sign again?

01:25:59.355 --> 01:26:01.303
STUDENT: Well not
the minus sign.

01:26:01.303 --> 01:26:03.738
I was just wondering because
in the previous problem

01:26:03.738 --> 01:26:07.147
you were doing the ellipse, you
started out with the equation

01:26:07.147 --> 01:26:08.608
with the negative y--

01:26:08.608 --> 01:26:10.556
MAGDALENA TODA:
For this one that's

01:26:10.556 --> 01:26:14.844
just the limit that says that
this is the go double integral

01:26:14.844 --> 01:26:18.920
of the area of the domain.

01:26:18.920 --> 01:26:23.120
It's just a consequence--
or correlate if you want.

01:26:23.120 --> 01:26:27.715
It's a consequence
of Green's theorem.

01:26:27.715 --> 01:26:31.100
When you forget that consequence
of Green's theorem and we say

01:26:31.100 --> 01:26:32.005
goodbye to that.

01:26:32.005 --> 01:26:35.810
But while you were out,
this is Green's theorem.

01:26:35.810 --> 01:26:38.635
The real Green's theorem,
the one that was a teacher.

01:26:38.635 --> 01:26:41.310
There are several
Greens I can give you.

01:26:41.310 --> 01:26:43.280
The famous Green
theorem is the one

01:26:43.280 --> 01:26:46.850
I said when you have--
this is what we apply here.

01:26:46.850 --> 01:26:50.972
The integral of M dx plus M dy.

01:26:50.972 --> 01:27:00.820
You have a double integral of
M sub x minus M sub y over c.

01:27:00.820 --> 01:27:03.450


01:27:03.450 --> 01:27:07.690
So I'm assuming we would have
had this case of maybe me not

01:27:07.690 --> 01:27:10.990
paying attention, or
being mean and not wanting

01:27:10.990 --> 01:27:14.484
to give you a simple problem.

01:27:14.484 --> 01:27:17.661
And what do you
do in such a case?

01:27:17.661 --> 01:27:19.660
It's not obvious to
everybody, but you will see.

01:27:19.660 --> 01:27:22.050
It's so pretty at some
point, if you know

01:27:22.050 --> 01:27:24.034
how to get out of the mess.

01:27:24.034 --> 01:27:27.010


01:27:27.010 --> 01:27:30.820
I was already thinking, but
I'm using polar coordinates.

01:27:30.820 --> 01:27:35.620
So that's arc of sine, so I
have to go back to the basics.

01:27:35.620 --> 01:27:40.000
If I go back to the
basics, ideas come to me.

01:27:40.000 --> 01:27:41.760
Right?

01:27:41.760 --> 01:27:45.930
So, OK.

01:27:45.930 --> 01:27:51.560
r-- let's put dr d theta,
just to get rid of it,

01:27:51.560 --> 01:27:53.780
because it's on my nerves.

01:27:53.780 --> 01:28:00.370
This is 0 to 2 pi,
this is 0 to r.

01:28:00.370 --> 01:28:03.340
And now, you say,
OK, in our mind,

01:28:03.340 --> 01:28:06.960
because we are lazy
people, plug in those

01:28:06.960 --> 01:28:10.970
and pull out what you can.

01:28:10.970 --> 01:28:14.970
One 3 out equals for what?

01:28:14.970 --> 01:28:18.400
Inside, you have r squared.

01:28:18.400 --> 01:28:20.720
Do you agree?

01:28:20.720 --> 01:28:29.510
And times your favorite
expression, which is cosine

01:28:29.510 --> 01:28:32.550
squared theta, minus
i squared theta.

01:28:32.550 --> 01:28:34.520
And you're going to ask me why.

01:28:34.520 --> 01:28:36.070
You shouldn't ask me why.

01:28:36.070 --> 01:28:40.305
You just square these
and subtract them,

01:28:40.305 --> 01:28:44.050
and see what in the world
you're going to get.

01:28:44.050 --> 01:28:48.290
Because you get r squared
times cosine squared,

01:28:48.290 --> 01:28:49.510
minus i squared.

01:28:49.510 --> 01:28:51.434
I'm too lazy to write
down the argument.

01:28:51.434 --> 01:28:52.850
But you know we
have trigonometry.

01:28:52.850 --> 01:28:55.670


01:28:55.670 --> 01:28:58.170
Yes, you see why it's
important for you

01:28:58.170 --> 01:29:02.010
to learn trigonometry
when you are little.

01:29:02.010 --> 01:29:05.730
You may be 50 or
60, in high school,

01:29:05.730 --> 01:29:09.188
or you may be freshman year.

01:29:09.188 --> 01:29:11.980
I don't care when, but you
have to learn that this is

01:29:11.980 --> 01:29:14.306
the cosine of the double angle.

01:29:14.306 --> 01:29:16.240
How many of you remember that?

01:29:16.240 --> 01:29:18.540
Maybe you learned that?

01:29:18.540 --> 01:29:19.460
Remember that?

01:29:19.460 --> 01:29:21.400
OK.

01:29:21.400 --> 01:29:25.800
I don't blame you at all
when you don't remember,

01:29:25.800 --> 01:29:30.920
because since I've been
the main checker of finals

01:29:30.920 --> 01:29:39.490
for the past five years--
it's a lot of finals.

01:29:39.490 --> 01:29:40.690
Yeah, the i is there.

01:29:40.690 --> 01:29:42.840
That's exactly what
I wanted to tell you,

01:29:42.840 --> 01:29:46.450
that's why I left some room.

01:29:46.450 --> 01:29:52.566
This data would be t.

01:29:52.566 --> 01:29:56.370
The double angle formula did
not appear on many finals.

01:29:56.370 --> 01:29:58.110
And I was thinking
it's a period.

01:29:58.110 --> 01:29:59.960
When I ask the
instructors, generally they

01:29:59.960 --> 01:30:06.550
say students have trouble
remembering or understanding

01:30:06.550 --> 01:30:10.130
this later on, by
avoiding the issue,

01:30:10.130 --> 01:30:14.012
you sort of bound to it for
the first time in Cal 2,

01:30:14.012 --> 01:30:16.868
because there are any
geometric formulas.

01:30:16.868 --> 01:30:21.634
And then, you bump again
inside it in Cal 3.

01:30:21.634 --> 01:30:23.310
And it never leaves you.

01:30:23.310 --> 01:30:27.560
So this, just knowing this
will help you so much.

01:30:27.560 --> 01:30:30.852
Let me put the r nicely here.

01:30:30.852 --> 01:30:34.123
And now finally, we know
how to solve it, because I'm

01:30:34.123 --> 01:30:35.289
going to go ahead and erase.

01:30:35.289 --> 01:30:44.660


01:30:44.660 --> 01:30:49.250
So why it is good for us is
that-- as Matthew observed

01:30:49.250 --> 01:30:52.810
a few moments ago,
whenever you have

01:30:52.810 --> 01:30:56.730
a product of a function, you
not only in a function in theta

01:30:56.730 --> 01:31:01.170
only, your life becomes easier
because you can separate them

01:31:01.170 --> 01:31:03.050
between the rhos.

01:31:03.050 --> 01:31:04.190
In two different products.

01:31:04.190 --> 01:31:05.950
So that's would be this theorem.

01:31:05.950 --> 01:31:11.270
And you have 3 times-- the
part that depends only on r,

01:31:11.270 --> 01:31:13.990
and the part that depends
only on theta, let's

01:31:13.990 --> 01:31:14.890
put them separate.

01:31:14.890 --> 01:31:22.170
We need theta, and
dr. And what do you

01:31:22.170 --> 01:31:24.520
integrate when you integrate?

01:31:24.520 --> 01:31:25.470
r cubed.

01:31:25.470 --> 01:31:29.100
Attention, do not do rr.

01:31:29.100 --> 01:31:30.850
From 0 to r.

01:31:30.850 --> 01:31:32.360
OK?

01:31:32.360 --> 01:31:33.651
STUDENT: And cosine theta?

01:31:33.651 --> 01:31:39.550
MAGDALENA TODA: And then you
have a 0 to 2 pi, cosine 2.

01:31:39.550 --> 01:31:42.126
now, let me give
you-- Let me tell you

01:31:42.126 --> 01:31:44.875
what it is, because when
I was young, I was naive

01:31:44.875 --> 01:31:48.100
and I always started with that.

01:31:48.100 --> 01:31:52.490
You should always start with the
part, the trig part in theta.

01:31:52.490 --> 01:31:54.030
Because that becomes 0.

01:31:54.030 --> 01:31:56.552
So no matter how
ugly this is, I've

01:31:56.552 --> 01:31:59.790
had professors who are
playing games with us,

01:31:59.790 --> 01:32:03.800
and they were giving us
some extremely ugly thing

01:32:03.800 --> 01:32:06.470
that would take you forever
for you to integrate.

01:32:06.470 --> 01:32:09.290
Or sometimes, it would have
been impossible to integrate.

01:32:09.290 --> 01:32:11.570
But then, the whole
thing would have been 0

01:32:11.570 --> 01:32:14.295
because when you
integrate cosine 2 theta,

01:32:14.295 --> 01:32:16.610
it goes to sine theta.

01:32:16.610 --> 01:32:20.600
Sine 2 theta at 2 pi and 0
are the same things, 0 minus 0

01:32:20.600 --> 01:32:21.120
equals z.

01:32:21.120 --> 01:32:23.800
So the answer is z.

01:32:23.800 --> 01:32:27.040
I cannot tell you how many
professors I've had who will

01:32:27.040 --> 01:32:28.569
play this game with us.

01:32:28.569 --> 01:32:30.360
They give us something
that discouraged us.

01:32:30.360 --> 01:32:34.010
No, it's not a piece of cake,
compared to what I have.

01:32:34.010 --> 01:32:37.450
Some integral value
will go over two lines,

01:32:37.450 --> 01:32:40.490
with a huge polynomial
or something.

01:32:40.490 --> 01:32:44.000
But in the end, the integral
was 0 for such a result. Yes?

01:32:44.000 --> 01:32:45.250
STUDENT: So I have a question.

01:32:45.250 --> 01:32:49.700
Could we take that force and
prove that it was conservative?

01:32:49.700 --> 01:32:54.940
MAGDALENA TODA: So now
that I'm questioning this,

01:32:54.940 --> 01:32:59.660
I'm not questioning
you, but I-- is

01:32:59.660 --> 01:33:05.630
the force, that is with you--
what is the original force

01:33:05.630 --> 01:33:08.360
that Alex is talking about?

01:33:08.360 --> 01:33:17.000
If I take y cubed i plus x cubed
j-- and you have to be careful.

01:33:17.000 --> 01:33:18.915
Is this conservative?

01:33:18.915 --> 01:33:22.670


01:33:22.670 --> 01:33:25.130
STUDENT: Yeah.

01:33:25.130 --> 01:33:27.740
MAGDALENA TODA: Really?

01:33:27.740 --> 01:33:30.936
Why would we pick
a conservative?

01:33:30.936 --> 01:33:33.814
STUDENT: Y squared plus
x squared over 2 is--

01:33:33.814 --> 01:33:35.605
MAGDALENA TODA: Why is
it not conservative?

01:33:35.605 --> 01:33:38.860


01:33:38.860 --> 01:33:40.560
IT doesn't pass the hole test.

01:33:40.560 --> 01:33:43.590


01:33:43.590 --> 01:33:48.280
So p sub y is not
equal to q sub x.

01:33:48.280 --> 01:33:52.090
If you primed this with respect
to y, you get that dy squared.

01:33:52.090 --> 01:33:54.640
Prime this with this respect
to x, you get 3x squared.

01:33:54.640 --> 01:33:57.430
So it's not concerned with him.

01:33:57.430 --> 01:34:00.642
And still, I'm
getting-- it's a loop,

01:34:00.642 --> 01:34:04.520
and I'm getting a 0, sort
of like I would expect it

01:34:04.520 --> 01:34:07.530
I had any dependence of that.

01:34:07.530 --> 01:34:09.250
What is the secret here?

01:34:09.250 --> 01:34:14.360
STUDENT: That is conservative,
given a condition.

01:34:14.360 --> 01:34:16.730
MAGDALENA TODA: Yes,
given a condition

01:34:16.730 --> 01:34:21.170
that your x and y are moving
on the serpent's circle.

01:34:21.170 --> 01:34:25.900
And that happens, because this
is a symmetric expression,

01:34:25.900 --> 01:34:28.420
and x and y are
moving on a circle,

01:34:28.420 --> 01:34:31.090
and one is the cosine theta
and one is sine theta.

01:34:31.090 --> 01:34:35.160
So in the end, it
simplifies out.

01:34:35.160 --> 01:34:40.280
But in general, if I would
have this kind of problem--

01:34:40.280 --> 01:34:43.950
if somebody asked me is this
conservative, the answer is no.

01:34:43.950 --> 01:34:46.125
Let me give you a
few more examples.

01:34:46.125 --> 01:34:57.620


01:34:57.620 --> 01:35:14.765
One example that maybe will look
hard to most people is here.

01:35:14.765 --> 01:35:36.545


01:35:36.545 --> 01:35:49.750
The vector value function
given by f of x, y incline,

01:35:49.750 --> 01:35:51.950
are two values.

01:35:51.950 --> 01:35:54.966
No, I mean define two
values of [INAUDIBLE].

01:35:54.966 --> 01:36:17.040


01:36:17.040 --> 01:36:19.170
A typical exam problem.

01:36:19.170 --> 01:36:22.980
And I saw it at
Texas A&M, as well.

01:36:22.980 --> 01:36:27.610
So maybe some people like this
kind of a, b, c, d problem.

01:36:27.610 --> 01:36:28.836
Is f conservative?

01:36:28.836 --> 01:36:37.220


01:36:37.220 --> 01:36:38.397
STUDENT: Yep

01:36:38.397 --> 01:36:39.855
MAGDALENA TODA:
You already did it?

01:36:39.855 --> 01:36:41.160
Good for you guys.

01:36:41.160 --> 01:36:45.335
So if I gave you one that
has three components what

01:36:45.335 --> 01:36:47.480
did you have to do?

01:36:47.480 --> 01:36:49.860
Compute the curl.

01:36:49.860 --> 01:36:52.460
You can, of course, compute
the curl also on this one

01:36:52.460 --> 01:36:55.115
and have 0 for the
third component.

01:36:55.115 --> 01:37:01.820
But the simplest thing
is to do f1 and f2.

01:37:01.820 --> 01:37:07.040
f1 prime with respect to y
equals f2 prime with respect

01:37:07.040 --> 01:37:08.090
to x.

01:37:08.090 --> 01:37:12.870
So I'm going to
make a smile here.

01:37:12.870 --> 01:37:16.310
And you realize that the authors
of such a problem, whether they

01:37:16.310 --> 01:37:21.270
are at Tech or at Texas
A&M. They do that on purpose

01:37:21.270 --> 01:37:28.390
so that you can use this
result to the next level.

01:37:28.390 --> 01:38:04.070
And they're saying compute
the happy u over the curve

01:38:04.070 --> 01:38:23.070
x cubed and y cubed equals 8 on
the path that connects points

01:38:23.070 --> 01:38:29.375
2, 1 and 1, 2 in [INAUDIBLE].

01:38:29.375 --> 01:38:39.275


01:38:39.275 --> 01:38:46.205
Does this integral depend on f?

01:38:46.205 --> 01:38:51.140


01:38:51.140 --> 01:38:52.112
State why.

01:38:52.112 --> 01:38:58.260


01:38:58.260 --> 01:39:04.540
And you see, they don't tell
you find the scalar potential.

01:39:04.540 --> 01:39:06.920
Which is bad, and
many of you will

01:39:06.920 --> 01:39:09.115
be able to see it
because you have

01:39:09.115 --> 01:39:14.000
good mathematical intuition,
and a computer process

01:39:14.000 --> 01:39:17.290
planning in the background
over all the other processes.

01:39:17.290 --> 01:39:18.900
We are very visual people.

01:39:18.900 --> 01:39:22.310
If you realize that every time
just there with each other

01:39:22.310 --> 01:39:25.720
through the classroom, there
are hundreds of distractions.

01:39:25.720 --> 01:39:28.165
There's the screen,
there is somebody

01:39:28.165 --> 01:39:31.210
who's next to you
who's sneezing,

01:39:31.210 --> 01:39:34.568
all sorts of distractions.

01:39:34.568 --> 01:39:37.860
Still, your computer
unit can still

01:39:37.860 --> 01:39:41.200
function, trying to
integrate and find the scalar

01:39:41.200 --> 01:39:42.450
potential, which is a miracle.

01:39:42.450 --> 01:39:46.250
I don't know how we managed
to do that after all.

01:39:46.250 --> 01:39:49.840
If you don't manage to do that,
what do you have to set up?

01:39:49.840 --> 01:39:54.870
You have to say, find is
there-- well, you know there is.

01:39:54.870 --> 01:39:59.900
So you're not going to question
the existence of the scalar

01:39:59.900 --> 01:40:03.680
potential You know it exists,
but you don't know what it is.

01:40:03.680 --> 01:40:09.750
What is f such that f sub
x would be 6xy plus 1,

01:40:09.750 --> 01:40:14.486
and m sub y will be 3x squared?

01:40:14.486 --> 01:40:18.315
And normally, you would
have to integrate backwards.

01:40:18.315 --> 01:40:21.350
Now, I'll give you 10 seconds.

01:40:21.350 --> 01:40:25.190
If in 10 seconds, you don't
find me a scalar potential,

01:40:25.190 --> 01:40:27.120
I'm going to make you
integrate backwards.

01:40:27.120 --> 01:40:31.360
So this is finding the scalar
potential by integration.

01:40:31.360 --> 01:40:34.455
The way you should, if
you weren't very smart.

01:40:34.455 --> 01:40:37.970
But I think you're
smart enough to smell

01:40:37.970 --> 01:40:42.410
the potential-- Very good.

01:40:42.410 --> 01:40:44.030
But what if you don't?

01:40:44.030 --> 01:40:46.050
OK I'm asking.

01:40:46.050 --> 01:40:50.475
So we had one or two
student who figured it out.

01:40:50.475 --> 01:40:51.225
What if you don't?

01:40:51.225 --> 01:40:55.240
If you don't, you can still do
perfectly fine on this problem.

01:40:55.240 --> 01:41:01.120
Let's see how we do it
without seeing or guessing.

01:41:01.120 --> 01:41:03.310
His brain was running
in the background.

01:41:03.310 --> 01:41:05.060
He came up with the answer.

01:41:05.060 --> 01:41:05.655
He's happy.

01:41:05.655 --> 01:41:09.640
He can move on to
the next level.

01:41:09.640 --> 01:41:12.270
STUDENT: Integrate both
sides with respect to r.

01:41:12.270 --> 01:41:15.970
MAGDALENA TODA: Right, and
then mix and match them.

01:41:15.970 --> 01:41:17.665
Make them in work.

01:41:17.665 --> 01:41:20.550
So try to integrate
with respect to x.

01:41:20.550 --> 01:41:25.970
6y-- or plus 1, I'm sorry guys.

01:41:25.970 --> 01:41:28.970
And once you get it,
you're going to get--

01:41:28.970 --> 01:41:32.130
STUDENT: 3x squared y plus x.

01:41:32.130 --> 01:41:34.690
MAGDALENA TODA: And
plus a c of what?

01:41:34.690 --> 01:41:37.970
And then take this fellow and
prime it with respect to y.

01:41:37.970 --> 01:41:41.480
And you're going to
get-- it's not hard.

01:41:41.480 --> 01:41:44.260
You're going to get dx
squared plus nothing,

01:41:44.260 --> 01:41:50.170
plus c from the y, and it's
good because I gave you

01:41:50.170 --> 01:41:51.330
a simple one.

01:41:51.330 --> 01:41:54.390
So sometimes you can
have something here,

01:41:54.390 --> 01:41:57.260
but in this case, it was just 0.

01:41:57.260 --> 01:42:00.400
So c is kappa as a constant.

01:42:00.400 --> 01:42:04.771
So instead of why we teach
found with a plus kappa here,

01:42:04.771 --> 01:42:07.937
and it still does it.

01:42:07.937 --> 01:42:12.810
So on such a problem,
I don't know,

01:42:12.810 --> 01:42:18.066
but I think I would give equal
weights to it, B and C. Compute

01:42:18.066 --> 01:42:20.874
the path integral
over the curve.

01:42:20.874 --> 01:42:24.200
This is horrible, as
an increasing curve.

01:42:24.200 --> 01:42:26.620
But I know that
there is a path that

01:42:26.620 --> 01:42:28.680
connects the points 2, 1 and 1.

01:42:28.680 --> 01:42:30.430
What I have to pay
attention to in my mind

01:42:30.430 --> 01:42:33.390
is that these points
actually are on the curve.

01:42:33.390 --> 01:42:36.170
And they are, because I
have 8 times 1 equals 8,

01:42:36.170 --> 01:42:37.810
1 times 8 equals 8.

01:42:37.810 --> 01:42:41.355
So while I was writing it,
I had to think a little bit

01:42:41.355 --> 01:42:42.580
on the problem.

01:42:42.580 --> 01:42:45.440
If you were to
draw-- well that's

01:42:45.440 --> 01:42:48.450
for you have to find
out when you go home.

01:42:48.450 --> 01:42:52.162
What do you think
this is going to be?

01:42:52.162 --> 01:42:55.080


01:42:55.080 --> 01:42:58.002
Actually, we have to
do it now, because it's

01:42:58.002 --> 01:43:01.770
a lot simpler than
you think it is.

01:43:01.770 --> 01:43:06.510
x and y will be positive,
I can also restrict that.

01:43:06.510 --> 01:43:09.060
It looks horrible, but
it's actually much easier

01:43:09.060 --> 01:43:09.990
than you think.

01:43:09.990 --> 01:43:15.770
So how do I compute that path
integral that makes the points?

01:43:15.770 --> 01:43:19.263
I'm going to have
fundamental there.

01:43:19.263 --> 01:43:22.756


01:43:22.756 --> 01:43:27.350
Which has f of x at q
minus f, with p, which

01:43:27.350 --> 01:43:31.280
says that little f is here.

01:43:31.280 --> 01:43:41.120
3x squared y plus
x at 2, 1 minus 3x

01:43:41.120 --> 01:43:47.300
squared y plus x at 1, 2.

01:43:47.300 --> 01:43:53.450
So all I have to do is
go ahead and-- do you

01:43:53.450 --> 01:43:56.980
see what I'm actually doing?

01:43:56.980 --> 01:43:57.790
It's funny.

01:43:57.790 --> 01:44:02.460
Which one is the origin, and
which one is the endpoint?

01:44:02.460 --> 01:44:03.830
The problem doesn't tell you.

01:44:03.830 --> 01:44:07.302
It tells you only you are
connecting the two points.

01:44:07.302 --> 01:44:10.164
But which one is the alpha,
and which one is the omega?

01:44:10.164 --> 01:44:10.955
Where do you start?

01:44:10.955 --> 01:44:12.550
You start here or
you start here?

01:44:12.550 --> 01:44:16.460


01:44:16.460 --> 01:44:17.070
OK.

01:44:17.070 --> 01:44:19.150
Sort of arbitrary.

01:44:19.150 --> 01:44:22.230
How do you handle this problem?

01:44:22.230 --> 01:44:26.150
Depending on the direction--
pick one direction you move on

01:44:26.150 --> 01:44:28.530
along the r, it's up to you.

01:44:28.530 --> 01:44:31.690
And then you get an answer, and
if you change the direction,

01:44:31.690 --> 01:44:34.250
what's going to happen
to the integral?

01:44:34.250 --> 01:44:37.970
It's just change the
sign and that's all.

01:44:37.970 --> 01:44:43.270
3 times 4, times 1, plus 2--
guys, keep an eye on my algebra

01:44:43.270 --> 01:44:48.104
please, because I
don't want to mess up.

01:44:48.104 --> 01:44:49.879
Am I right, here?

01:44:49.879 --> 01:44:50.420
STUDENT: Yes.

01:44:50.420 --> 01:44:52.300
MAGDALENA TODA: So how much?

01:44:52.300 --> 01:44:53.990
14, is it?

01:44:53.990 --> 01:44:56.315
STUDENT: It's 7.

01:44:56.315 --> 01:44:57.315
MAGDALENA TODA: Minus 7.

01:44:57.315 --> 01:45:06.491


01:45:06.491 --> 01:45:06.990
Good.

01:45:06.990 --> 01:45:07.880
Wonderful.

01:45:07.880 --> 01:45:11.670
So we know what to get,
and we know this does not

01:45:11.670 --> 01:45:13.180
depend on the fact.

01:45:13.180 --> 01:45:16.770
How much blah, blah,
blah does the instructor

01:45:16.770 --> 01:45:20.887
expect for you to get full
credit on the problem?

01:45:20.887 --> 01:45:22.220
STUDENT: Just enough to explain.

01:45:22.220 --> 01:45:23.845
MAGDALENA TODA: Just
enough to explain.

01:45:23.845 --> 01:45:29.810
About 2 lines or 1 line saying
you can say anything really.

01:45:29.810 --> 01:45:34.470
You can say this is the theorem
that either shows independence

01:45:34.470 --> 01:45:35.861
of that integral.

01:45:35.861 --> 01:45:43.310
If the force F vector value
function is conservative,

01:45:43.310 --> 01:45:46.746
then this is what
you have to write.

01:45:46.746 --> 01:45:49.252
This doesn't depend
on the path c.

01:45:49.252 --> 01:45:51.450
And you apply the
fundamental theorem

01:45:51.450 --> 01:45:54.290
of path integrals for
the scalar potential.

01:45:54.290 --> 01:45:57.760
And that scalar potential
depends on the endpoints

01:45:57.760 --> 01:45:59.660
that you're taking.

01:45:59.660 --> 01:46:02.150
And the value of
the work depends--

01:46:02.150 --> 01:46:06.290
the work depends only on the
scalar potential and the two

01:46:06.290 --> 01:46:07.510
points.

01:46:07.510 --> 01:46:08.320
That's enough.

01:46:08.320 --> 01:46:09.780
That's more than enough.

01:46:09.780 --> 01:46:13.816
What if somebody's
not good with wording?

01:46:13.816 --> 01:46:17.170
I'm not going to write
her all that explanation.

01:46:17.170 --> 01:46:21.650
I'm just going to say whatever.

01:46:21.650 --> 01:46:25.100
I'm going to give
her the theorem

01:46:25.100 --> 01:46:27.210
in mathematical compressed way.

01:46:27.210 --> 01:46:30.960
And I don't care if she
understands it or not.

01:46:30.960 --> 01:46:34.620
Even if you write this
formula with not much wording,

01:46:34.620 --> 01:46:36.870
I still give you credit.

01:46:36.870 --> 01:46:38.560
But I would prefer
that you give me

01:46:38.560 --> 01:46:41.640
some sort of-- some
sort of explanation.

01:46:41.640 --> 01:46:42.802
Yes, sir.

01:46:42.802 --> 01:46:44.093
STUDENT: You said answer was 0.

01:46:44.093 --> 01:46:45.801
Then it would have
been path independent?

01:46:45.801 --> 01:46:50.860


01:46:50.860 --> 01:46:53.630
MAGDALENA TODA: No, the
answer would not be for sure 0

01:46:53.630 --> 01:46:56.770
if it was a longer loop.

01:46:56.770 --> 01:46:58.870
If it were a longer
closed curve,

01:46:58.870 --> 01:47:03.570
that way where it
starts, it ends.

01:47:03.570 --> 01:47:07.470
Even if I take a weekly
road between the two points,

01:47:07.470 --> 01:47:09.170
I still get 7, right?

01:47:09.170 --> 01:47:11.290
That's the whole idea.

01:47:11.290 --> 01:47:12.530
Am I clear about that?

01:47:12.530 --> 01:47:14.710
Are we clear about that?

01:47:14.710 --> 01:47:21.360
Let me ask you though,
how do you find out?

01:47:21.360 --> 01:47:25.760
Because I don't know how
many of you figured out

01:47:25.760 --> 01:47:28.526
what kind of curve that is.

01:47:28.526 --> 01:47:32.600
And it looks like an enemy
to you, but there is a catch.

01:47:32.600 --> 01:47:38.520
It's an old friend of
yours and you don't see it.

01:47:38.520 --> 01:47:40.379
So what is the curve?

01:47:40.379 --> 01:47:41.265
What is the curve?

01:47:41.265 --> 01:47:46.650
And what is this arc of a
curve between 2, 1 and 1, 2?

01:47:46.650 --> 01:47:48.230
Can we find out what that is?

01:47:48.230 --> 01:47:49.350
Of course, or cubic.

01:47:49.350 --> 01:47:50.450
It's a fake cubic.

01:47:50.450 --> 01:47:53.780
It's a fake cubic--

01:47:53.780 --> 01:47:56.224
STUDENT: To function together?

01:47:56.224 --> 01:47:58.150
MAGDALENA TODA: Let's
see what this is.

01:47:58.150 --> 01:48:03.060
xy cubed minus 2 cubed equals 0.

01:48:03.060 --> 01:48:06.315
We were in fourth grade--
well, our teachers--

01:48:06.315 --> 01:48:13.640
I think our teachers teach us
when we were little that this,

01:48:13.640 --> 01:48:16.660
if you divided by a
minus- I wasn't little.

01:48:16.660 --> 01:48:19.236
I was in high school.

01:48:19.236 --> 01:48:21.132
Well, 14-year-old.

01:48:21.132 --> 01:48:22.080
STUDENT: A cubed.

01:48:22.080 --> 01:48:23.280
STUDENT: A squared.

01:48:23.280 --> 01:48:24.370
MAGDALENA TODA: A squared.

01:48:24.370 --> 01:48:26.010
STUDENT: Minus 2AB.

01:48:26.010 --> 01:48:27.080
Plus 2AB.

01:48:27.080 --> 01:48:28.960
MAGDALENA TODA: Very good.

01:48:28.960 --> 01:48:30.870
Plus AB, not 2AB.

01:48:30.870 --> 01:48:31.630
STUDENT: Oh, darn.

01:48:31.630 --> 01:48:33.750
MAGDALENA TODA: Plus B squared.

01:48:33.750 --> 01:48:34.890
Suppose you don't believe.

01:48:34.890 --> 01:48:36.730
That proves this.

01:48:36.730 --> 01:48:38.060
Let's multiply.

01:48:38.060 --> 01:48:41.660
A cubed plus A squared
B plus AB squared.

01:48:41.660 --> 01:48:44.020
I'm done with the
first multiplication.

01:48:44.020 --> 01:48:50.366
Minus BA squared minus
AB squared minus B cubed.

01:48:50.366 --> 01:48:52.290
Do they cancel out?

01:48:52.290 --> 01:48:53.733
Yes.

01:48:53.733 --> 01:48:55.180
Good.

01:48:55.180 --> 01:48:57.605
Cancel out.

01:48:57.605 --> 01:49:00.040
And cancel out.

01:49:00.040 --> 01:49:01.560
Out, poof.

01:49:01.560 --> 01:49:02.850
We've proved it, why?

01:49:02.850 --> 01:49:08.700
Because maybe some of you--
nobody gave it to proof before.

01:49:08.700 --> 01:49:11.590


01:49:11.590 --> 01:49:17.510
So as an application,
what is this?

01:49:17.510 --> 01:49:18.010
There.

01:49:18.010 --> 01:49:19.100
Who is A and who is B?

01:49:19.100 --> 01:49:23.790
A is xy, B is 2.

01:49:23.790 --> 01:49:34.440
So you have xy minus 2 times
all this fluffy guy, xy

01:49:34.440 --> 01:49:42.381
squared plus 2xy plus--

01:49:42.381 --> 01:49:44.510
STUDENT: 4.

01:49:44.510 --> 01:49:45.260
MAGDALENA TODA: 4.

01:49:45.260 --> 01:49:49.292
And I also said, because
I was sneaky, that's why.

01:49:49.292 --> 01:49:54.510
To make your life easier
or harder. xy is positive.

01:49:54.510 --> 01:49:57.783
When I said xy was positive,
what was I intending?

01:49:57.783 --> 01:50:03.320
I was intending for you to see
that this cannot be 0 ever.

01:50:03.320 --> 01:50:07.678
So the only possible
for you to have 0 here

01:50:07.678 --> 01:50:10.070
is when xy equals 2.

01:50:10.070 --> 01:50:14.200
And xy equals 2 is a
much simpler curve.

01:50:14.200 --> 01:50:17.720
And I want to know
if you realize

01:50:17.720 --> 01:50:22.160
that this will have the points
2,1 and 1, 2 staring at you.

01:50:22.160 --> 01:50:23.370
Have a nice day today.

01:50:23.370 --> 01:50:25.201
Take care.

01:50:25.201 --> 01:50:26.632
And good luck.

01:50:26.632 --> 01:50:31.880


01:50:31.880 --> 01:50:34.075
What is it?

01:50:34.075 --> 01:50:34.950
STUDENT: [INAUDIBLE].

01:50:34.950 --> 01:50:36.760
MAGDALENA TODA:
Some sort of animal.

01:50:36.760 --> 01:50:38.380
It's a curve, a linear curve.

01:50:38.380 --> 01:50:42.276
It's not a line.

01:50:42.276 --> 01:50:43.250
What is it?

01:50:43.250 --> 01:50:47.633
Talking about conics because
I was talking a little bit

01:50:47.633 --> 01:50:49.581
with Casey about conics.

01:50:49.581 --> 01:50:52.503
Is this a conic?

01:50:52.503 --> 01:50:53.477
Yeah.

01:50:53.477 --> 01:50:55.430
What is a conic?

01:50:55.430 --> 01:50:59.970
A conic is any kind of
curve that looks like this.

01:50:59.970 --> 01:51:04.530
In general form--
oh my god, ABCD.

01:51:04.530 --> 01:51:08.190
Now I got my ABC
plus f equals 0.

01:51:08.190 --> 01:51:10.030
This is a conic in plane.

01:51:10.030 --> 01:51:13.650
My conic is missing
everything else.

01:51:13.650 --> 01:51:16.020
And B is 0.

01:51:16.020 --> 01:51:18.830
And there is a way where
you-- I showed you how you

01:51:18.830 --> 01:51:21.730
know what kind of conic it is.

01:51:21.730 --> 01:51:28.130
A, A, B, B, C. A is
positive is-- no, A is 0,

01:51:28.130 --> 01:51:31.740
B is-- it should be 2 here.

01:51:31.740 --> 01:51:34.158
So you split this in half.

01:51:34.158 --> 01:51:37.110
1/2, 1/2, and 0.

01:51:37.110 --> 01:51:41.086
The determinant of this is
negative, the discriminant.

01:51:41.086 --> 01:51:43.653
That's why we call it
discriminant about the conic.

01:51:43.653 --> 01:51:44.900
So it cannot be an ellipse.

01:51:44.900 --> 01:51:46.491
So what the heck is it?

01:51:46.491 --> 01:51:47.365
STUDENT: [INAUDIBLE].

01:51:47.365 --> 01:51:48.698
MAGDALENA TODA: Well, I'm silly.

01:51:48.698 --> 01:51:50.180
I should have pulled out for y.

01:51:50.180 --> 01:51:53.040


01:51:53.040 --> 01:51:56.725
And I knew that it
goes down like 1/x.

01:51:56.725 --> 01:52:00.590
But I'm asking you, why in
the world is that a conic?

01:52:00.590 --> 01:52:01.670
Because you say, wait.

01:52:01.670 --> 01:52:03.230
Wait a minute.

01:52:03.230 --> 01:52:09.900
I know this curve since I was
five year old in kindergarten.

01:52:09.900 --> 01:52:13.150
And this is the point 2, 1.

01:52:13.150 --> 01:52:16.510


01:52:16.510 --> 01:52:17.240
It's on it.

01:52:17.240 --> 01:52:22.930
And there is a symmetric
point for your pleasure here.

01:52:22.930 --> 01:52:25.120
1, 2.

01:52:25.120 --> 01:52:26.640
And between the
two points, there

01:52:26.640 --> 01:52:31.530
is just one arc of a curve.

01:52:31.530 --> 01:52:34.120
And this is the path that
you are dragging some object

01:52:34.120 --> 01:52:35.390
with force f.

01:52:35.390 --> 01:52:37.980
You are computing
the work of a-- maybe

01:52:37.980 --> 01:52:40.990
you're computing the work of
a neutron between those two

01:52:40.990 --> 01:52:42.930
locations.

01:52:42.930 --> 01:52:43.786
It's a--

01:52:43.786 --> 01:52:44.642
STUDENT: Hyperbola?

01:52:44.642 --> 01:52:46.000
MAGDALENA TODA: Hyperbola.

01:52:46.000 --> 01:52:47.030
Why Nitish?

01:52:47.030 --> 01:52:47.768
Yes, sir.

01:52:47.768 --> 01:52:49.518
STUDENT: I was just
wondering, couldn't we

01:52:49.518 --> 01:52:51.696
have gone to xy equals 2 plane?

01:52:51.696 --> 01:52:52.820
STUDENT: Yeah, way quicker.

01:52:52.820 --> 01:52:55.259
STUDENT: x cubed, y
cubed equals 2 cubed.

01:52:55.259 --> 01:52:56.550
Then you'd just do both sides--

01:52:56.550 --> 01:52:57.250
MAGDALENA TODA:
That's what I did.

01:52:57.250 --> 01:52:57.865
STUDENT: The cubed root.

01:52:57.865 --> 01:52:59.400
MAGDALENA TODA:
Didn't I do that?

01:52:59.400 --> 01:53:02.840
No, because in
general, it's not--

01:53:02.840 --> 01:53:07.482
you cannot say if and only
if xy equals 2 in general.

01:53:07.482 --> 01:53:10.600
You have to write to
decompose the polynomial.

01:53:10.600 --> 01:53:12.450
You were lucky
this was positive.

01:53:12.450 --> 01:53:15.049
STUDENT: Well, because
we divided by x cubed.

01:53:15.049 --> 01:53:16.590
We could have just
divided everything

01:53:16.590 --> 01:53:18.610
by x cubed, and then taken
the cube root of both sides.

01:53:18.610 --> 01:53:20.401
MAGDALENA TODA: He's
saying the same thing.

01:53:20.401 --> 01:53:23.931
But in mathematics, we don't--
let me show you something.

01:53:23.931 --> 01:53:25.472
STUDENT: It would
work for this case,

01:53:25.472 --> 01:53:26.860
but not necessarily
for all cases?

01:53:26.860 --> 01:53:27.735
MAGDALENA TODA: Yeah.

01:53:27.735 --> 01:53:39.140
Let me show you some other
example where you just-- how

01:53:39.140 --> 01:53:41.130
do you solve this equation?

01:53:41.130 --> 01:53:46.020
By the way, a math
field test is coming.

01:53:46.020 --> 01:53:48.270
No, only if you're a math major.

01:53:48.270 --> 01:53:50.510
Sorry, junior or senior.

01:53:50.510 --> 01:53:53.360
In one math field test,
you don't have to take it.

01:53:53.360 --> 01:53:56.880
But some people who
go to graduate school,

01:53:56.880 --> 01:54:01.495
if they take the math field
test, that replaces the GRE,

01:54:01.495 --> 01:54:03.475
if the school agrees.

01:54:03.475 --> 01:54:06.450
So there was this questions,
how many roots does it have

01:54:06.450 --> 01:54:07.770
and what kind?

01:54:07.770 --> 01:54:11.380
Two are imaginary
and one is real.

01:54:11.380 --> 01:54:14.741
But everybody said
it only had one root.

01:54:14.741 --> 01:54:17.030
How can it have one root
if it's a cubic equation?

01:54:17.030 --> 01:54:18.580
So one root.

01:54:18.580 --> 01:54:20.596
x1 is 1.

01:54:20.596 --> 01:54:23.081
The other two are imaginary.

01:54:23.081 --> 01:54:24.330
This is the case in this also.

01:54:24.330 --> 01:54:26.450
You have some imaginary roots.

01:54:26.450 --> 01:54:31.440
So those roots
are funny, but you

01:54:31.440 --> 01:54:35.740
would have to
solve this equation

01:54:35.740 --> 01:54:42.100
because this is x minus 1
times x squared plus x plus 1.

01:54:42.100 --> 01:54:45.580
So the roots are minus
1, plus minus square root

01:54:45.580 --> 01:54:52.230
of b squared minus 4ac
over 2, which are minus 1

01:54:52.230 --> 01:54:57.290
plus minus square
root of 3i over 2.

01:54:57.290 --> 01:55:01.230
Do you guys know
how they are called?

01:55:01.230 --> 01:55:05.600
You know them because in
some countries we learn them.

01:55:05.600 --> 01:55:07.982
But do you know the notations?

01:55:07.982 --> 01:55:09.190
STUDENT: What they call them?

01:55:09.190 --> 01:55:10.065
MAGDALENA TODA: Yeah.

01:55:10.065 --> 01:55:12.190


01:55:12.190 --> 01:55:14.150
There is a Greek letter.

01:55:14.150 --> 01:55:15.872
STUDENT: Iota.

01:55:15.872 --> 01:55:17.330
MAGDALENA TODA: In
India, probably.

01:55:17.330 --> 01:55:19.060
In my country, it was omega.

01:55:19.060 --> 01:55:19.874
But I don't think--

01:55:19.874 --> 01:55:20.874
STUDENT: In India, iota.

01:55:20.874 --> 01:55:22.875


01:55:22.875 --> 01:55:25.810
MAGDALENA TODA: But we call
them omega and omega squared.

01:55:25.810 --> 01:55:28.640
Because one is the
square of the other.

01:55:28.640 --> 01:55:30.140
They are, of course,
both imaginary.

01:55:30.140 --> 01:55:35.662
And we call this the
cubic roots of unity.

01:55:35.662 --> 01:55:39.110


01:55:39.110 --> 01:55:41.970
You say Magdalena, why would
you talk about imaginary numbers

01:55:41.970 --> 01:55:43.880
when everything is real?

01:55:43.880 --> 01:55:44.500
OK.

01:55:44.500 --> 01:55:48.140
It's real for the time being
while you are still with me.

01:55:48.140 --> 01:55:50.330
The moment you're going
to say goodbye to me

01:55:50.330 --> 01:55:55.120
and you know in 3350 your
life is going to change.

01:55:55.120 --> 01:55:57.340
In that course,
they will ask you

01:55:57.340 --> 01:56:02.571
to solve this equation just like
we asked all our 3350 students.

01:56:02.571 --> 01:56:05.050
To our surprise,
the students don't

01:56:05.050 --> 01:56:06.880
know what imaginary roots are.

01:56:06.880 --> 01:56:07.940
Many, you know.

01:56:07.940 --> 01:56:10.376
You will refresh your memory.

01:56:10.376 --> 01:56:12.000
But the majority of
the students didn't

01:56:12.000 --> 01:56:15.140
know how to get to
those imaginary numbers.

01:56:15.140 --> 01:56:20.360
You're going to need to not
only use them, but also express

01:56:20.360 --> 01:56:22.557
these in terms of trigonometry.

01:56:22.557 --> 01:56:25.540


01:56:25.540 --> 01:56:31.730
So just out of curiosity, since
I am already talking to you,

01:56:31.730 --> 01:56:34.885
and since I've preparing you a
little bit for the differential

01:56:34.885 --> 01:56:38.662
equations class where you
have lots of electric circuits

01:56:38.662 --> 01:56:41.430
and applications
of trigonometry,

01:56:41.430 --> 01:56:45.780
these imaginary numbers
can also be put-- they

01:56:45.780 --> 01:56:50.726
are in general of the form
a plus ib. a plus minus ib.

01:56:50.726 --> 01:56:54.868
And we agree that in
3350 you have to do that.

01:56:54.868 --> 01:56:56.945
Out of curiosity,
is there anybody

01:56:56.945 --> 01:57:02.926
who knows the trigonometric
form of these complex numbers?

01:57:02.926 --> 01:57:06.300
STUDENT: Isn't it r e to the j--

01:57:06.300 --> 01:57:10.170


01:57:10.170 --> 01:57:14.240
MAGDALENA TODA: So you would
have exactly what he says here.

01:57:14.240 --> 01:57:18.485
This number will
be-- if it's plus.

01:57:18.485 --> 01:57:20.949
r e to the i theta.

01:57:20.949 --> 01:57:26.220
He knows a little bit
more than most students.

01:57:26.220 --> 01:57:34.280
And that is cosine
theta plus i sine theta.

01:57:34.280 --> 01:57:36.810
Can you find me the
angle theta if I

01:57:36.810 --> 01:57:42.870
want to write cosine theta
plus i sine theta or cosine

01:57:42.870 --> 01:57:46.390
theta minus i sine theta?

01:57:46.390 --> 01:57:50.210
Can you find me
the angle of theta?

01:57:50.210 --> 01:57:50.965
Is it hard?

01:57:50.965 --> 01:57:53.290
Is it easy?

01:57:53.290 --> 01:57:54.990
What in the world is it?

01:57:54.990 --> 01:57:59.810


01:57:59.810 --> 01:58:01.400
Think like this.

01:58:01.400 --> 01:58:04.440
We are done with this
example, but I'm just

01:58:04.440 --> 01:58:08.170
saying some things that
will help you in 3350.

01:58:08.170 --> 01:58:11.860
If you want cosine
theta to be minus 1/2

01:58:11.860 --> 01:58:20.553
and you want sine theta to be
root 3 over 2, which quadrant?

01:58:20.553 --> 01:58:22.900
Which quadrant are you in?

01:58:22.900 --> 01:58:24.120
STUDENT: Second.

01:58:24.120 --> 01:58:25.620
MAGDALENA TODA: The
second quadrant.

01:58:25.620 --> 01:58:27.030
Very good.

01:58:27.030 --> 01:58:28.030
All right.

01:58:28.030 --> 01:58:31.465
So think cosine.

01:58:31.465 --> 01:58:36.420
If cosine would be a half and
sine would be root 3 over 2,

01:58:36.420 --> 01:58:38.390
it would be in first quadrant.

01:58:38.390 --> 01:58:40.420
And what angle would that be?

01:58:40.420 --> 01:58:40.922
STUDENT: 60.

01:58:40.922 --> 01:58:41.755
STUDENT: That's 60--

01:58:41.755 --> 01:58:45.720
MAGDALENA TODA: 60 degrees,
which is pi over 3, right?

01:58:45.720 --> 01:58:52.900
But pi over 3 is your
friend, so he's happy.

01:58:52.900 --> 01:58:54.900
Well, he is there somewhere.

01:58:54.900 --> 01:58:58.900


01:58:58.900 --> 01:59:00.770
STUDENT: 120.

01:59:00.770 --> 01:59:05.300
MAGDALENA TODA: Where you
are here, you are at what?

01:59:05.300 --> 01:59:07.200
How much is 120-- very good.

01:59:07.200 --> 01:59:09.210
How much is 120 pi?

01:59:09.210 --> 01:59:10.769
STUDENT: 4 pi?

01:59:10.769 --> 01:59:11.560
MAGDALENA TODA: No.

01:59:11.560 --> 01:59:11.850
STUDENT: 2 pi over 3.

01:59:11.850 --> 01:59:13.155
MAGDALENA TODA: 2 pi over 3.

01:59:13.155 --> 01:59:14.460
Excellent.

01:59:14.460 --> 01:59:16.840
So 2 pi over 3.

01:59:16.840 --> 01:59:21.130
This would be if you
were to think about it--

01:59:21.130 --> 01:59:22.390
this is in radians.

01:59:22.390 --> 01:59:24.320
Let me write radians.

01:59:24.320 --> 01:59:28.470
In degrees, that's 120 degrees.

01:59:28.470 --> 01:59:38.820
So to conclude my detour
to introduction to 3350.

01:59:38.820 --> 01:59:47.104
When they will ask you to solve
this equation, x cubed minus 1,

01:59:47.104 --> 01:59:49.580
you have to tell them like that.

01:59:49.580 --> 01:59:53.142
They will ask you to put
it in trigonometric form.

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.

02:00:04.140 --> 02:00:07.330
And the other one
is x3 equals cosine

02:00:07.330 --> 02:00:15.682
of 2 pi over 3 minus
i sine of 2 pi over 3.

02:00:15.682 --> 02:00:16.840
The last thing.

02:00:16.840 --> 02:00:18.660
Because I should let you go.

02:00:18.660 --> 02:00:20.015
There was no break.

02:00:20.015 --> 02:00:23.440
I squeezed your brains
really bad today.

02:00:23.440 --> 02:00:26.370
We still have like 150 minutes.

02:00:26.370 --> 02:00:29.190
I stole from you--
no, I stole really big

02:00:29.190 --> 02:00:33.330
because we would have-- yeah,
we still have 15 minutes.

02:00:33.330 --> 02:00:37.294
But the break was 10 minutes,
so I didn't give you a break.

02:00:37.294 --> 02:00:40.030
What would this be if
you wanted to express it

02:00:40.030 --> 02:00:43.030
in terms of another angle?

02:00:43.030 --> 02:00:47.142
That's the last thing
I'm asking of you.

02:00:47.142 --> 02:00:48.603
STUDENT: [INAUDIBLE].

02:00:48.603 --> 02:00:50.551
MAGDALENA TODA: Not minus.

02:00:50.551 --> 02:00:53.170
Like cosine of an angle
plus i sine of an angle.

02:00:53.170 --> 02:00:55.942
You would need to go to
another quadrant, right?

02:00:55.942 --> 02:00:57.570
And which quadrant?

02:00:57.570 --> 02:00:58.407
STUDENT: 4.

02:00:58.407 --> 02:00:59.990
MAGDALENA TODA:
You've said it before.

02:00:59.990 --> 02:01:02.570
That would be 4 pi over 3.

02:01:02.570 --> 02:01:05.230
And 4 pi over 3.

02:01:05.230 --> 02:01:10.160
Keep in mind these things
with imaginary numbers because

02:01:10.160 --> 02:01:13.540
in 3350, they will rely on
you knowing these things.

02:01:13.540 --> 02:01:15.709


02:01:15.709 --> 02:01:17.750
STUDENT: Then you apply
Euler's formula up there.

02:01:17.750 --> 02:01:21.429


02:01:21.429 --> 02:01:22.470
MAGDALENA TODA: Oh, yeah.

02:01:22.470 --> 02:01:24.261
By the way, this is
called Euler's formula.

02:01:24.261 --> 02:01:27.535


02:01:27.535 --> 02:01:30.930
STUDENT: In middle
school, they teach you,

02:01:30.930 --> 02:01:33.840
and they tell you when
discriminant is small,

02:01:33.840 --> 02:01:35.885
there's no solutions.

02:01:35.885 --> 02:01:36.760
MAGDALENA TODA: Yeah.

02:01:36.760 --> 02:01:37.890
STUDENT: And you
go to [INAUDIBLE].

02:01:37.890 --> 02:01:40.393
MAGDALENA TODA: When the
discriminant is less than 0,

02:01:40.393 --> 02:01:42.357
there are no real solutions.

02:01:42.357 --> 02:01:44.321
But you have in pairs
imaginary solutions.

02:01:44.321 --> 02:01:46.285
They always come in pairs.

02:01:46.285 --> 02:01:50.230


02:01:50.230 --> 02:01:52.496
Do you want me to
show you probably

02:01:52.496 --> 02:01:55.240
the most important problem
in 3350 in 2 minutes,

02:01:55.240 --> 02:01:58.568
and then I'll let you go?

02:01:58.568 --> 02:02:01.003
STUDENT: Sure.

02:02:01.003 --> 02:02:07.690
MAGDALENA TODA: So somebody
gives you the equation

02:02:07.690 --> 02:02:10.390
of the harmonic oscillator.

02:02:10.390 --> 02:02:12.010
And you say, what
the heck is that?

02:02:12.010 --> 02:02:17.420
You have a little spring
and you pull that spring.

02:02:17.420 --> 02:02:19.360
And it's going to come back.

02:02:19.360 --> 02:02:21.860
You displace it, it comes back.

02:02:21.860 --> 02:02:24.360
It oscillates back and forth,
oscillates back and forth.

02:02:24.360 --> 02:02:27.950
If you were to write the
solutions of the harmonic

02:02:27.950 --> 02:02:29.800
oscillator in electric
circuits, there

02:02:29.800 --> 02:02:31.300
would be oscillating functions.

02:02:31.300 --> 02:02:36.530
So it has to do with sine and
cosine, so they must be trig.

02:02:36.530 --> 02:02:38.910
If somebody gives
you this equation,

02:02:38.910 --> 02:02:57.056
let's say ax squared-- y
double prime of x minus b.

02:02:57.056 --> 02:02:59.040
Plus.

02:02:59.040 --> 02:03:04.000
Equals to 0.

02:03:04.000 --> 02:03:09.470
And here is a y equals 0.

02:03:09.470 --> 02:03:12.500
Why would that
show up like that?

02:03:12.500 --> 02:03:19.370
Well, Hooke's law tells
you that there is a force.

02:03:19.370 --> 02:03:21.680
And there is a
force and the force

02:03:21.680 --> 02:03:23.430
is mass times acceleration.

02:03:23.430 --> 02:03:27.485
And acceleration is like this
type of second derivative

02:03:27.485 --> 02:03:30.830
of the displacement.

02:03:30.830 --> 02:03:37.580
And F and the displacement
are proportional,

02:03:37.580 --> 02:03:41.230
when you write F
equals displacement,

02:03:41.230 --> 02:03:43.980
let's call it y of x.

02:03:43.980 --> 02:03:47.535
When you have y of x, x is time.

02:03:47.535 --> 02:03:49.297
That's the displacement.

02:03:49.297 --> 02:03:50.005
That's the force.

02:03:50.005 --> 02:03:50.630
That's the k.

02:03:50.630 --> 02:03:53.060
So you have a certain
Hooke's constant.

02:03:53.060 --> 02:03:54.880
Hooke's law constant.

02:03:54.880 --> 02:03:56.870
So when you write
this, Hooke's law

02:03:56.870 --> 02:03:58.590
is going to become like that.

02:03:58.590 --> 02:04:04.930
Mass times y double prime of
x equals-- this is the force.

02:04:04.930 --> 02:04:06.725
k times y of x.

02:04:06.725 --> 02:04:09.990


02:04:09.990 --> 02:04:16.540
But it depends because
you can have plus minus.

02:04:16.540 --> 02:04:18.580
So you can have plus or minus.

02:04:18.580 --> 02:04:20.146
And these are
positive functions.

02:04:20.146 --> 02:04:27.500


02:04:27.500 --> 02:04:31.920
You have two equations
in that case.

02:04:31.920 --> 02:04:39.642
One equation is the form y
double prime plus-- give me

02:04:39.642 --> 02:04:40.470
a number.

02:04:40.470 --> 02:04:43.405
Cy equals 0.

02:04:43.405 --> 02:04:49.570
And the other one would be y
double prime minus cy equals 0.

02:04:49.570 --> 02:04:51.371
All right.

02:04:51.371 --> 02:04:54.976
Now, how hard is to
guess your solutions?

02:04:54.976 --> 02:04:59.916


02:04:59.916 --> 02:05:02.170
Can you guess the
solutions with naked eyes?

02:05:02.170 --> 02:05:04.760


02:05:04.760 --> 02:05:05.635
STUDENT: e to the x--

02:05:05.635 --> 02:05:09.080


02:05:09.080 --> 02:05:13.920
MAGDALENA TODA: So if you have--
you have e to the something.

02:05:13.920 --> 02:05:17.780
If you didn't have a c, it
would make your life easier.

02:05:17.780 --> 02:05:18.700
Forget about the c.

02:05:18.700 --> 02:05:20.910
The c will act the
same in the end.

02:05:20.910 --> 02:05:25.796
So here, what are the
possible solutions?

02:05:25.796 --> 02:05:27.012
STUDENT: e to the--

02:05:27.012 --> 02:05:29.220
MAGDALENA TODA: e to the t
is one of them. e to the x

02:05:29.220 --> 02:05:32.750
is one of them, right?

02:05:32.750 --> 02:05:35.540
So in the end, to
solve such a problem

02:05:35.540 --> 02:05:37.300
they teach you the method.

02:05:37.300 --> 02:05:39.660
You take the equation.

02:05:39.660 --> 02:05:42.080
And for that, you associate
the so-called characteristic

02:05:42.080 --> 02:05:44.480
equation.

02:05:44.480 --> 02:05:47.250
For power 2, you put r squared.

02:05:47.250 --> 02:05:51.474
Then you minus n for-- this
is how many times is it prime?

02:05:51.474 --> 02:05:52.250
No times.

02:05:52.250 --> 02:05:53.080
0 times.

02:05:53.080 --> 02:05:55.250
So you put a 1.

02:05:55.250 --> 02:05:58.010
If it's prime one times,
y prime is missing.

02:05:58.010 --> 02:06:01.770
It's prime 1 time,
you would put minus r.

02:06:01.770 --> 02:06:02.880
Equals 0.

02:06:02.880 --> 02:06:06.950
And then you look at
the two roots of that.

02:06:06.950 --> 02:06:07.820
And what are they?

02:06:07.820 --> 02:06:08.710
Plus minus 1.

02:06:08.710 --> 02:06:11.490
So r1 is 1, r2 is 2.

02:06:11.490 --> 02:06:13.370
And there is a
theorem that says--

02:06:13.370 --> 02:06:15.260
STUDENT: r2 is minus 1.

02:06:15.260 --> 02:06:17.125
MAGDALENA TODA: r2 is minus 1.

02:06:17.125 --> 02:06:19.620
Excuse me.

02:06:19.620 --> 02:06:23.970
OK, there's a theorem that
says all the solutions

02:06:23.970 --> 02:06:28.160
of this equation come as
linear combinations of e

02:06:28.160 --> 02:06:31.230
to the r1t and e to the r2t.

02:06:31.230 --> 02:06:33.106
So linear combination
means you can

02:06:33.106 --> 02:06:39.150
take any number a and any
number b, or c1 and c2, anything

02:06:39.150 --> 02:06:40.140
like that.

02:06:40.140 --> 02:06:44.650
So all the solutions of
this will look like e

02:06:44.650 --> 02:06:48.230
to the t with an a in
front plus e to the minus

02:06:48.230 --> 02:06:50.055
t with a b in front.

02:06:50.055 --> 02:06:53.300
Could you have seen
that with naked eye?

02:06:53.300 --> 02:06:54.350
Well, yeah.

02:06:54.350 --> 02:06:57.270
I mean, you are smart
and you guessed one.

02:06:57.270 --> 02:06:59.250
An you said e to
the t satisfied.

02:06:59.250 --> 02:07:02.320
Because if you put e to the
p and prime it as many times

02:07:02.320 --> 02:07:04.870
as you want, you
still get e to the t.

02:07:04.870 --> 02:07:06.050
So you get 0.

02:07:06.050 --> 02:07:09.180
But nobody thought of-- or maybe
some people thought about e

02:07:09.180 --> 02:07:10.483
to the minus t.

02:07:10.483 --> 02:07:11.066
STUDENT: Yeah.

02:07:11.066 --> 02:07:11.880
I was about to go
through that one.

02:07:11.880 --> 02:07:13.030
MAGDALENA TODA: You were about.

02:07:13.030 --> 02:07:13.800
STUDENT: That's for a selection.

02:07:13.800 --> 02:07:16.008
MAGDALENA TODA: So even if
you take e to the minus t,

02:07:16.008 --> 02:07:17.240
you get the same answer.

02:07:17.240 --> 02:07:19.790
And you get this thing.

02:07:19.790 --> 02:07:24.040
All right, all the combinations
will satisfy the same equation

02:07:24.040 --> 02:07:24.660
as well.

02:07:24.660 --> 02:07:26.630
This is a superposition
principle.

02:07:26.630 --> 02:07:28.622
With this, it was easy.

02:07:28.622 --> 02:07:31.610
But this is the so-called
harmonic oscillator equation.

02:07:31.610 --> 02:07:36.100


02:07:36.100 --> 02:07:40.290
So either you have it simplified
y double prime plus y equals 0,

02:07:40.290 --> 02:07:46.250
or you have some constant c.

02:07:46.250 --> 02:07:48.695
Well, what do you
do in that case?

02:07:48.695 --> 02:07:50.766
Let's assume you have 1.

02:07:50.766 --> 02:07:53.946
Who can guess the solutions?

02:07:53.946 --> 02:07:55.920
STUDENT: 0 and cosine--

02:07:55.920 --> 02:07:57.982
MAGDALENA TODA: No, 0
is the trivial solution

02:07:57.982 --> 02:07:59.160
and it's not going to count.

02:07:59.160 --> 02:08:03.860
You can get it from the
combination of the--

02:08:03.860 --> 02:08:05.350
STUDENT: y equals sine t.

02:08:05.350 --> 02:08:06.850
MAGDALENA TODA:
Sine t is a solution

02:08:06.850 --> 02:08:11.665
because sine t prime is cosine.

02:08:11.665 --> 02:08:13.770
When you prime it
again, it's minus sine.

02:08:13.770 --> 02:08:16.680
When you add sine and
minus sine, you get 0.

02:08:16.680 --> 02:08:19.410
So you just guessed
1 and you're right.

02:08:19.410 --> 02:08:20.670
Make a face.

02:08:20.670 --> 02:08:21.790
Do you see another one?

02:08:21.790 --> 02:08:22.770
STUDENT: Cosine t.

02:08:22.770 --> 02:08:23.728
MAGDALENA TODA: Cosine.

02:08:23.728 --> 02:08:26.155


02:08:26.155 --> 02:08:28.600
They are independent,
linear independent.

02:08:28.600 --> 02:08:31.220
And so the multitude
of solutions

02:08:31.220 --> 02:08:34.270
for that-- I taught you
a whole chapter in 3350.

02:08:34.270 --> 02:08:37.140
Now you don't have
to take it anymore--

02:08:37.140 --> 02:08:39.717
is going to be a equals sine t--

02:08:39.717 --> 02:08:41.050
STUDENT: How about e to the i t?

02:08:41.050 --> 02:08:42.300
MAGDALENA TODA: Plus b sine t.

02:08:42.300 --> 02:08:43.440
I tell you in a second.

02:08:43.440 --> 02:08:46.530
All right, we have to
do an e to the i t.

02:08:46.530 --> 02:08:47.520
OK.

02:08:47.520 --> 02:08:51.350
So you guessed that all the
solutions will be combinations

02:08:51.350 --> 02:08:56.020
like-- on the monitor when you
have cosine and sine, if you

02:08:56.020 --> 02:08:58.870
add them up-- multiply
and add them up,

02:08:58.870 --> 02:09:02.390
you get something like the
monitor thing at the hospital.

02:09:02.390 --> 02:09:04.480
So any kind of
oscillation like that

02:09:04.480 --> 02:09:07.686
is a combination of this kind.

02:09:07.686 --> 02:09:13.050
Maybe with some different
phases and amplitudes.

02:09:13.050 --> 02:09:16.830
You have cosine of 70 or
cosine of 5t or something.

02:09:16.830 --> 02:09:18.650
But let me show
you what they are

02:09:18.650 --> 02:09:24.640
going to show you [INAUDIBLE]
for the harmonic oscillator

02:09:24.640 --> 02:09:27.270
equation how the method goes.

02:09:27.270 --> 02:09:29.250
You solve for the
characteristic equation.

02:09:29.250 --> 02:09:34.600
So you have r squared
plus 1 equals 0.

02:09:34.600 --> 02:09:38.860
Now, here's where most of
the students in 3350 fail.

02:09:38.860 --> 02:09:40.468
They understand that.

02:09:40.468 --> 02:09:43.880
And some of them say, OK,
this has no solutions.

02:09:43.880 --> 02:09:46.990
Some of them even say this
has solutions plus minus 1.

02:09:46.990 --> 02:09:48.870
I mean, crazy stuff.

02:09:48.870 --> 02:09:51.490
Now, what are the
solutions of that?

02:09:51.490 --> 02:09:53.455
Because the theory
in this case says

02:09:53.455 --> 02:09:57.330
if your solutions are
imaginary, then y1

02:09:57.330 --> 02:10:00.980
would be e to the ax cosine bx.

02:10:00.980 --> 02:10:05.450
And y2 will be e
to the ax sine bx

02:10:05.450 --> 02:10:09.446
where your imaginary
solutions are a plus minus ib.

02:10:09.446 --> 02:10:13.990
It has a lot to do with
Euler's formula in a way.

02:10:13.990 --> 02:10:21.000
So if you knew the theory in
3350 and not be just very smart

02:10:21.000 --> 02:10:24.130
and get these by yourselves
by guessing them,

02:10:24.130 --> 02:10:26.520
how are you supposed
to know that?

02:10:26.520 --> 02:10:30.580
Well, r squared
equals minus 1, right?

02:10:30.580 --> 02:10:34.036
The square root of minus 1 is i.

02:10:34.036 --> 02:10:34.910
STUDENT: Or negative.

02:10:34.910 --> 02:10:36.159
MAGDALENA TODA: Or negative i.

02:10:36.159 --> 02:10:40.600
So r1 is 0 plus minus i.

02:10:40.600 --> 02:10:42.460
So who is a?

02:10:42.460 --> 02:10:44.181
a is 0.

02:10:44.181 --> 02:10:45.710
Who is b?

02:10:45.710 --> 02:10:46.720
b is 1.

02:10:46.720 --> 02:10:53.010
So the solutions are e to
the 0x equal cosine 1x and e

02:10:53.010 --> 02:10:59.040
to the 0x sine 1x, which
is cosine x, sine x.

02:10:59.040 --> 02:11:03.210
Now you know why you can
do everything formalized

02:11:03.210 --> 02:11:06.170
and you get all these
solutions from a method.

02:11:06.170 --> 02:11:10.286
This method is an
entire chapter.

02:11:10.286 --> 02:11:12.400
It's so much easier than in 350.

02:11:12.400 --> 02:11:14.670
So much easier than Calculus 3.

02:11:14.670 --> 02:11:16.420
You will say this is easy.

02:11:16.420 --> 02:11:17.720
It's a pleasure.

02:11:17.720 --> 02:11:22.790
You spend about one
fourth of the semester

02:11:22.790 --> 02:11:24.562
just on this method.

02:11:24.562 --> 02:11:26.270
So now you don't have
to take it anymore.

02:11:26.270 --> 02:11:29.080
You can learn it all by
yourself and you're going

02:11:29.080 --> 02:11:33.060
to be ready for the next thing.

02:11:33.060 --> 02:11:34.760
So I'm just giving you courage.

02:11:34.760 --> 02:11:40.350
If you do really, really well in
Calc 3, 3350 will be a breeze.

02:11:40.350 --> 02:11:42.290
You can breeze through that.

02:11:42.290 --> 02:11:46.170
You only have the probability
in stats for most engineers

02:11:46.170 --> 02:11:48.595
to take.

02:11:48.595 --> 02:11:54.210
Math is not so complicated.