0:00:00.000,0:00:01.792


0:00:01.792,0:00:05.020
MAGDALENA TODA: Welcome[br]to our review of 13.1.

0:00:05.020,0:00:07.410
How many of you didn't[br]get your exams back?

0:00:07.410,0:00:10.380
I have your exam, and yours.

0:00:10.380,0:00:11.370
And you have to wait.

0:00:11.370,0:00:12.607
I don't have it with me.

0:00:12.607,0:00:15.825
I have it in my office.

0:00:15.825,0:00:19.591
If you have questions[br]about the score,

0:00:19.591,0:00:23.028
why don't you go ahead and[br]email me right after class.

0:00:23.028,0:00:31.375
Chapter 13 is a very[br]physical chapter.

0:00:31.375,0:00:34.262
It has a lot to do with[br]mechanical engineering,

0:00:34.262,0:00:36.970
with mechanics,[br]physics, electricity.

0:00:36.970,0:00:40.670


0:00:40.670,0:00:44.980
You're going to see things,[br]weird things like work.

0:00:44.980,0:00:47.110
You've already seen work.

0:00:47.110,0:00:49.780
Do you remember the definition?

0:00:49.780,0:00:52.962
So we define the work[br]as a path integral

0:00:52.962,0:00:55.120
along the regular curve.

0:00:55.120,0:00:58.600
And by regular curve-- I'm[br]sorry if I'm repeating myself,

0:00:58.600,0:01:02.095
but this is part of the[br]deal-- R is the position

0:01:02.095,0:01:10.005
vector in R3 that is class C1.

0:01:10.005,0:01:15.243
That means differentiable and[br]derivatives are continuous.

0:01:15.243,0:01:18.624
Plus you are not[br]allowed to stop.

0:01:18.624,0:01:24.580
So no matter how drunk,[br]the bug has to keep flying,

0:01:24.580,0:01:27.800
and not even for a[br]fraction of a second is he

0:01:27.800,0:01:31.090
or she allowed to[br]have velocity 0.

0:01:31.090,0:01:34.200
At no point I want[br]to have velocity 0.

0:01:34.200,0:01:36.300
And that's the position vector.

0:01:36.300,0:01:43.620
And then you have some force[br]field acting on you-- no,

0:01:43.620,0:01:47.530
acting on the particle[br]at every moment.

0:01:47.530,0:01:54.010
So you have an F that is[br]acting at location xy.

0:01:54.010,0:01:58.140
Maybe if you are in space,[br]let's talk about the xyz,

0:01:58.140,0:02:02.130
where x is a function of[br]t, y is a functional of t,

0:02:02.130,0:02:06.520
z is a function of t,[br]which is the same as saying

0:02:06.520,0:02:11.910
that R of t, which is the given[br]position vector, is x of t

0:02:11.910,0:02:12.705
y of t.

0:02:12.705,0:02:15.580
Let me put angular bracket,[br]although I hate them,

0:02:15.580,0:02:20.140
because you like angular[br]brackets for vectors.

0:02:20.140,0:02:22.910
F is also a nice function.

0:02:22.910,0:02:24.490
How nice?

0:02:24.490,0:02:26.580
We discussed a[br]little bit last time.

0:02:26.580,0:02:28.860
It really doesn't[br]have to be continuous.

0:02:28.860,0:02:30.650
The book assumes it continues.

0:02:30.650,0:02:33.422
It has to be[br]integrable, so maybe it

0:02:33.422,0:02:35.720
could be piecewise continuous.

0:02:35.720,0:02:42.596
So I had nice enough, was[br]it continues piecewise.

0:02:42.596,0:02:46.390


0:02:46.390,0:02:51.480
And we define the work as[br]being the path integral over c.

0:02:51.480,0:02:54.730
I keep repeating, because[br]that's going to be on the final

0:02:54.730,0:02:55.790
as well.

0:02:55.790,0:02:58.780
So all the notions[br]that are important

0:02:58.780,0:03:02.650
should be given enough[br]attention in this class.

0:03:02.650,0:03:03.430
Hi.

0:03:03.430,0:03:09.270
So do you guys remember[br]how we denoted F?

0:03:09.270,0:03:15.015
F was, in general, three[br]components in our F1, F2, F3.

0:03:15.015,0:03:18.827
They are functions of[br]the position vector,

0:03:18.827,0:03:22.010
or the position xyz.

0:03:22.010,0:03:24.740
And the position is[br]a function of time.

0:03:24.740,0:03:28.380
So all in all, after[br]you do all the work,

0:03:28.380,0:03:35.230
keep in mind that when you[br]multiply with a dot product,

0:03:35.230,0:03:40.090
the integral will give you what?

0:03:40.090,0:03:42.420
A time integral?

0:03:42.420,0:03:47.210
From a time T0 to a time[br]T1, you are here at time T0

0:03:47.210,0:03:49.330
and you are here at time T1.

0:03:49.330,0:03:51.870


0:03:51.870,0:03:55.647
Maybe your curve is[br]piecewise, differentiable,

0:03:55.647,0:03:56.730
you don't know what it is.

0:03:56.730,0:04:01.890
But let's assume just a[br]very nice, smooth arc here.

0:04:01.890,0:04:03.030
Of what?

0:04:03.030,0:04:11.010
Of F1 times what is that?[br]x prime of t plus F2 times

0:04:11.010,0:04:18.160
y prime of t, plus F3[br]times z prime of t dt.

0:04:18.160,0:04:21.440
So keep in mind that[br]Mr. dR is your friend.

0:04:21.440,0:04:23.420
And he was-- what was he?

0:04:23.420,0:04:29.110
Was defined as the[br]velocity vector multiplied

0:04:29.110,0:04:32.380
by the infinitesimal element dt.

0:04:32.380,0:04:35.840
Say again, the[br]velocity vector prime

0:04:35.840,0:04:41.610
was a vector in F3 quantified[br]by the infinitesimal element dt.

0:04:41.610,0:04:46.140
So we reduce this Calc[br]3 notion path integral

0:04:46.140,0:04:53.200
to a Calc 1 notion, which was a[br]simple integral from t0 to t1.

0:04:53.200,0:04:55.160
And we've done a[br]lot of applications.

0:04:55.160,0:04:56.630
What else have we done?

0:04:56.630,0:05:00.430
We've done some[br]integral of this type

0:05:00.430,0:05:03.850
over another curve, script c.

0:05:03.850,0:05:06.030
I'm repeating mostly for Alex.

0:05:06.030,0:05:10.040
You're caught in the process.

0:05:10.040,0:05:14.310
And there are two or three[br]people who need an update.

0:05:14.310,0:05:19.380
Maybe I have another[br]function of g and ds.

0:05:19.380,0:05:25.296
And this is an integral that[br]in the end will depend on s.

0:05:25.296,0:05:27.700
But s itself depends on t.

0:05:27.700,0:05:31.410
So if I were to re-express[br]this in terms of d,

0:05:31.410,0:05:34.910
how would I re-express[br]the whole thing?

0:05:34.910,0:05:40.555
g of s, of t, whatever that[br]is, then Mr. ds was what?

0:05:40.555,0:05:42.040
STUDENT: s prime of t.

0:05:42.040,0:05:42.956
MAGDALENA TODA: Right.

0:05:42.956,0:05:47.556
So this was the-- that s[br]prime of t was the speed.

0:05:47.556,0:05:51.298
The speed of the arc of a curve.

0:05:51.298,0:05:59.310
So you have an R of[br]t and R3, a vector.

0:05:59.310,0:06:04.150
And the speed was,[br]by definition,

0:06:04.150,0:06:07.245
arc length element was[br]by definition integral

0:06:07.245,0:06:09.170
from 2t0 to t.

0:06:09.170,0:06:14.745
Of the speed R prime[br]magnitude d tau.

0:06:14.745,0:06:17.710
I'll have you put tau[br]because I'm Greek,

0:06:17.710,0:06:19.090
and it's all Greek to me.

0:06:19.090,0:06:23.360
So the tau, some people call[br]the tau the dummy variable.

0:06:23.360,0:06:24.930
I don't like to call it dumb.

0:06:24.930,0:06:27.680
It's a very smart variable.

0:06:27.680,0:06:31.160
It goes from t0 to t, so what[br]you have is a function of t.

0:06:31.160,0:06:32.900
This guy is speed.

0:06:32.900,0:06:38.880
So when you do that[br]here, ds becomes speed,

0:06:38.880,0:06:43.230
R prime of t times dt.

0:06:43.230,0:06:45.290
This was your old friend ds.

0:06:45.290,0:06:50.750
And let me put it on top[br]of this guy with speed.

0:06:50.750,0:06:53.633
Because he was so[br]important to you,

0:06:53.633,0:06:56.339
you cannot forget about him.

0:06:56.339,0:07:03.846
So that was review of--[br]reviewing of 13.1 and 13.2

0:07:03.846,0:07:10.235
There were some things in[br]13.3 that I pointed out

0:07:10.235,0:07:12.400
to you are important.

0:07:12.400,0:07:16.990
13.3 was independence of path.

0:07:16.990,0:07:18.920
Everybody write, magic-- no.

0:07:18.920,0:07:20.800
Magic section.

0:07:20.800,0:07:22.470
No, have to be serious.

0:07:22.470,0:07:31.950
So that's independence of path[br]of certain type of integrals,

0:07:31.950,0:07:35.080
of some integrals.

0:07:35.080,0:07:37.730
And an integral like[br]that, a path integral

0:07:37.730,0:07:41.060
is independent of path.

0:07:41.060,0:07:44.530
When would such an animal--[br]look at this pink animal,

0:07:44.530,0:07:47.030
inside-- when would[br]this not depend

0:07:47.030,0:07:51.234
on the path you are taking[br]between two given points?

0:07:51.234,0:07:55.130
So I can move on another[br]arc and another arc

0:07:55.130,0:07:59.320
and another regular arc, and[br]all sorts of regular arcs.

0:07:59.320,0:08:02.398
It doesn't matter[br]which path I'm taking--

0:08:02.398,0:08:04.120
STUDENT: If that[br]force is conservative.

0:08:04.120,0:08:05.995
MAGDALENA TODA: If the[br]force is conservative.

0:08:05.995,0:08:07.165
Excellent, Alex.

0:08:07.165,0:08:12.770
And what did it mean for a[br]force to be conservative?

0:08:12.770,0:08:15.880
How many of you[br]know-- it's no shame.

0:08:15.880,0:08:17.350
Just raise hands.

0:08:17.350,0:08:19.750
If you forgot what it is,[br]don't raise your hand.

0:08:19.750,0:08:23.350
But if you remember what it[br]means for a force F force

0:08:23.350,0:08:27.340
field-- may the force be[br]with you-- be conservative,

0:08:27.340,0:08:30.157
then what do you do?

0:08:30.157,0:08:33.150
Say F is conservative[br]by definition.

0:08:33.150,0:08:38.751


0:08:38.751,0:08:50.084
When, if and only, F[br]there is a so-called--

0:08:50.084,0:08:50.750
STUDENT: Scalar.

0:08:50.750,0:08:51.916
MAGDALENA TODA: --potential.

0:08:51.916,0:08:53.527
Scalar potential, thank you.

0:08:53.527,0:08:54.110
I'll fix that.

0:08:54.110,0:09:01.770
A scalar potential function f.

0:09:01.770,0:09:07.130


0:09:07.130,0:09:09.850
Instead of there is, I[br]didn't want to put this.

0:09:09.850,0:09:11.650
Because a few[br]people told me they

0:09:11.650,0:09:14.570
got scared about[br]the symbolistics.

0:09:14.570,0:09:17.146
This means "there exists."

0:09:17.146,0:09:22.380
OK, smooth potential[br]such that-- at least

0:09:22.380,0:09:26.800
is differential [INAUDIBLE][br]1 such that the nabla of f--

0:09:26.800,0:09:28.240
what the heck is that?

0:09:28.240,0:09:32.180
The gradient of this little[br]f will be the given F.

0:09:32.180,0:09:38.500
And we saw all sorts of wizards[br]here, like, Harry Potter,

0:09:38.500,0:09:42.930
[INAUDIBLE] well,[br]there are many,

0:09:42.930,0:09:47.670
Alex, Erin, many,[br]many-- Matthew.

0:09:47.670,0:09:49.540
So what did they do?

0:09:49.540,0:09:50.961
They guessed the[br]scalar potential.

0:09:50.961,0:09:53.370
I had to stop because[br]there are 10 of them.

0:09:53.370,0:09:55.940
It's a whole school[br]of Harry Potter.

0:09:55.940,0:09:58.760
How do they find the little f?

0:09:58.760,0:10:00.790
Through witchcraft.

0:10:00.790,0:10:01.860
No.

0:10:01.860,0:10:02.735
Normally you should--

0:10:02.735,0:10:04.818
STUDENT: I've actually[br]done it through witchcraft.

0:10:04.818,0:10:05.506
Tell you that?

0:10:05.506,0:10:06.505
MAGDALENA TODA: You did.

0:10:06.505,0:10:08.770
I think you can do it[br]through witchcraft.

0:10:08.770,0:10:15.080
But practically everybody[br]has the ability to guess.

0:10:15.080,0:10:17.512
Why do we have the ability[br]to guess and check?

0:10:17.512,0:10:21.210
Because our brain does[br]the integration for you.

0:10:21.210,0:10:24.160
Whether you tell your[br]brain to stop or not,

0:10:24.160,0:10:27.250
when your brain, for example,[br]sees is kind of function--

0:10:27.250,0:10:30.340
and now I'm gonna[br]test your magic skills

0:10:30.340,0:10:32.040
on a little harder one.

0:10:32.040,0:10:36.040
I didn't want to do an[br]R2 value vector function.

0:10:36.040,0:10:38.070
Let me go to R3.

0:10:38.070,0:10:42.070
But I know that you have[br]your witchcraft handy.

0:10:42.070,0:10:47.650
So let's say somebody[br]gave you a force field

0:10:47.650,0:10:55.620
that is yz i plus xzj plus xyk.

0:10:55.620,0:10:58.900
And you're going to jump and[br]say this is a piece of cake.

0:10:58.900,0:11:03.680
I can see the scalar potential[br]and just wave my magic wand,

0:11:03.680,0:11:06.950
and I get it.

0:11:06.950,0:11:08.099
STUDENT: [INAUDIBLE]

0:11:08.099,0:11:09.390
MAGDALENA TODA: Oh my god, yes.

0:11:09.390,0:11:11.000
Guys, you saw it fast.

0:11:11.000,0:11:12.780
OK, I should be proud of you.

0:11:12.780,0:11:14.300
And I am proud of you.

0:11:14.300,0:11:18.610
I've had made classes[br]where the students couldn't

0:11:18.610,0:11:21.930
see any of the scalar[br]potentials that I gave them,

0:11:21.930,0:11:23.930
that I asked them to guess.

0:11:23.930,0:11:25.380
How did you deal with it?

0:11:25.380,0:11:27.500
You integrate this[br]with respect to F?

0:11:27.500,0:11:29.927
In the back of[br]your mind you did.

0:11:29.927,0:11:31.510
And then you guessed[br]one, and then you

0:11:31.510,0:11:34.120
said, OK so should be xyz.

0:11:34.120,0:11:36.790
Does it verify my[br]other two conditions?

0:11:36.790,0:11:38.460
And you say, oh yeah, it does.

0:11:38.460,0:11:42.550
Because of I prime with respect[br]to y, I have exactly xz.

0:11:42.550,0:11:46.510
If I prime with respect to c I[br]have exactly xy, so I got it.

0:11:46.510,0:11:49.560
And even if somebody[br]said xyz plus 7,

0:11:49.560,0:11:51.160
they would still be right.

0:11:51.160,0:11:56.060
In the end you can have[br]any xyz plus a constant.

0:11:56.060,0:11:58.260
In general it's not[br]so easy to guess.

0:11:58.260,0:12:02.020
But there are lots of examples[br]of conservative forces where

0:12:02.020,0:12:06.590
you simply cannot see the scalar[br]potential or cannot deduce it

0:12:06.590,0:12:09.850
like in a few seconds.

0:12:09.850,0:12:12.970
Expect something easy,[br]though, like that,

0:12:12.970,0:12:14.945
something that you can see.

0:12:14.945,0:12:15.820
Let's see an example.

0:12:15.820,0:12:19.260
Assume this is your force field[br]acting on a particle that's

0:12:19.260,0:12:23.510
moving on a curving space.

0:12:23.510,0:12:29.130
And it's stubborn and it[br]decides to move on a helix,

0:12:29.130,0:12:32.560
because it's a-- I don't[br]know what kind of particle

0:12:32.560,0:12:34.696
would move on a[br]helix, but suppose

0:12:34.696,0:12:39.490
a lot of particles, just a[br]little train or a drunken bug

0:12:39.490,0:12:40.690
or something.

0:12:40.690,0:12:45.070
And you were moving[br]on another helix.

0:12:45.070,0:12:52.170
Now suppose that helix will[br]be R of t equals cosine t

0:12:52.170,0:13:01.110
sine t and t where you[br]have t as 0 to start with.

0:13:01.110,0:13:02.352
What do I have at 0?

0:13:02.352,0:13:05.640
The point 1, 0, 0.

0:13:05.640,0:13:09.120
That's the point,[br]let's call it A.

0:13:09.120,0:13:12.870
And let's call this B. I[br]don't know what I want to do.

0:13:12.870,0:13:16.580
I'll just do a[br]complete rotation,

0:13:16.580,0:13:18.510
just to make my life easier.

0:13:18.510,0:13:23.120
And this is B. And that[br]will be A at t equals 0

0:13:23.120,0:13:25.160
and B equals 2 pi.

0:13:25.160,0:13:32.840


0:13:32.840,0:13:35.352
So what will this be at B?

0:13:35.352,0:13:37.120
STUDENT: 1, 0, 2 pi.

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

0:13:40.430,0:13:44.320
So you perform a complete[br]rotation and come back.

0:13:44.320,0:13:49.750
Now, if your force is[br]conservative, you are lucky.

0:13:49.750,0:13:53.100
Because you know the theorem[br]that says in that case

0:13:53.100,0:13:57.820
the work integral will[br]be independent of path.

0:13:57.820,0:14:03.910
And due to the theorem in-- what[br]section was that again-- 13.3,

0:14:03.910,0:14:06.890
independence of path,[br]you know that this

0:14:06.890,0:14:11.720
is going to be-- let me rewrite[br]it one more time with gradient

0:14:11.720,0:14:16.150
of f instead of big F.

0:14:16.150,0:14:19.615
And this will become what,[br]f of the q-- not the q.

0:14:19.615,0:14:24.065
In the book it's f of q minus[br]f of q. f of B minus f of A,

0:14:24.065,0:14:24.565
right?

0:14:24.565,0:14:28.050


0:14:28.050,0:14:29.030
What does this mean?

0:14:29.030,0:14:33.410
You have to measure[br]the-- to evaluate

0:14:33.410,0:14:39.440
the coordinates of[br]this function xyz

0:14:39.440,0:14:50.080
where t equals 2 pi minus[br]xyz where t equals what?

0:14:50.080,0:14:52.404
0.

0:14:52.404,0:14:56.830
And now I have to be[br]careful, because I

0:14:56.830,0:14:58.200
have to evaluate them.

0:14:58.200,0:15:06.632
So when t is 0 I have x[br]is 1, y is 0, and t is 0.

0:15:06.632,0:15:08.100
In the end it doesn't matter.

0:15:08.100,0:15:12.530
I can get 0-- I[br]can get 0 for this

0:15:12.530,0:15:14.700
and get 0 for that as well.

0:15:14.700,0:15:20.600
So when this is 2 pi I get[br]x equals 1, y equals 0,

0:15:20.600,0:15:23.210
and t equals 2 pi.

0:15:23.210,0:15:26.650
So in the end, both products[br]are 0 and I got a 0.

0:15:26.650,0:15:31.890
So although the [INAUDIBLE][br]works very hard-- I mean,

0:15:31.890,0:15:36.450
works hard in our perception[br]to get from a point

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

0:15:38.730,0:15:39.380
Why?

0:15:39.380,0:15:42.470
Because it's a vector[br]value thing inside.

0:15:42.470,0:15:46.560
And there are some[br]annihilations going on.

0:15:46.560,0:15:50.473
So that reminds me[br]of another example.

0:15:50.473,0:15:52.828
So we are done[br]with this example.

0:15:52.828,0:15:55.654
Let's go back to our washer.

0:15:55.654,0:15:57.560
I was just doing[br]laundry last night

0:15:57.560,0:16:01.360
and I was thinking of[br]the washer example.

0:16:01.360,0:16:04.960
And I thought of a small[br]variation of the washer

0:16:04.960,0:16:09.430
example, just assuming that[br]I would give you a pop quiz.

0:16:09.430,0:16:12.120
And I'm not giving you[br]a pop quiz right now.

0:16:12.120,0:16:14.940
But if I gave you[br]a pop quiz now,

0:16:14.940,0:16:20.850
I would ask you example[br]two, the washer.

0:16:20.850,0:16:24.412


0:16:24.412,0:16:27.770
It is performing[br]a circular motion,

0:16:27.770,0:16:30.000
and I want to know[br]the work performed

0:16:30.000,0:16:36.320
by the centrifugal force[br]between various points.

0:16:36.320,0:16:48.180
So have the circular motion,[br]the centrifugal force.

0:16:48.180,0:16:50.892
This is the[br]centrifugal, I'm sorry.

0:16:50.892,0:16:54.065
I'll take the centrifugal force.

0:16:54.065,0:16:56.480
And that was last[br]time we discussed

0:16:56.480,0:17:04.960
that, that was extending[br]the radius of the initial--

0:17:04.960,0:17:07.220
the vector value position.

0:17:07.220,0:17:11.560
So you have that in[br]every point, xi plus yj.

0:17:11.560,0:17:14.670
And you want F to[br]be able xi plus yj.

0:17:14.670,0:17:20.368
But it points outside[br]from the point

0:17:20.368,0:17:22.965
on the circular trajectory.

0:17:22.965,0:17:26.040


0:17:26.040,0:17:31.350
And I asked you, find[br]out what you performed

0:17:31.350,0:17:38.686
by F in one full rotation.

0:17:38.686,0:17:43.760


0:17:43.760,0:17:48.975
We gave the equation of motion,[br]being cosine t y sine t,

0:17:48.975,0:17:51.560
if you remember from last time.

0:17:51.560,0:17:57.390
And then W2, let's[br]say, is performed by F

0:17:57.390,0:18:01.910
from t equals 0 to t equals pi.

0:18:01.910,0:18:03.496
I want that as well.

0:18:03.496,0:18:12.570
And W2 performed by F from[br]t-- that makes t0 to t

0:18:12.570,0:18:20.030
equals pi-- t equals 0[br]to t equals pi over 4.

0:18:20.030,0:18:21.830
These are all very[br]easy questions,

0:18:21.830,0:18:24.970
and you should be able to[br]answer them in no time.

0:18:24.970,0:18:28.430
Now, let me tell you something.

0:18:28.430,0:18:29.710
We are in plane, not in space.

0:18:29.710,0:18:30.790
But it doesn't matter.

0:18:30.790,0:18:35.440
It's like the third quadrant[br]would be 0, piece of cake.

0:18:35.440,0:18:39.997
Your eye should be so[br]well-trained that when

0:18:39.997,0:18:41.580
you look at the force[br]field like that,

0:18:41.580,0:18:44.159
and people talk about what[br]you should ask yourself,

0:18:44.159,0:18:44.950
is it conservative?

0:18:44.950,0:18:48.390


0:18:48.390,0:18:51.010
And it is conservative.

0:18:51.010,0:18:53.830
And that means little f is what?

0:18:53.830,0:18:56.900


0:18:56.900,0:18:59.700
Nitish said that yesterday.

0:18:59.700,0:19:00.860
Why did you go there?

0:19:00.860,0:19:02.630
You want to sleep today?

0:19:02.630,0:19:05.510
I'm just teasing you.

0:19:05.510,0:19:08.470
I got so comfortable with[br]you sitting in the front row.

0:19:08.470,0:19:10.070
STUDENT: I took his spot.

0:19:10.070,0:19:12.071
STUDENT: She doesn't like[br]you sitting over here.

0:19:12.071,0:19:13.070
MAGDALENA TODA: It's OK.

0:19:13.070,0:19:13.920
It's fine.

0:19:13.920,0:19:16.710
I still give him credit[br]for what he said last time.

0:19:16.710,0:19:19.570
So do you guys remember,[br]he gave us this answer?

0:19:19.570,0:19:23.070
x squared plus y squared over[br]2, and he found the scalar

0:19:23.070,0:19:26.980
potential through witchcraft[br]in about a second and a half?

0:19:26.980,0:19:28.190
OK.

0:19:28.190,0:19:31.430
We are gonna conclude something.

0:19:31.430,0:19:36.420
Do you remember that I found the[br]answer by find the explanation?

0:19:36.420,0:19:39.160
I got W to be 0.

0:19:39.160,0:19:45.340
But if I were to find another[br]explanation why the work would

0:19:45.340,0:19:50.180
be 0 in this case, it[br]would have been 0 anyway

0:19:50.180,0:19:53.220
for any force field.

0:19:53.220,0:19:58.117
Even if I took the F[br]to be something else.

0:19:58.117,0:20:03.320
Assume that F would be[br]G. Really wild, crazy,

0:20:03.320,0:20:06.970
but still differentiable[br]vector value function.

0:20:06.970,0:20:08.700
G differential.

0:20:08.700,0:20:15.050
Would the work that we[br]want be the same for G?

0:20:15.050,0:20:15.727
STUDENT: Yeah.

0:20:15.727,0:20:16.560
MAGDALENA TODA: Why?

0:20:16.560,0:20:18.310
STUDENT: Because of[br]displacement scenario.

0:20:18.310,0:20:20.680
MAGDALENA TODA: Since[br]it's conservative,

0:20:20.680,0:20:23.150
you have a closed loop.

0:20:23.150,0:20:25.800
So the closed loop[br]will say, thick F

0:20:25.800,0:20:30.340
at that terminal point minus[br]thick F at the initial point.

0:20:30.340,0:20:33.620
But if a loop motion,[br]your terminal point

0:20:33.620,0:20:35.290
is the initial point.

0:20:35.290,0:20:36.040
Duh.

0:20:36.040,0:20:40.902
So you have the[br]same point, the P

0:20:40.902,0:20:44.490
equals qe if it's[br]a closed curve.

0:20:44.490,0:20:48.294
So for a closed curve--[br]we also call that a loop.

0:20:48.294,0:20:50.210
With a basketball, it[br]would have been too easy

0:20:50.210,0:20:54.920
and you would have gotten a[br]dollar for free like that.

0:20:54.920,0:20:57.930
So any closed curve[br]is called a loop.

0:20:57.930,0:21:01.610
If your force field is[br]conservative-- attention,

0:21:01.610,0:21:05.390
you might have examples[br]like that in the exams--

0:21:05.390,0:21:08.990
then it doesn't matter[br]who little f is,

0:21:08.990,0:21:12.510
if p equals q you get 0 anyway.

0:21:12.510,0:21:15.750
But the reason why I[br]said you would get 0

0:21:15.750,0:21:20.950
on the example of last time[br]was a slightly different one.

0:21:20.950,0:21:24.367
What does the engineer[br]say to himself?

0:21:24.367,0:21:25.700
STUDENT: Force is perpendicular.

0:21:25.700,0:21:26.360
MAGDALENA TODA: Yeah.

0:21:26.360,0:21:27.000
Very good.

0:21:27.000,0:21:28.990
Whenever the force[br]is perpendicular

0:21:28.990,0:21:33.180
to the trajectory, I'm going[br]to get 0 for the force.

0:21:33.180,0:21:36.710
Because at every[br]moment the dot product

0:21:36.710,0:21:41.160
between the force and the[br]displacement direction,

0:21:41.160,0:21:45.190
which would be like dR, the[br]tangent to the displacement,

0:21:45.190,0:21:46.750
would be [INAUDIBLE].

0:21:46.750,0:21:50.190
And cosine of [INAUDIBLE] is 0.

0:21:50.190,0:21:50.770
Duh.

0:21:50.770,0:21:52.810
So that's another reason.

0:21:52.810,0:21:59.822
Reason of last time[br]was F perpendicular

0:21:59.822,0:22:05.330
to the R prime[br]direction, R prime

0:22:05.330,0:22:11.030
being the velocity-- look,[br]when I'm moving in a circle,

0:22:11.030,0:22:13.030
this is the force.

0:22:13.030,0:22:14.520
And I'm moving.

0:22:14.520,0:22:17.660
This is my velocity, is[br]the tangent to the circle.

0:22:17.660,0:22:22.460
And the velocity and the normal[br]are always perpendicular,

0:22:22.460,0:22:23.110
at every point.

0:22:23.110,0:22:24.000
That's why I have 0.

0:22:24.000,0:22:26.600


0:22:26.600,0:22:32.100
So note that even if I[br]didn't take a close look,

0:22:32.100,0:22:36.020
why would the answer[br]be from 0 to pi?

0:22:36.020,0:22:38.670
Still?

0:22:38.670,0:22:42.685
0 because of that.

0:22:42.685,0:22:43.640
0.

0:22:43.640,0:22:47.000
How about from 0 to pi over 4?

0:22:47.000,0:22:49.890
Still 0.

0:22:49.890,0:22:52.170
And of course if somebody[br]would not believe them,

0:22:52.170,0:22:54.860
if somebody would not[br]understand the theory,

0:22:54.860,0:22:57.605
they would do the work and[br]they would get to the answer

0:22:57.605,0:23:01.046
and say, oh my[br]god, yeah, I got 0.

0:23:01.046,0:23:02.660
All right?

0:23:02.660,0:23:03.670
OK.

0:23:03.670,0:23:09.796
Now, what if somebody--[br]and I want to spray this.

0:23:09.796,0:23:11.730
Can I go ahead and[br]erase the board

0:23:11.730,0:23:15.160
and move onto example[br]three or whatever?

0:23:15.160,0:23:16.420
Yes?

0:23:16.420,0:23:17.210
OK.

0:23:17.210,0:23:19.081
All right.

0:23:19.081,0:23:21.520
STUDENT: Could you say[br]non-conservative force?

0:23:21.520,0:23:23.570
MAGDALENA TODA: Yeah,[br]that's what I-- exactly.

0:23:23.570,0:23:26.371
You are a mind reader.

0:23:26.371,0:23:31.838
You are gonna guess my mind.

0:23:31.838,0:23:44.263


0:23:44.263,0:23:46.800
And I'm going to[br]pick a nasty one.

0:23:46.800,0:23:49.845
And since I'm doing[br]review anyway,

0:23:49.845,0:23:50.970
you may have one like that.

0:23:50.970,0:23:55.250
And you may have both one that[br]involves a conservative force

0:23:55.250,0:24:00.110
field and one that does not[br]involve a conservative force

0:24:00.110,0:24:00.690
field.

0:24:00.690,0:24:07.450
And we can ask you, find us the[br]work belong to different path.

0:24:07.450,0:24:11.690
And I've done this[br]type of example before.

0:24:11.690,0:24:15.450
Let's take F of[br]x and y in plane.

0:24:15.450,0:24:29.050
In our two I take xyi[br]plus x squared y of j.

0:24:29.050,0:24:34.650
And the problem would[br]involve my favorite picture,

0:24:34.650,0:24:39.000
y equals x squared and y[br]equals x, our two paths.

0:24:39.000,0:24:40.520
One is the straight path.

0:24:40.520,0:24:43.700
One is the [INAUDIBLE] path.

0:24:43.700,0:24:46.280
They go from 0,[br]0 to 1, 1 anyway.

0:24:46.280,0:24:49.220


0:24:49.220,0:24:58.250
And I'm asking you to[br]find W1 along path one

0:24:58.250,0:25:01.250
and W2 along path two.

0:25:01.250,0:25:06.280
And of course,[br]example three, if this

0:25:06.280,0:25:10.270
were conservative[br]you would say, oh,

0:25:10.270,0:25:11.895
it doesn't matter[br]what path I'm taking,

0:25:11.895,0:25:14.930
I'm still getting[br]the same answer.

0:25:14.930,0:25:17.499
But is this conservative?

0:25:17.499,0:25:18.425
STUDENT: No.

0:25:18.425,0:25:20.177
Because you said it wasn't.

0:25:20.177,0:25:21.260
MAGDALENA TODA: Very good.

0:25:21.260,0:25:22.870
So how do you know?

0:25:22.870,0:25:26.220
That's one test[br]when you are in two.

0:25:26.220,0:25:30.708
There is the magic test that[br]says-- let's say this is M,

0:25:30.708,0:25:36.700
and let's say this is N. You[br]would have to check if M sub--

0:25:36.700,0:25:37.271
STUDENT: y.

0:25:37.271,0:25:38.020
MAGDALENA TODA: y.

0:25:38.020,0:25:38.519
Very good.

0:25:38.519,0:25:39.640
I'm proud of you.

0:25:39.640,0:25:42.260
You're ready for[br]3350, by the way.

0:25:42.260,0:25:44.030
Is equal to N sub x.

0:25:44.030,0:25:46.600
M sub y is x.

0:25:46.600,0:25:49.180
N sub x is 2xy.

0:25:49.180,0:25:51.530
They are not equal.

0:25:51.530,0:25:55.240
So that's me crying that I have[br]to do the work twice and get--

0:25:55.240,0:25:57.634
probably I'll get two[br]different examples.

0:25:57.634,0:26:00.970


0:26:00.970,0:26:03.510
If you read the book--[br]I'm afraid to ask

0:26:03.510,0:26:09.320
how many of you opened the[br]book at section 13.2, 13.3.

0:26:09.320,0:26:12.000
But did you read[br]it, any of them?

0:26:12.000,0:26:13.000
STUDENT: Nitish read it.

0:26:13.000,0:26:15.400
MAGDALENA TODA: Oh, good.

0:26:15.400,0:26:20.140
There is another criteria for[br]a force to be conservative.

0:26:20.140,0:26:22.120
If you are, it's piece of cake.

0:26:22.120,0:26:23.296
You do that, right?

0:26:23.296,0:26:24.337
MAGDALENA TODA: Yes, sir?

0:26:24.337,0:26:25.620
STUDENT: Curl has frequency 0.

0:26:25.620,0:26:27.494
MAGDALENA TODA: The curl[br]criteria, excellent.

0:26:27.494,0:26:29.000
The curl has to be zero.

0:26:29.000,0:26:38.977
So if F in R 3 is[br]conservative, then you'll

0:26:38.977,0:26:40.060
get different order curve.

0:26:40.060,0:26:42.052
Curl F is 0.

0:26:42.052,0:26:44.640
Now let's check what[br]the heck was curl.

0:26:44.640,0:26:47.610
You see, mathematics[br]is not a bunch

0:26:47.610,0:26:51.910
of these joint discussions[br]like other sciences.

0:26:51.910,0:26:55.340
In mathematics, if you don't[br]know a section or you skipped

0:26:55.340,0:26:58.450
it, you are sick, you[br]have a date that day,

0:26:58.450,0:27:02.700
you didn't study, then it's[br]all over because you cannot

0:27:02.700,0:27:06.650
understand how to work out the[br]problems and materials if you

0:27:06.650,0:27:08.000
skip the section.

0:27:08.000,0:27:12.020
Curl was the one[br]where we learned

0:27:12.020,0:27:16.150
that we used the determinant.

0:27:16.150,0:27:17.300
That's the easiest story.

0:27:17.300,0:27:20.740
It came with a t-shirt,[br]but that t-shirt really

0:27:20.740,0:27:25.800
doesn't help because[br]it's easier to,

0:27:25.800,0:27:28.270
instead of memorizing[br]the formula,

0:27:28.270,0:27:31.440
you set out the determinant.

0:27:31.440,0:27:33.865
So you have the operator[br]derivative with respect

0:27:33.865,0:27:40.760
to x, y z followed by what?

0:27:40.760,0:27:43.100
F1, F2, F3.

0:27:43.100,0:27:46.945
Now in your case, I'm[br]asking you if you did it

0:27:46.945,0:27:51.680
for this F, what is[br]the third component?

0:27:51.680,0:27:52.510
STUDENT: The 0.

0:27:52.510,0:27:54.800
MAGDALENA TODA: The[br]0, so this guy is 0.

0:27:54.800,0:27:59.740
This guy is X squared[br]Y, and this guy is xy.

0:27:59.740,0:28:01.690
And it should be[br]a piece of cake,

0:28:01.690,0:28:03.990
but I want to do[br]it one more time.

0:28:03.990,0:28:08.520
I times the minor derivative[br]of 0 with respect to y

0:28:08.520,0:28:11.540
is 0 minus derivative[br]of x squared

0:28:11.540,0:28:15.410
y respect to 0, all[br]right, plus j minus

0:28:15.410,0:28:17.860
j because I'm alternating.

0:28:17.860,0:28:19.965
You've known enough[br]in your algebra

0:28:19.965,0:28:22.840
to know why I'm expanding[br]along the first row.

0:28:22.840,0:28:25.700
I have a minus, all[br]right, then the x

0:28:25.700,0:28:33.600
of 0, 0 derivative of xy[br]respect to the 0 plus k times

0:28:33.600,0:28:37.550
the minor corresponding[br]to k derivative 2xy.

0:28:37.550,0:28:45.050


0:28:45.050,0:28:46.325
Oh, and the derivative--

0:28:46.325,0:28:49.062


0:28:49.062,0:28:52.504
STUDENT: Yeah, this[br]is the n equals 0.

0:28:52.504,0:28:54.170
MAGDALENA TODA: Oh,[br]yeah, that's the one

0:28:54.170,0:28:58.700
where it's not a because[br]that's not conservative.

0:28:58.700,0:28:59.830
So what do you get.

0:28:59.830,0:29:04.950
You get 2xy minus x, right?

0:29:04.950,0:29:07.100
But I don't know how to[br]write it better than that.

0:29:07.100,0:29:08.100
Well, it doesn't matter.

0:29:08.100,0:29:09.510
Leave it like that.

0:29:09.510,0:29:18.080
So this would be 0 if it only[br]if x would be 0, but otherwise y

0:29:18.080,0:29:18.860
was 1/2.

0:29:18.860,0:29:22.850
But in general, it[br]is not a 0, good.

0:29:22.850,0:29:30.470
So F is not[br]conservative, and then we

0:29:30.470,0:29:32.400
can say goodbye[br]to the whole thing

0:29:32.400,0:29:39.680
here and move on to[br]computing the works.

0:29:39.680,0:29:42.020
What is the only[br]way we can do that?

0:29:42.020,0:29:46.491
By parameterizing[br]the first path,

0:29:46.491,0:29:48.926
but I didn't say which[br]one is the first path.

0:29:48.926,0:29:52.578
This is the first path, so[br]x of t equals t, and y of t

0:29:52.578,0:29:55.257
equals t is your[br]parameterization.

0:29:55.257,0:30:03.910
The simplest one, and then[br]W1 will be integral of-- I'm

0:30:03.910,0:30:10.135
too lazy to write down x of t,[br]y of t, but this is what it is.

0:30:10.135,0:30:14.540
Times x prime of[br]t plus x squared

0:30:14.540,0:30:21.550
y times y prime of t dt where--

0:30:21.550,0:30:31.372
STUDENT: Isn't that just[br]xy dx y-- never mind.

0:30:31.372,0:30:33.565
MAGDALENA TODA: This is F2.

0:30:33.565,0:30:36.560
And this is x prime,[br]and this is y prime

0:30:36.560,0:30:40.394
because this thing is[br]just-- I have no idea.

0:30:40.394,0:30:41.852
STUDENT: Right,[br]but what I'm asking

0:30:41.852,0:30:46.719
is that not the same as just[br]F1 dx because we're going

0:30:46.719,0:30:49.630
to do a chain rule anyway.

0:30:49.630,0:30:53.750
MAGDALENA TODA: If I put[br]the x, I cannot put this.

0:30:53.750,0:30:57.440
OK, this times that is dx.

0:30:57.440,0:30:59.900
This guy times this guy is dx.

0:30:59.900,0:31:01.400
STUDENT: But then[br]you can't use your

0:31:01.400,0:31:03.980
MAGDALENA TODA: Then I[br]cannot use the t's then.

0:31:03.980,0:31:06.486
STUDENT: All right, there we go.

0:31:06.486,0:31:11.160
MAGDALENA TODA: All right, so[br]I have integral from 0 to 1 t,

0:31:11.160,0:31:15.390
t times 1 t squared.

0:31:15.390,0:31:18.260
If I make a mistake, that would[br]be a silly algebra mistake

0:31:18.260,0:31:18.760
[INAUDIBLE].

0:31:18.760,0:31:21.595


0:31:21.595,0:31:23.980
All right, class.

0:31:23.980,0:31:33.540
t cubed times 1dt,[br]how much is this?

0:31:33.540,0:31:38.002
t cubed over 3 plus t[br]to the fourth over 4.

0:31:38.002,0:31:39.252
STUDENT: It's just 2-- oh, no.

0:31:39.252,0:31:45.100


0:31:45.100,0:31:48.120
MAGDALENA TODA: Very good.

0:31:48.120,0:31:52.500
Do not expect that we kill you[br]with computations on the exams,

0:31:52.500,0:31:55.010
but that's not what we want.

0:31:55.010,0:31:58.250
We want to test if you have[br]the basic understanding of what

0:31:58.250,0:32:03.267
this is all about, not to[br]kill you with, OK, that.

0:32:03.267,0:32:05.350
I'm not going to say that[br]in front of the cameras,

0:32:05.350,0:32:06.805
but everybody knows that.

0:32:06.805,0:32:08.270
There are professors[br]who would like

0:32:08.270,0:32:09.690
to kill you with computations.

0:32:09.690,0:32:12.180
Now, we're living in[br]a different world.

0:32:12.180,0:32:15.590
If I gave you a long[br]polynomial sausage here

0:32:15.590,0:32:17.660
and I ask you to[br]work with it, that

0:32:17.660,0:32:21.090
doesn't mean that I'm smart[br]because MATLAB can do it.

0:32:21.090,0:32:25.100
Mathematica, you get some[br]very nice simplifications

0:32:25.100,0:32:28.500
over there, so I'm[br]just trying to see

0:32:28.500,0:32:35.730
if rather than being able[br]to compute with no error,

0:32:35.730,0:32:40.670
you are having the basic[br]understanding of the concept.

0:32:40.670,0:32:44.520
And the rest can been done[br]by the mathematical software,

0:32:44.520,0:32:49.650
which, nowadays, most[br]mathematicians are using.

0:32:49.650,0:32:52.780
If you asked me 15[br]years ago, I think

0:32:52.780,0:32:57.660
I knew colleagues at all the[br]ranks in academia who would not

0:32:57.660,0:33:01.594
touch Mathematica or[br]MATLAB or Maple say

0:33:01.594,0:33:04.810
that's like tool from[br]evil or something,

0:33:04.810,0:33:08.072
but now everybody uses.

0:33:08.072,0:33:10.760
Engineers use mostly[br]MATLAB as I told you.

0:33:10.760,0:33:15.910
Mathematicians use both[br]MATLAB and Mathematica.

0:33:15.910,0:33:19.450
Some of them use Maple,[br]especially the ones who

0:33:19.450,0:33:23.002
have demos for K-12[br]level teachers,

0:33:23.002,0:33:26.135
but MATLAB is a wonderful[br]tool, very pretty powerful

0:33:26.135,0:33:27.583
in many ways.

0:33:27.583,0:33:30.830
If you are doing any kind[br]of linear algebra project--

0:33:30.830,0:33:33.760
I noticed three or four of you[br]are taking linear algebra-- you

0:33:33.760,0:33:39.070
can always rely on MATLAB being[br]the best of all of the above.

0:33:39.070,0:33:40.230
OK, W2.

0:33:40.230,0:33:42.970


0:33:42.970,0:33:49.830
For W2, I have a parabola, and[br]it's, again, a piece of cake.

0:33:49.830,0:33:54.545
X prime will be 1,[br]y prime will be 2t.

0:33:54.545,0:33:56.485
When I write down[br]the whole thing,

0:33:56.485,0:33:58.667
I have to pay a little[br]bit of attention

0:33:58.667,0:34:02.480
when I substitute[br]especially when I'm

0:34:02.480,0:34:04.595
taking an exam under pressure.

0:34:04.595,0:34:08.860


0:34:08.860,0:34:13.909
x squared is t[br]squared, y is t squared

0:34:13.909,0:34:17.270
times y prime, which is 2t.

0:34:17.270,0:34:19.570
So now this is x prime.

0:34:19.570,0:34:21.139
This is y prime.

0:34:21.139,0:34:24.704
Let me change colors.

0:34:24.704,0:34:26.924
All politicians change colors.

0:34:26.924,0:34:29.300
But I'm not a[br]politician, but I'm

0:34:29.300,0:34:34.030
thinking it's useful for you[br]to see who everybody was.

0:34:34.030,0:34:38.690
This is the F1 in terms of t.

0:34:38.690,0:34:46.214
That's the idea of what that[br]is, and this is F2 in terms of t

0:34:46.214,0:34:48.121
as well.

0:34:48.121,0:34:49.810
Oh, my God, another answer?

0:34:49.810,0:34:53.540
Absolutely, I'm going to[br]get an another answer.

0:34:53.540,0:34:57.276
Is it obviously to everybody[br]I'm going to get another answer?

0:34:57.276,0:34:58.080
STUDENT: Yeah.

0:34:58.080,0:35:01.495
MAGDALENA TODA: So I don't[br]have to put the t's here,

0:35:01.495,0:35:03.790
but I thought it was[br]sort of neat to see

0:35:03.790,0:35:05.950
that t goes from 0 to 1.

0:35:05.950,0:35:08.760
And what do I get?

0:35:08.760,0:35:16.355
This whole lot of them is t[br]cubed plus 2 t to the fifth.

0:35:16.355,0:35:19.040


0:35:19.040,0:35:26.002
So when I do the integration,[br]I get t to the 4 over 4 plus--

0:35:26.002,0:35:27.335
shut up, Magdalena, get people--

0:35:27.335,0:35:29.745


0:35:29.745,0:35:30.620
STUDENT: [INAUDIBLE].

0:35:30.620,0:35:32.040
MAGDALENA TODA: Very good.

0:35:32.040,0:35:36.010
Yeah, he's done[br]the simplification.

0:35:36.010,0:35:37.570
STUDENT: You get[br]the same values.

0:35:37.570,0:35:40.330


0:35:40.330,0:35:45.340
Plug in 1, you get 7/12 again.

0:35:45.340,0:35:48.190
MAGDALENA TODA: So I'm[br]asking you-- OK, what was it?

0:35:48.190,0:35:56.830
Solve 0, 1-- so I'm[br]asking why do you

0:35:56.830,0:36:00.888
think we get the same value?

0:36:00.888,0:36:03.310
Because the force[br]is not conservative,

0:36:03.310,0:36:06.870
and I went on another path.

0:36:06.870,0:36:10.075
I went on one path, and[br]I went on another path.

0:36:10.075,0:36:15.656
And look, obviously my[br]expression was different.

0:36:15.656,0:36:18.870
It's like one of those[br]math games or UIL games.

0:36:18.870,0:36:20.970
And look at the algebra.

0:36:20.970,0:36:23.521
The polynomials are different.

0:36:23.521,0:36:25.960
What was my luck here?

0:36:25.960,0:36:27.143
I took 1.

0:36:27.143,0:36:27.726
STUDENT: Yeah.

0:36:27.726,0:36:29.490
MAGDALENA TODA: I[br]could have taken 2.

0:36:29.490,0:36:35.990
So if instead of 1, I would[br]have taken another number,

0:36:35.990,0:36:38.330
then the higher the power,[br]the bigger the number

0:36:38.330,0:36:39.490
would have been.

0:36:39.490,0:36:40.407
I could have taken 2--

0:36:40.407,0:36:42.114
STUDENT: You could[br]have taken negative 1,

0:36:42.114,0:36:44.240
and you still wouldn't[br]have got the same answer.

0:36:44.240,0:36:48.940
MAGDALENA TODA: Yeah, there[br]are many reasons why that is.

0:36:48.940,0:36:53.900
But anyway, know that when you[br]take 1, 1 to every power is 1.

0:36:53.900,0:36:55.470
And yeah, you were lucky.

0:36:55.470,0:36:58.430
But in general, keep in[br]mind that if the force is

0:36:58.430,0:37:01.500
conservative, in[br]general, in most examples

0:37:01.500,0:37:04.510
you're not going to get the[br]same answer for the work

0:37:04.510,0:37:10.680
because it does depend on[br]the path you want to take.

0:37:10.680,0:37:18.210
I think I have reviewed quite[br]everything that I wanted.

0:37:18.210,0:37:27.330


0:37:27.330,0:37:29.870
So I should be ready[br]to move forward.

0:37:29.870,0:37:32.630


0:37:32.630,0:37:42.326
So I'm saying we are done[br]with sections 13.1, 13.2,

0:37:42.326,0:37:48.920
and 13.3, which was my[br]favorite because it's not

0:37:48.920,0:37:50.970
just the integral of[br]the path that I like,

0:37:50.970,0:37:54.600
but it's the so-called[br]fundamental theorem of calculus

0:37:54.600,0:38:04.870
3, which says, fundamental[br]theorem of the path integral

0:38:04.870,0:38:12.030
saying that you have f of the[br]endpoint minus f of the origin,

0:38:12.030,0:38:14.430
where little f is[br]that scalar potential

0:38:14.430,0:38:17.310
as the linear function[br]was concerned.

0:38:17.310,0:38:24.380
I'm going to call it the[br]fundamental theorem of path

0:38:24.380,0:38:26.060
integral.

0:38:26.060,0:38:29.230
Last time I told you the[br]fundamental theorem of calculus

0:38:29.230,0:38:31.800
is Federal Trade Commission.

0:38:31.800,0:38:34.620
We refer to that in Calc 1.

0:38:34.620,0:38:39.080
But this one is the fundamental[br]theorem of path integral.

0:38:39.080,0:38:42.815
Remember it because at[br]least one problem out of 15

0:38:42.815,0:38:44.648
or something on the[br]final, and there are not

0:38:44.648,0:38:45.564
going to be very many.

0:38:45.564,0:38:48.815
It's going to ask you to[br]know that result. This is

0:38:48.815,0:38:51.950
an important theorem.

0:38:51.950,0:38:55.970
And another important theorem[br]that is starting right now

0:38:55.970,0:38:57.810
is Green's theorem.

0:38:57.810,0:39:02.690
Green's theorem is[br]a magic result. I

0:39:02.690,0:39:04.905
have a t-shirt with it.

0:39:04.905,0:39:06.410
I didn't bring it today.

0:39:06.410,0:39:08.290
Maybe I'm going to bring[br]it next time First,

0:39:08.290,0:39:12.376
I want you to see[br]the result, and then

0:39:12.376,0:39:15.230
I'll bring the t-shirt[br]to the exam, so OK.

0:39:15.230,0:39:18.080


0:39:18.080,0:39:24.870
Assume that you have a[br]soup called Jordan curve.

0:39:24.870,0:39:27.960


0:39:27.960,0:39:32.380
You see, mathematicians don't[br]follow mathematical objects

0:39:32.380,0:39:34.370
by their names.

0:39:34.370,0:39:37.370
We are crazy people, but[br]we don't have a big ego.

0:39:37.370,0:39:41.890
We would not say a theorem[br]of myself or whatever.

0:39:41.890,0:39:45.340
We never give our names to that.

0:39:45.340,0:39:50.930
But all through calculus you[br]saw all sorts of results.

0:39:50.930,0:39:57.090
Like you see the Jordan[br]curve is a terminology,

0:39:57.090,0:40:00.200
but then you see[br]everywhere the Linus rule.

0:40:00.200,0:40:02.360
Did Linus get to[br]call it his own rule?

0:40:02.360,0:40:06.320
No, but Euler's[br]number, these are

0:40:06.320,0:40:08.515
things that were[br]discovered, and in honor

0:40:08.515,0:40:11.640
of that particular[br]mathematician,

0:40:11.640,0:40:13.290
we call them names.

0:40:13.290,0:40:15.593
We call them the name[br]of the mathematician.

0:40:15.593,0:40:18.430


0:40:18.430,0:40:22.560
Out of curiosity for[br]0.5 extra credit points,

0:40:22.560,0:40:25.140
find out who Jordan was.

0:40:25.140,0:40:32.810
Jordan curve is a closed[br]curve that, in general,

0:40:32.810,0:40:34.595
could be piecewise continuous.

0:40:34.595,0:40:40.610


0:40:40.610,0:40:43.230
So you have a closed[br]loop over here.

0:40:43.230,0:40:50.170
So in general, I could[br]have something like that

0:40:50.170,0:40:54.102
that does not enclose.

0:40:54.102,0:40:56.542
That encloses a[br]domain without holes.

0:40:56.542,0:41:04.350


0:41:04.350,0:41:07.540
Holes are functions[br]of the same thing.

0:41:07.540,0:41:10.240
STUDENT: So doesn't it[br]need to be continuous?

0:41:10.240,0:41:12.440
MAGDALENA TODA:[br]No, I said it is.

0:41:12.440,0:41:13.775
STUDENT: You said, piecewise.

0:41:13.775,0:41:15.170
MAGDALENA TODA: Ah, piecewise.

0:41:15.170,0:41:16.448
This is piecewise.

0:41:16.448,0:41:17.762
STUDENT: Oh, so it's piecewise.

0:41:17.762,0:41:18.262
OK.

0:41:18.262,0:41:20.430
MAGDALENA TODA: So you[br]have a bunch of arcs.

0:41:20.430,0:41:23.050
Finitely many, let's[br]say, in your case.

0:41:23.050,0:41:26.230
Finitely many arcs,[br]they have corners,

0:41:26.230,0:41:29.740
but you can see define the[br]integral along such a path.

0:41:29.740,0:41:33.676


0:41:33.676,0:41:37.530
Oh, and also for another[br]0.5 extra credit,

0:41:37.530,0:41:40.140
find out who Mr.[br]Green was because he

0:41:40.140,0:41:43.490
has several theorems that[br]are through mathematics

0:41:43.490,0:41:46.360
and free mechanics and[br]variation calculus.

0:41:46.360,0:41:50.860
There are several identities[br]that are called Greens.

0:41:50.860,0:41:52.420
There is this famous[br]Green's theorem,

0:41:52.420,0:41:54.890
but there are Green's[br]first identity,

0:41:54.890,0:41:57.844
Green's second identity,[br]and all sorts of things.

0:41:57.844,0:42:01.825
And find out who Mr.[br]Green was, and as a total,

0:42:01.825,0:42:03.826
you have 1 point extra credit.

0:42:03.826,0:42:08.870
And you can turn in a regular[br]essay like a two-page thing.

0:42:08.870,0:42:12.602
You want biography of these[br]mathematicians if you want,

0:42:12.602,0:42:15.500
just a few paragraphs.

0:42:15.500,0:42:19.210
So what does Green's theorem do?

0:42:19.210,0:42:26.200
Green's theorem is[br]a remarkable result

0:42:26.200,0:42:31.083
which links the path integral[br]to the double integral.

0:42:31.083,0:42:38.268
It's a remarkable[br]and famous result.

0:42:38.268,0:42:48.330
And that links the path[br]integral on the closed

0:42:48.330,0:43:07.262
curve to a double integral[br]over the domain enclosed.

0:43:07.262,0:43:09.761
I can see the domain[br]inside, but you

0:43:09.761,0:43:15.160
have to understand it's[br]enclosed by the curve.

0:43:15.160,0:43:20.740


0:43:20.740,0:43:24.444
All right, and assume[br]that you have--

0:43:24.444,0:43:35.970
M and N are C1 functions of[br]x and y, what does it mean?

0:43:35.970,0:43:37.620
M is a function of xy.

0:43:37.620,0:43:40.300
N is a function of xy in plane.

0:43:40.300,0:43:43.360
Both of them are differentiable[br]with continuous derivative.

0:43:43.360,0:43:46.692


0:43:46.692,0:43:47.830
They are differentiable.

0:43:47.830,0:43:49.515
You can take the[br]partial derivatives,

0:43:49.515,0:43:51.840
and all the partial[br]derivatives are continuous.

0:43:51.840,0:43:55.496
That's what we mean[br]by being C1 functions.

0:43:55.496,0:43:58.610
And there the magic[br]happens, so let me show you

0:43:58.610,0:44:02.320
where the magic happens.

0:44:02.320,0:44:06.360
This in the box,[br]the path integral

0:44:06.360,0:44:20.637
over c of M dx plus Ndy is[br]equal to the double integral

0:44:20.637,0:44:22.390
over the domain enclosed.

0:44:22.390,0:44:24.250
OK, this is the c.

0:44:24.250,0:44:27.160
On the boundary you[br]go counterclockwise

0:44:27.160,0:44:29.415
like any respectable[br]mathematician

0:44:29.415,0:44:33.880
would go in a trigonometric[br]sense, just counterclockwise.

0:44:33.880,0:44:36.776
And this is the domain[br]being closed by c.

0:44:36.776,0:44:40.060


0:44:40.060,0:44:44.260
And you put here the[br]integral, which is magic.

0:44:44.260,0:44:46.240
This is easy to[br]remember for you.

0:44:46.240,0:44:48.170
This is not easy to[br]remember unless I

0:44:48.170,0:44:49.980
take the t-shirt to[br]the exam, and you

0:44:49.980,0:44:52.068
cheat by looking at my t-shirt.

0:44:52.068,0:44:54.589
No, by the time of the[br]exam, I promised you

0:44:54.589,0:44:59.310
you are going to have at least[br]one week, seven days or more,

0:44:59.310,0:45:03.310
10-day period in which[br]we will study samples,

0:45:03.310,0:45:05.692
various samples of old finals.

0:45:05.692,0:45:08.320
I'm going to go ahead and[br]send you some by email.

0:45:08.320,0:45:11.440
Do you mind?

0:45:11.440,0:45:13.676
In the next week[br]after this week, we

0:45:13.676,0:45:15.400
are going to start reviewing.

0:45:15.400,0:45:20.240
And by dA I mean dxdy, the[br]usual area limit in Cartesian

0:45:20.240,0:45:23.710
coordinates the way you[br]are used to it the most.

0:45:23.710,0:45:27.320


0:45:27.320,0:45:29.720
And then, Alex is looking[br]at it and said, well,

0:45:29.720,0:45:32.380
then I tell her that[br]the most elegant way

0:45:32.380,0:45:34.690
is to put it with dxdy.

0:45:34.690,0:45:38.480
This is what we call a[br]one form in mathematics.

0:45:38.480,0:45:39.750
What is a one form.

0:45:39.750,0:45:43.945
It is a linear combination of[br]this infinitesimal elements

0:45:43.945,0:45:47.490
dxdy in plane with some[br]scalar functions of x

0:45:47.490,0:45:49.491
and y in front of her.

0:45:49.491,0:45:50.750
OK, so what do we do?

0:45:50.750,0:45:52.505
We integrate the one form.

0:45:52.505,0:45:57.460
The book doesn't talk about one[br]forms because the is actually

0:45:57.460,0:46:00.727
written for the average[br]student, the average freshman

0:46:00.727,0:46:04.920
or the average[br]sophomore, but I think

0:46:04.920,0:46:08.250
we have an exposure to[br]the notion of one form,

0:46:08.250,0:46:10.770
so I can get a little bit[br]more elegant and more rigorous

0:46:10.770,0:46:12.290
in my speech.

0:46:12.290,0:46:15.752
If you are a graduate[br]student, you most likely

0:46:15.752,0:46:18.140
would know this is a one form.

0:46:18.140,0:46:22.140
That's actually the[br]definition of a one form.

0:46:22.140,0:46:23.475
And you'll say, what is this?

0:46:23.475,0:46:27.284
This is actually two[br]form, but you are

0:46:27.284,0:46:28.450
going to say, wait a minute.

0:46:28.450,0:46:30.610
I have a scalar[br]function, whatever

0:46:30.610,0:46:34.430
that is, from the integration[br]in front of the dxdy

0:46:34.430,0:46:39.740
you want but you never said[br]that dxdy is a two form.

0:46:39.740,0:46:44.990
Actually, I did, and I[br]didn't call it a two form.

0:46:44.990,0:46:47.280
Do you remember that[br]I introduced to you

0:46:47.280,0:46:50.434
some magic wedge product?

0:46:50.434,0:46:53.640


0:46:53.640,0:46:57.750
And we said, this is[br]a tiny displacement.

0:46:57.750,0:46:59.750
Dx infinitesimal is small.

0:46:59.750,0:47:02.360
Imagine how much the[br]video we'll there

0:47:02.360,0:47:04.960
is an infinitesimal[br]displacement dx

0:47:04.960,0:47:07.570
and an infinitesimal[br]displacement dy,

0:47:07.570,0:47:10.700
and you have some[br]sort of a sign area.

0:47:10.700,0:47:15.130
So we said, we don't[br]just take dxdy,

0:47:15.130,0:47:19.070
but we take a product[br]between dxdy with a wedge,

0:47:19.070,0:47:21.700
meaning that if I[br]change the order,

0:47:21.700,0:47:24.140
I'm going to have minus dy here.

0:47:24.140,0:47:29.140
This is typical exterior[br]derivative theory-- exterior

0:47:29.140,0:47:31.456
derivative theory.

0:47:31.456,0:47:34.940
And it's a theory that[br]starts more or less

0:47:34.940,0:47:36.450
at the graduate level.

0:47:36.450,0:47:39.510
And many people get their[br]master's degree in math

0:47:39.510,0:47:43.452
and never get to see it, and[br]I pity them, but this life.

0:47:43.452,0:47:47.470
On the other hand, when[br]you have dx, which dx--

0:47:47.470,0:47:50.546
the area between dx and dx is 0.

0:47:50.546,0:47:53.400
So we're all very happy[br]I get rid of those.

0:47:53.400,0:47:55.850
When I have the sign[br]between the displacement,

0:47:55.850,0:47:57.570
dy and itself is 0.

0:47:57.570,0:47:59.990
So these are the[br]basic properties

0:47:59.990,0:48:05.420
that we started[br]about the sign area.

0:48:05.420,0:48:07.720
I want to show you what happens.

0:48:07.720,0:48:16.360
I'm going to-- yeah,[br]I'm going to erase here.

0:48:16.360,0:48:22.690


0:48:22.690,0:48:25.856
I'm going to show[br]you later I'm going

0:48:25.856,0:48:32.700
to prove this theorem to you[br]later using these tricks that I

0:48:32.700,0:48:35.096
just showed you here.

0:48:35.096,0:48:53.656
I will provide proof[br]to this formula, OK?

0:48:53.656,0:48:57.500
And let's take a look at[br]that, and we say, well,

0:48:57.500,0:49:00.010
can I memorize that by[br]the time of the final?

0:49:00.010,0:49:01.660
Yes, you can.

0:49:01.660,0:49:13.070
What is beautiful about[br]this, it can actually

0:49:13.070,0:49:18.510
help you solve problems that you[br]didn't think would be possible.

0:49:18.510,0:49:20.920
For example, example[br]1, and I say,

0:49:20.920,0:49:26.400
that would be one of[br]the most basic ones.

0:49:26.400,0:49:38.710
Find the geometric meaning of[br]the integral over a c where

0:49:38.710,0:49:39.890
c is a closed loop.

0:49:39.890,0:49:41.920
OK, c is a loop.

0:49:41.920,0:49:47.388
Piecewise define Jordan[br]curve-- Jordan curve.

0:49:47.388,0:49:49.645
And I integrate out[br]of something weird.

0:49:49.645,0:49:51.144
And you say, oh, my God.

0:49:51.144,0:49:51.950
Look at her.

0:49:51.950,0:49:59.360
She picked some weird function[br]where the path from the dx

0:49:59.360,0:50:05.970
is M, and the path in front of[br]dy is N, the M and N functions.

0:50:05.970,0:50:07.780
Why would pick like that?

0:50:07.780,0:50:11.260
You wouldn't know yet, but[br]if you apply Green's theorem,

0:50:11.260,0:50:14.040
assuming you believe[br]it's true, you

0:50:14.040,0:50:18.271
have double integral over the[br]domain enclosed by this loop.

0:50:18.271,0:50:24.820
The loop is enclosing[br]this domain of what?

0:50:24.820,0:50:32.080
Now, I'm trying to shut up,[br]and I'm want you to talk.

0:50:32.080,0:50:35.542
What am I going to[br]write over here?

0:50:35.542,0:50:36.940
STUDENT: 1 plus 1.

0:50:36.940,0:50:40.600
MAGDALENA TODA: 1 plus[br]1, how fun is that?

0:50:40.600,0:50:46.130
Y minus 1, 1 plus 1 equals[br]2 last time I checked,

0:50:46.130,0:50:49.370
and this is dA.

0:50:49.370,0:50:52.900
And what do you think[br]this animal would be?

0:50:52.900,0:50:56.090
The cast of 2 always can escape.

0:50:56.090,0:51:00.566
So if we don't want[br]it, just kick it out.

0:51:00.566,0:51:04.480
What is the remaining[br]double integral for d of DA?

0:51:04.480,0:51:07.310
You have seen this guy all[br]through the Calculus 3 course.

0:51:07.310,0:51:09.850
You're tired of it.

0:51:09.850,0:51:13.780
You said, I cannot wait for[br]this semester to be over

0:51:13.780,0:51:19.060
because this is the double[br]integral of 1dA over d.

0:51:19.060,0:51:21.940
What in the world is that?

0:51:21.940,0:51:24.364
That is the--

0:51:24.364,0:51:25.330
STUDENT: --area.

0:51:25.330,0:51:26.700
MAGDALENA TODA: Area, very good.

0:51:26.700,0:51:31.150
This is the area of the[br]domain d inside the curve.

0:51:31.150,0:51:34.980
The shaded area is this.

0:51:34.980,0:51:39.060
So you have discovered[br]something beautiful

0:51:39.060,0:51:46.530
that the area of the domain[br]enclosed by a Jordan curve

0:51:46.530,0:51:51.040
equals 1/2 because you pull[br]the two out in front here,

0:51:51.040,0:51:56.320
it's going to be 1/2 of the path[br]integrals over the boundary.

0:51:56.320,0:51:58.590
This is called[br]boundary of a domain.

0:51:58.590,0:52:00.380
c is the boundary of the domain.

0:52:00.380,0:52:04.990


0:52:04.990,0:52:06.940
Some mathematicians--[br]I don't know

0:52:06.940,0:52:10.950
how far you want to go with your[br]education, but in a few years

0:52:10.950,0:52:13.300
you might become[br]graduate students.

0:52:13.300,0:52:18.700
And even some engineers use this[br]notation boundary of d, del d.

0:52:18.700,0:52:22.230
That means the boundaries,[br]the frontier of a domain.

0:52:22.230,0:52:24.430
The fence of a ranch.

0:52:24.430,0:52:27.050
That is the del d, but[br]don't tell the rancher

0:52:27.050,0:52:30.598
because he will take his gun[br]out and shoot you thinking

0:52:30.598,0:52:33.860
that you are off the hook or[br]you are after something weird.

0:52:33.860,0:52:38.340
So that's the boundary[br]of the domain.

0:52:38.340,0:52:42.444
And then you have[br]minus ydx plus xdy.

0:52:42.444,0:52:46.210


0:52:46.210,0:52:48.210
MAGDALENA TODA: We discover[br]something beautiful.

0:52:48.210,0:52:50.210
Something important.

0:52:50.210,0:52:52.760
And now I'm asking,[br]with this exercise--

0:52:52.760,0:52:59.480
one which I could even--[br]I could even call a lemma.

0:52:59.480,0:53:03.964
Lemma is not quite a[br]theorem, because it's based--

0:53:03.964,0:53:05.130
could be based on a theorem.

0:53:05.130,0:53:09.460
It's a little result that can[br]be proved in just a few lines

0:53:09.460,0:53:12.480
without being something[br]sophisticated based

0:53:12.480,0:53:15.800
on something you[br]knew from before.

0:53:15.800,0:53:19.950
So this is called a lemma.

0:53:19.950,0:53:26.250
When you have a sophisticated[br]area to compute--

0:53:26.250,0:53:30.020
or even can you prove-- if you[br]believe in Green's theorem,

0:53:30.020,0:53:33.410
can you prove that the[br]area inside the circle

0:53:33.410,0:53:34.780
is pi r squared?

0:53:34.780,0:53:39.966
Can you prove that the[br]area inside of an ellipse

0:53:39.966,0:53:41.490
is-- I don't know what.

0:53:41.490,0:53:44.370
Do you know the area[br]inside of an ellipse?

0:53:44.370,0:53:47.561
Nobody taught me in school.

0:53:47.561,0:53:50.800
I don't know why[br]it's so beautiful.

0:53:50.800,0:53:55.150
I learned what an ellipse[br]was in eleventh grade

0:53:55.150,0:53:59.190
in high school and again[br]a review as a freshman

0:53:59.190,0:54:01.310
analytic geometry.

0:54:01.310,0:54:03.070
So we've seen conics again--

0:54:03.070,0:54:04.945
STUDENT: I think we did[br]conics in 10th grade.

0:54:04.945,0:54:06.684
We might have seen it.

0:54:06.684,0:54:08.100
MAGDALENA TODA:[br]But nobody told me

0:54:08.100,0:54:10.170
like-- I give you an ellipse.

0:54:10.170,0:54:12.040
Compute the area inside.

0:54:12.040,0:54:13.170
I had no idea.

0:54:13.170,0:54:15.420
And I didn't know[br]the formula until I

0:54:15.420,0:54:17.820
became an assistant professor.

0:54:17.820,0:54:19.490
I was already in my thirties.

0:54:19.490,0:54:23.970
That's a shame to see that[br]thing for the first time OK.

0:54:23.970,0:54:27.900
So let's see if we believe[br]this lemma, and the Green's

0:54:27.900,0:54:28.736
theorem of course.

0:54:28.736,0:54:31.624
But let's apply the[br]lemma, primarily

0:54:31.624,0:54:33.492
from the Green's theorem.

0:54:33.492,0:54:36.600
Can we actually prove[br]that the area of the disk

0:54:36.600,0:54:40.890
is pi r squared and the[br]area of the ellipse--

0:54:40.890,0:54:43.330
inside the ellipse[br]will be god knows what.

0:54:43.330,0:54:47.180
And we will discover[br]that by ourselves.

0:54:47.180,0:54:49.385
I think that's the[br]beauty of mathematics.

0:54:49.385,0:54:53.630
Because every now and then[br]even if you discover things

0:54:53.630,0:54:56.280
that people have known[br]for hundreds of years,

0:54:56.280,0:54:58.080
it still gives you[br]the satisfaction

0:54:58.080,0:55:01.840
that you did something by[br]yourself-- all on yourself.

0:55:01.840,0:55:06.350
Like, when you feel[br]build a helicopter or you

0:55:06.350,0:55:07.730
build a table.

0:55:07.730,0:55:09.898
There are many more[br]beautiful tables

0:55:09.898,0:55:12.050
that were built before[br]you, but still it's

0:55:12.050,0:55:14.980
a lot of satisfaction that[br]you do all by yourself.

0:55:14.980,0:55:16.850
It's the same with mathematics.

0:55:16.850,0:55:23.386
So let's see what we can[br]do now for the first time.

0:55:23.386,0:55:24.590
Not for the first time.

0:55:24.590,0:55:28.030
We do it in other ways.

0:55:28.030,0:55:37.608
Can you prove using the lemma[br]or Green's theorem-- which

0:55:37.608,0:55:43.125
is the same thing-- either[br]one-- that the area of the disk

0:55:43.125,0:55:47.850
of radius r-- this is the r.

0:55:47.850,0:55:52.330
so this the radius[br]r is pi r squared.

0:55:52.330,0:55:56.290


0:55:56.290,0:55:57.530
I hope so.

0:55:57.530,0:55:59.700
And the answer is, I hope so.

0:55:59.700,0:56:00.660
And that's all.

0:56:00.660,0:56:03.540


0:56:03.540,0:56:09.460
Area of the disk of radius r.

0:56:09.460,0:56:10.240
Oh my god.

0:56:10.240,0:56:12.620
So you go, well.

0:56:12.620,0:56:19.420
If I knew the parameterization[br]of that boundary C,

0:56:19.420,0:56:20.600
it would be a piece of cake.

0:56:20.600,0:56:25.640
Because I would just-- I know[br]how to do a path integral now.

0:56:25.640,0:56:27.620
I've learned in the[br]previous sections,

0:56:27.620,0:56:30.460
so this should be easy.

0:56:30.460,0:56:32.690
Can we do that?

0:56:32.690,0:56:33.330
So let's see.

0:56:33.330,0:56:36.360


0:56:36.360,0:56:38.290
Without computing[br]the double integral,

0:56:38.290,0:56:41.270
because I can always do[br]that with polar coordinates.

0:56:41.270,0:56:42.880
And we are going to do that.

0:56:42.880,0:56:47.640


0:56:47.640,0:56:49.544
Let's do that as[br]well, as practice.

0:56:49.544,0:56:53.801
Because so you[br]review for the exam.

0:56:53.801,0:56:57.100


0:56:57.100,0:57:00.940
But another way to[br]do it would be what?

0:57:00.940,0:57:06.640
1/2 integral over the circle.

0:57:06.640,0:57:13.900
And how do I parametrize a[br]circle of fixed radius r?

0:57:13.900,0:57:14.830
Who tells me?

0:57:14.830,0:57:18.020
x of t will be--[br]that was Chapter 10.

0:57:18.020,0:57:20.872
Everything is a[br]circle in mathematics.

0:57:20.872,0:57:21.800
STUDENT: r cosine t.

0:57:21.800,0:57:22.925
MAGDALENA TODA: r cosine t.

0:57:22.925,0:57:24.210
Excellent.

0:57:24.210,0:57:25.786
y of t is?

0:57:25.786,0:57:26.640
STUDENT: r sine t.

0:57:26.640,0:57:29.420
MAGDALENA TODA: r sine t.

0:57:29.420,0:57:32.360
So, finally I'm going to[br]go ahead and use this one.

0:57:32.360,0:57:37.065
And I'm going to say, well,[br]minus y to be plugged in.

0:57:37.065,0:57:39.800


0:57:39.800,0:57:43.000
This is minus y.

0:57:43.000,0:57:44.580
Multiply by dx.

0:57:44.580,0:57:47.550
Well, you say, wait a[br]minute. dx with respect.

0:57:47.550,0:57:48.750
What is dx?

0:57:48.750,0:57:52.410
dx is just x prime dt.

0:57:52.410,0:57:54.690
Dy is just y prime dt.

0:57:54.690,0:57:56.370
And t goes out.

0:57:56.370,0:57:57.520
It's banished.

0:57:57.520,0:57:59.620
No, he's the most important guy.

0:57:59.620,0:58:02.970
So t goes from something[br]to something else.

0:58:02.970,0:58:05.260
We will see that later.

0:58:05.260,0:58:06.970
What is x prime dt?

0:58:06.970,0:58:12.850
X prime is minus r sine[br]theta-- sine t, Magdalena.

0:58:12.850,0:58:15.740
Minus r sine t.

0:58:15.740,0:58:18.260
That was x prime.

0:58:18.260,0:58:19.430
Change the color.

0:58:19.430,0:58:23.240
Give people some[br]variation in their life.

0:58:23.240,0:58:32.056
Plus r cosine t,[br]because this x--

0:58:32.056,0:58:32.930
STUDENT: [INAUDIBLE].

0:58:32.930,0:58:39.310


0:58:39.310,0:58:46.440
MAGDALENA TODA: --times[br]the y, which is r cosine t.

0:58:46.440,0:58:49.150
So it suddenly became beautiful.

0:58:49.150,0:58:52.480
It looks-- first it looks ugly,[br]but now it became beautiful.

0:58:52.480,0:58:52.980
Why?

0:58:52.980,0:58:54.365
How come it became beautiful?

0:58:54.365,0:58:56.740
STUDENT: Because you got sine[br]squared plus cosine square.

0:58:56.740,0:58:58.281
MAGDALENA TODA:[br]Because I got a plus.

0:58:58.281,0:59:01.850
If you pay attention, plus sine[br]squared plus cosine squared.

0:59:01.850,0:59:04.970
So I have, what is sine[br]squared plus cosine squared?

0:59:04.970,0:59:07.804
I heard that our[br]students in trig--

0:59:07.804,0:59:11.859
Poly told me-- who still don't[br]know that this is the most

0:59:11.859,0:59:13.650
important thing you[br]learn in trigonometry--

0:59:13.650,0:59:15.150
is Pythagorean theorem.

0:59:15.150,0:59:15.860
Right?

0:59:15.860,0:59:24.236
So you have 1/2 integral

0:59:24.236,0:59:27.075
STUDENT: r squared--

0:59:27.075,0:59:28.450
MAGDALENA TODA:[br]r-- no, I'm lazy.

0:59:28.450,0:59:30.920
I'm going slow-- r.

0:59:30.920,0:59:32.730
dt.

0:59:32.730,0:59:34.916
T from what to what?

0:59:34.916,0:59:36.780
From 0 times 0.

0:59:36.780,0:59:39.880
I'm starting whatever[br]I want, actually.

0:59:39.880,0:59:44.214
I go counterclockwise[br]I'm into pi.

0:59:44.214,0:59:45.986
STUDENT: Why is[br]that not r squared?

0:59:45.986,0:59:47.808
It should be r squared.

0:59:47.808,0:59:49.176
MAGDALENA TODA: I'm sorry, guys.

0:59:49.176,0:59:50.090
I'm sorry.

0:59:50.090,0:59:53.550
I don't know what[br]I am-- r squared.

0:59:53.550,0:59:57.060
1/2 r squared times 2 pi.

0:59:57.060,1:00:00.740


1:00:00.740,1:00:03.410
So we have pi r squared.

1:00:03.410,1:00:06.870
And if you did not[br]tell me it's r squared,

1:00:06.870,1:00:09.710
we wouldn't have[br]gotten the answer.

1:00:09.710,1:00:10.210
That's good.

1:00:10.210,1:00:14.130


1:00:14.130,1:00:16.257
What's the other way to do it?

1:00:16.257,1:00:18.530
If a problem on[br]the final would ask

1:00:18.530,1:00:22.172
you prove in two different[br]ways that the rubber

1:00:22.172,1:00:25.130
disk is pi r squared using[br]Calc 3, or whatever--

1:00:25.130,1:00:26.130
STUDENT: Would require--

1:00:26.130,1:00:28.350
MAGDALENA TODA: The[br]double integral, right?

1:00:28.350,1:00:28.850
Right?

1:00:28.850,1:00:31.141
STUDENT: Could have done[br]Cartesian coordinates as well.

1:00:31.141,1:00:33.105
If that counts as a second way.

1:00:33.105,1:00:33.980
MAGDALENA TODA: Yeah.

1:00:33.980,1:00:34.830
You can-- OK.

1:00:34.830,1:00:36.360
What could this be?

1:00:36.360,1:00:37.140
Oh my god.

1:00:37.140,1:00:42.280
This would be minus 1[br]to 1 minus square root

1:00:42.280,1:00:45.966
1 minus x squared to square[br]root 1 minus x squared.

1:00:45.966,1:00:46.849
Am i right guys?

1:00:46.849,1:00:47.390
STUDENT: Yep.

1:00:47.390,1:00:48.700
MAGDALENA TODA: 1 dy dx.

1:00:48.700,1:00:51.165
Of course it's a pain.

1:00:51.165,1:00:53.706
STUDENT: You could double that[br]and set the bottoms both equal

1:00:53.706,1:00:55.074
to 0.

1:00:55.074,1:00:55.990
MAGDALENA TODA: Right.

1:00:55.990,1:01:01.420
So we can do by symmetry--

1:01:01.420,1:01:02.720
STUDENT: Yeah.

1:01:02.720,1:01:05.060
MAGDALENA TODA: I'm--[br]shall I erase or leave it.

1:01:05.060,1:01:07.790
Are you understand[br]what Alex is saying?

1:01:07.790,1:01:12.192
This is 2i is the integral[br]that you will get.

1:01:12.192,1:01:13.650
STUDENT: Just write[br]it next to it--

1:01:13.650,1:01:14.990
MAGDALENA TODA: I tell[br]you four times, you

1:01:14.990,1:01:16.490
see, Alex, because you have--

1:01:16.490,1:01:16.820
STUDENT: Oh, yeah.

1:01:16.820,1:01:19.069
MAGDALENA TODA: --symmetry[br]with respect to the x-axis,

1:01:19.069,1:01:21.420
and symmetry with[br]respect to y-axis.

1:01:21.420,1:01:26.725
And you can take 0[br]to 1 and 0 to that.

1:01:26.725,1:01:29.210
And you have x from 0 to 1.

1:01:29.210,1:01:33.740
You have y from 0 to stop.

1:01:33.740,1:01:35.440
Square root of 1 minus x square.

1:01:35.440,1:01:37.465
Like the strips.

1:01:37.465,1:01:41.600
And you have 4[br]times that A1, which

1:01:41.600,1:01:44.925
would be the area of[br]the first quadratic.

1:01:44.925,1:01:46.470
You can do that, too.

1:01:46.470,1:01:46.970
It's easier.

1:01:46.970,1:01:50.100
But the best way to do that is[br]not in Cartesian coordinates.

1:01:50.100,1:01:52.770
The best way is to do[br]it in polar coordinates.

1:01:52.770,1:01:56.570
Always remember[br]your Jacobian is r.

1:01:56.570,1:02:00.892
So if you have[br]Jacobian r-- erase.

1:02:00.892,1:02:03.420
Let's put r here again.

1:02:03.420,1:02:08.270
And then dr d theta.

1:02:08.270,1:02:10.280
But now you say, wait[br]a minute, Magdalena.

1:02:10.280,1:02:11.780
You said r is fixed.

1:02:11.780,1:02:12.439
Yes.

1:02:12.439,1:02:13.980
And that's why I[br]need to learn Greek,

1:02:13.980,1:02:15.762
because it's all Greek to me.

1:02:15.762,1:02:18.860
Instead of r I put[br]rho as a variable.

1:02:18.860,1:02:23.560
And I say, rho is[br]between 0 and r.

1:02:23.560,1:02:25.300
r is fixed.

1:02:25.300,1:02:27.200
That's my [INAUDIBLE].

1:02:27.200,1:02:32.130
Big r is not usually written[br]as a variable from 0 to some.

1:02:32.130,1:02:33.485
I cannot use that.

1:02:33.485,1:02:37.260
So I have to us a Greek letter,[br]whether I like it or not.

1:02:37.260,1:02:39.580
And theta is from 0 to 2 pi.

1:02:39.580,1:02:41.750
And I still get the same thing.

1:02:41.750,1:02:47.050
I get r-- rho squared[br]over 2 between 0 and r.

1:02:47.050,1:02:48.610
And I have 2 pi.

1:02:48.610,1:02:53.350
And in the end that means[br]pi r squared, and I'm back.

1:02:53.350,1:02:56.260
And you say, wait,[br]this is Example 4.

1:02:56.260,1:02:57.462
Whatever example.

1:02:57.462,1:02:59.180
Is it Example 4, 5?

1:02:59.180,1:03:01.090
You say, this is[br]a piece of cake.

1:03:01.090,1:03:05.610
I have two methods showing[br]me that area of the disk

1:03:05.610,1:03:06.965
is so pi r squared.

1:03:06.965,1:03:08.204
It's so trivial.

1:03:08.204,1:03:12.140
Yeah, then let's move[br]on and do the ellipse.

1:03:12.140,1:03:14.740
Or we could have been[br]smart and done the ellipse

1:03:14.740,1:03:16.710
from the beginning.

1:03:16.710,1:03:18.600
And then the circular[br]disk would have

1:03:18.600,1:03:23.010
been just a trivial, particular[br]example of the ellipse.

1:03:23.010,1:03:25.010
But let's do the ellipse[br]with this magic formula

1:03:25.010,1:03:26.740
that I just taught you.

1:03:26.740,1:03:29.830


1:03:29.830,1:03:34.250
In the finals-- I'm going to[br]send you a bunch of finals.

1:03:34.250,1:03:36.630
You're going to be[br]amused, because you're

1:03:36.630,1:03:38.460
going to look at[br]them and you say,

1:03:38.460,1:03:41.650
regardless of the year and[br]semester when the final was

1:03:41.650,1:03:44.220
given for Calc 3,[br]there was always

1:03:44.220,1:03:49.040
one of the problems at the[br]end using direct application

1:03:49.040,1:03:50.960
of Green's theorem.

1:03:50.960,1:03:53.290
So Green's theorem[br]is an obsession,

1:03:53.290,1:03:55.214
and not only at Tech.

1:03:55.214,1:03:58.780
I was looking UT Austin,[br]A&M, other schools--

1:03:58.780,1:04:05.990
California Berkley-- all the[br]Calc 3 courses on the final

1:04:05.990,1:04:10.620
have at least one application--[br]direct application

1:04:10.620,1:04:12.360
applying principal.

1:04:12.360,1:04:12.860
OK.

1:04:12.860,1:04:17.250


1:04:17.250,1:04:18.960
So what did I say?

1:04:18.960,1:04:21.715
I said that we have[br]to draw an ellipse.

1:04:21.715,1:04:25.426
How do we draw an ellipse[br]without making it up?

1:04:25.426,1:04:26.845
That's the question.

1:04:26.845,1:04:28.737
STUDENT: Draw a circle.

1:04:28.737,1:04:30.156
MAGDALENA TODA: Draw a circle.

1:04:30.156,1:04:32.060
Good answer.

1:04:32.060,1:04:33.580
OK.

1:04:33.580,1:04:35.060
All right.

1:04:35.060,1:04:40.010
And guys this[br]started really bad.

1:04:40.010,1:04:43.433
So I'm doing what I can.

1:04:43.433,1:04:46.391


1:04:46.391,1:04:49.349
I should have tried[br]more coffee today,

1:04:49.349,1:04:52.460
because I'm getting[br]insecure and very shaky.

1:04:52.460,1:04:52.960
OK.

1:04:52.960,1:04:58.100
So I have the ellipse[br]in standard form

1:04:58.100,1:05:01.620
of center O, x squared over[br]x squared plus y squared

1:05:01.620,1:05:05.300
over B squared equals 1.

1:05:05.300,1:05:07.890
And now you are going to[br]me who is A and who is B?

1:05:07.890,1:05:08.920
What are they called?

1:05:08.920,1:05:10.314
Semi--

1:05:10.314,1:05:11.105
STUDENT: Semiotics.

1:05:11.105,1:05:12.188
MAGDALENA TODA: Semiotics.

1:05:12.188,1:05:15.480
A and B. Good.

1:05:15.480,1:05:18.900
Find the area.

1:05:18.900,1:05:21.810
I don't like-- OK.

1:05:21.810,1:05:27.080
Let's put B inside, and let's[br]put C outside the boundary.

1:05:27.080,1:05:42.690
So area of the ellipse domain[br]D will be-- by the lemma-- 1/2

1:05:42.690,1:05:46.010
integral over C.

1:05:46.010,1:05:47.250
This is C. Is not f.

1:05:47.250,1:05:48.200
Don't confuse it.

1:05:48.200,1:05:50.616
It is my beautiful[br]script C. I've

1:05:50.616,1:05:52.420
tried to use it many times.

1:05:52.420,1:05:55.240
Going to be minus y dx plus xdy.

1:05:55.240,1:05:58.380


1:05:58.380,1:05:59.490
Again, why was that?

1:05:59.490,1:06:04.530
Because we said this[br]is M and this is N,

1:06:04.530,1:06:09.110
and Green's theorem will give[br]you double integral of N sub x

1:06:09.110,1:06:10.570
minus M sub y.

1:06:10.570,1:06:13.910
So you have 1 minus[br]minus 1, which is 2.

1:06:13.910,1:06:15.910
And 2 knocked that out.

1:06:15.910,1:06:16.410
OK.

1:06:16.410,1:06:19.090
That's how we prove it.

1:06:19.090,1:06:19.720
OK.

1:06:19.720,1:06:24.210
Problem is that I do not the[br]parametrization of the ellipse.

1:06:24.210,1:06:28.220
And if somebody doesn't help me,[br]I'm going to be in big trouble.

1:06:28.220,1:06:32.620


1:06:32.620,1:06:34.420
And I'll start[br]cursing and I'm not

1:06:34.420,1:06:37.070
allowed to curse in[br]front of the classroom.

1:06:37.070,1:06:40.760
But you can help me on[br]that, because this reminds

1:06:40.760,1:06:46.040
you of a famous Greek identity.

1:06:46.040,1:06:48.520
The fundamental trig identity.

1:06:48.520,1:06:51.610
If this would be cosine[br]squared of theta,

1:06:51.610,1:06:55.260
and this would be sine squared[br]of theta, as two animals,

1:06:55.260,1:06:56.710
their sum would be 1.

1:06:56.710,1:07:00.630
And whenever you have sums[br]of sum squared thingies,

1:07:00.630,1:07:03.834
then you have to think trig.

1:07:03.834,1:07:06.710
So, what would be[br]good as a parameter?

1:07:06.710,1:07:07.300
OK.

1:07:07.300,1:07:10.240
What would be good[br]as a parametrization

1:07:10.240,1:07:12.214
to make this come true?

1:07:12.214,1:07:15.012
STUDENT: You have the cosine[br]of theta would equal x over x.

1:07:15.012,1:07:15.970
MAGDALENA TODA: Uh-huh.

1:07:15.970,1:07:18.126
So then x would be A times--

1:07:18.126,1:07:19.410
STUDENT: The cosine of theta.

1:07:19.410,1:07:21.590
MAGDALENA TODA:[br]Do you like theta?

1:07:21.590,1:07:23.690
You don't, because[br]you're not Greek.

1:07:23.690,1:07:25.050
That's the problem.

1:07:25.050,1:07:26.840
If you were Greek,[br]you would like it.

1:07:26.840,1:07:29.260
We had a colleague who[br]is not here anymore.

1:07:29.260,1:07:30.790
Greek from Cypress.

1:07:30.790,1:07:38.250
And he could claim that the[br]most important-- most important

1:07:38.250,1:07:40.230
alphabet is the[br]Greek one, and that's

1:07:40.230,1:07:44.210
why the mathematicians[br]adopted it.

1:07:44.210,1:07:45.150
OK?

1:07:45.150,1:07:47.150
B sine t.

1:07:47.150,1:07:48.270
How do you check?

1:07:48.270,1:07:49.290
You always think, OK.

1:07:49.290,1:07:51.460
This over that is cosine.

1:07:51.460,1:07:53.490
This over this is sine.

1:07:53.490,1:07:54.340
I square them.

1:07:54.340,1:07:56.210
I get exactly that[br]and I get a 1.

1:07:56.210,1:07:56.710
Good.

1:07:56.710,1:07:57.670
I'm in good shape.

1:07:57.670,1:08:01.380
I know that this[br]implicit equation--

1:08:01.380,1:08:04.710
this is an implicit[br]equation-- happens if and only

1:08:04.710,1:08:11.080
if I have this system of[br]the parametrization with t

1:08:11.080,1:08:17.000
between-- anything I want,[br]including the basic 0 to 2

1:08:17.000,1:08:18.420
pi interval.

1:08:18.420,1:08:22.380
And then if I were to move[br]all around for time real t

1:08:22.380,1:08:26.274
I would wind around that the[br]circle infinitely many times.

1:08:26.274,1:08:29.270
Between time equals[br]minus infinity--

1:08:29.270,1:08:33.060
that nobody remembers-- and[br]time equals plus infinity--

1:08:33.060,1:08:36.180
that nobody will[br]ever get to know.

1:08:36.180,1:08:38.550
So those are the values of it.

1:08:38.550,1:08:41.366
All the real values, actually.

1:08:41.366,1:08:44.960
I only needed from 0 to 2[br]pi to wind one time around.

1:08:44.960,1:08:46.660
And this is the idea.

1:08:46.660,1:08:48.514
I wind one time around.

1:08:48.514,1:08:51.149
Now people-- you're going[br]to see mathematicians

1:08:51.149,1:08:52.740
are not the greatest people.

1:08:52.740,1:09:01.254
I've seen engineers and[br]physicists use a lot this sign.

1:09:01.254,1:09:02.420
Do you know what this means?

1:09:02.420,1:09:04.420
STUDENT: It means[br]one full revolution.

1:09:04.420,1:09:06.550
MAGDALENA TODA: It[br]means a full revolution.

1:09:06.550,1:09:10.410
You're going to have[br]a loop-- loops, that's

1:09:10.410,1:09:11.240
whatever you want.

1:09:11.240,1:09:13.380
Here and goes counterclockwise.

1:09:13.380,1:09:15.720
And they put this[br]little sign showing

1:09:15.720,1:09:21.790
I'm going counterclockwise on[br]a closed curved, or a loop.

1:09:21.790,1:09:22.509
All right.

1:09:22.509,1:09:24.439
Don't think they are crazy.

1:09:24.439,1:09:27.479
This was used in lots[br]of scientific papers

1:09:27.479,1:09:30.810
in math, physics, and[br]engineering, and so on.

1:09:30.810,1:09:31.310
OK.

1:09:31.310,1:09:34.850


1:09:34.850,1:09:36.660
Let's do it then.

1:09:36.660,1:09:38.279
Can we do it by ourselves?

1:09:38.279,1:09:39.210
I think so.

1:09:39.210,1:09:39.810
That's see.

1:09:39.810,1:09:42.370
1/2 is 1.

1:09:42.370,1:09:45.310
And I don't like[br]the pink marker.

1:09:45.310,1:09:47.270
Integral log.

1:09:47.270,1:09:51.930
Time from 0 to 2 pi[br]should be measured.

1:09:51.930,1:09:55.740
y minus B sine t.

1:09:55.740,1:10:01.730


1:10:01.730,1:10:04.345
dx-- what tells me that?

1:10:04.345,1:10:06.810
STUDENT: B minus--

1:10:06.810,1:10:07.223


1:10:07.223,1:10:08.306
MAGDALENA TODA: Very good.

1:10:08.306,1:10:09.800
Minus A sine t.

1:10:09.800,1:10:10.796
How hard is that?

1:10:10.796,1:10:16.274
It's a piece of cake Plus x--

1:10:16.274,1:10:18.179
STUDENT: A cosine.

1:10:18.179,1:10:19.262
MAGDALENA TODA: Very good.

1:10:19.262,1:10:21.760
A cosine t.

1:10:21.760,1:10:23.090
TImes--

1:10:23.090,1:10:25.630
STUDENT: B cosine t.

1:10:25.630,1:10:28.790
MAGDALENA TODA: --B cosine t.

1:10:28.790,1:10:30.130
And dt.

1:10:30.130,1:10:32.750
And this thing-- look at it.

1:10:32.750,1:10:33.460
It's huge.

1:10:33.460,1:10:35.700
It looks huge, but it's[br]so beautiful, because--

1:10:35.700,1:10:36.569
STUDENT: AB.

1:10:36.569,1:10:37.360
MAGDALENA TODA: AB.

1:10:37.360,1:10:38.805
Why is it AB?

1:10:38.805,1:10:44.030
It's AB because sine squared[br]plus cosine squared inside

1:10:44.030,1:10:46.180
becomes 1.

1:10:46.180,1:10:49.990
And I have plus AB,[br]plus AB, AB out.

1:10:49.990,1:10:52.020
Kick out the AB.

1:10:52.020,1:10:57.090
Kick out the A and[br]the B and you get

1:10:57.090,1:11:01.750
something beautiful-- sine[br]squared t plus cosine squared

1:11:01.750,1:11:03.420
t is your old friend.

1:11:03.420,1:11:04.840
And he says, I'm 1.

1:11:04.840,1:11:08.049
Look how beautiful[br]life is for you.

1:11:08.049,1:11:09.498
Finally, we proved it.

1:11:09.498,1:11:10.947
What did we prove?

1:11:10.947,1:11:11.913
We are almost there.

1:11:11.913,1:11:12.879
We got a 1/2.

1:11:12.879,1:11:15.780


1:11:15.780,1:11:18.600
A constant value kick out, AB.

1:11:18.600,1:11:21.504


1:11:21.504,1:11:22.472
STUDENT: Times 2 pi.

1:11:22.472,1:11:23.597
MAGDALENA TODA: Times 2 pi.

1:11:23.597,1:11:26.840


1:11:26.840,1:11:28.010
Good.

1:11:28.010,1:11:30.110
2 goes away.

1:11:30.110,1:11:33.300
And we got a magic thing that[br]nobody taught us in school,

1:11:33.300,1:11:34.730
because they were mean.

1:11:34.730,1:11:37.360
They really didn't want[br]us to learn too much.

1:11:37.360,1:11:38.760
That's the thingy.

1:11:38.760,1:11:40.540
AB pi.

1:11:40.540,1:11:45.220
AB pi is what we were[br]hoping for, because, look.

1:11:45.220,1:11:47.820
I mean it's almost[br]too good to be true.

1:11:47.820,1:11:53.610
Well, it's a disk of radius[br]r, A and B are equal.

1:11:53.610,1:11:55.870
And they are the[br]radius of the disk.

1:11:55.870,1:11:58.730
And that's why we[br]have pi r squared

1:11:58.730,1:12:00.972
as a particular[br]example of the disk

1:12:00.972,1:12:04.924
of the area of this ellipse.

1:12:04.924,1:12:07.641
When I saw it the first[br]time, I was like, well,

1:12:07.641,1:12:12.500
I'm glad that I lived to be[br]30 or something to learn this.

1:12:12.500,1:12:17.510
Because nobody had shown it[br]to me in K-12 or in college.

1:12:17.510,1:12:22.534
And I was a completing-- I was[br]a PhD and I didn't know it.

1:12:22.534,1:12:25.378
And then I said, oh,[br]that's why-- pi AB.

1:12:25.378,1:12:26.800
Yes, OK.

1:12:26.800,1:12:27.770
All right.

1:12:27.770,1:12:31.870
So it's so easy to[br]understand once you-- well.

1:12:31.870,1:12:33.493
Once you learn the section.

1:12:33.493,1:12:34.909
If you don't learn[br]the section you

1:12:34.909,1:12:38.970
will not be able to understand.

1:12:38.970,1:12:39.470
OK.

1:12:39.470,1:12:39.970
All right.

1:12:39.970,1:12:42.300
I'm going to go[br]ahead and erase this.

1:12:42.300,1:12:44.950
And I'll show you[br]an example that

1:12:44.950,1:12:49.500
was popping up like an obsession[br]with the numbers changed

1:12:49.500,1:12:53.190
in most of the final exams[br]that happen in the last three

1:12:53.190,1:12:58.900
years, regardless of[br]who wrote the exam.

1:12:58.900,1:13:04.590
Because this problem really[br]matches the learning outcomes,

1:13:04.590,1:13:08.530
oh, just about any university--[br]any good university

1:13:08.530,1:13:10.690
around the world.

1:13:10.690,1:13:12.240
So you'll say, wow.

1:13:12.240,1:13:13.110
It's so easy.

1:13:13.110,1:13:16.754
I could not believe it[br]that-- how easy it is.

1:13:16.754,1:13:25.664
But once you see it, you[br]will-- you'll say, wow.

1:13:25.664,1:13:26.660
It's easy.

1:13:26.660,1:13:34.626


1:13:34.626,1:13:35.126
OK.

1:13:35.126,1:13:41.102


1:13:41.102,1:13:44.120
[CHATTER]

1:13:44.120,1:13:46.034
Let's try this one.

1:13:46.034,1:13:48.519
You have a circle.

1:13:48.519,1:13:57.770
and the circle will be[br]a circle radius r given

1:13:57.770,1:14:03.590
and origin 0 of 4, 9, 0, and 0.

1:14:03.590,1:14:08.520


1:14:08.520,1:14:17.290
And I'm going to[br]write-- I'm going

1:14:17.290,1:14:19.970
to give you-- first I'm going[br]to give you a very simple one.

1:14:19.970,1:14:31.171


1:14:31.171,1:14:37.989
Compute in the[br]simplest possible way.

1:14:37.989,1:14:41.570
If you don't want to[br]parametrize the circle--

1:14:41.570,1:14:43.400
you can always[br]parametrize the circle.

1:14:43.400,1:14:44.302
Right?

1:14:44.302,1:14:45.420
But you don't want to.

1:14:45.420,1:14:49.270
You want to do it the[br]fastest possible way

1:14:49.270,1:14:51.440
without parameterizing[br]the circle.

1:14:51.440,1:14:53.880
Without writing down[br]what I'm writing down.

1:14:53.880,1:14:55.110
You are in a hurry.

1:14:55.110,1:14:58.980
You have 20-- 15 minutes[br]left of your final.

1:14:58.980,1:15:00.700
And you're looking[br]at me and say, I

1:15:00.700,1:15:02.220
hope I get an A in this final.

1:15:02.220,1:15:05.887
So what do you have to[br]remember when you look at that?

1:15:05.887,1:15:09.660


1:15:09.660,1:15:14.770
M and M. M and M.[br]No, M and N. OK.

1:15:14.770,1:15:18.690
And you have to remember[br]that you are over a circle

1:15:18.690,1:15:20.150
so you have a closed loop.

1:15:20.150,1:15:21.780
And that's a Jordan curve.

1:15:21.780,1:15:24.130
That's enclosing a disk.

1:15:24.130,1:15:28.070
So you have a relationship[br]between the path

1:15:28.070,1:15:34.650
integral along the C and the[br]area along the D-- over D.

1:15:34.650,1:15:36.350
Which is of what?

1:15:36.350,1:15:38.720
Is N sub x minus M sub y.

1:15:38.720,1:15:41.300
So let me write it[br]in this form, which

1:15:41.300,1:15:46.120
is the same thing my students[br]mostly prefer to write it as.

1:15:46.120,1:15:48.710
N sub x minus M sub y.

1:15:48.710,1:15:51.790
The t-shirt I have[br]has it written

1:15:51.790,1:15:56.960
like that, because it was[br]bought from nerdytshirt.com

1:15:56.960,1:16:01.220
And it was especially[br]created to impress nerds.

1:16:01.220,1:16:04.300
And of course if you[br]look at the del notation

1:16:04.300,1:16:07.380
that gives you that kind[br]of snobbish attitude

1:16:07.380,1:16:11.670
that you aren't a scientist.

1:16:11.670,1:16:12.220
OK.

1:16:12.220,1:16:16.332
So what is this[br]going to be then?

1:16:16.332,1:16:19.390
Double integral over d.

1:16:19.390,1:16:22.390
And sub x is up[br]here so it gave 5.

1:16:22.390,1:16:24.750
And sub y is a piece of cake.

1:16:24.750,1:16:36.530
3 dx dy equals 2 out times[br]the area of the disk, which

1:16:36.530,1:16:38.466
is something you know.

1:16:38.466,1:16:40.886
And I'm not going to ask you[br]to prove that all over again.

1:16:40.886,1:16:42.760
So you have to say 2.

1:16:42.760,1:16:46.586
I know the area of the[br]disk-- pi r squared.

1:16:46.586,1:16:48.002
And that's the answer.

1:16:48.002,1:16:49.418
And you leave the room.

1:16:49.418,1:16:50.362
And that's it.

1:16:50.362,1:16:52.260
It's almost too[br]easy to believe it,

1:16:52.260,1:16:58.280
but it was always there in[br]the simplest possible way.

1:16:58.280,1:17:02.610
And now I'm wondering, if I[br]were to give you something hard,

1:17:02.610,1:17:08.040
because-- you know my theory[br]that when you practice

1:17:08.040,1:17:11.810
at something in[br]the classroom you

1:17:11.810,1:17:16.650
have to be working on harder[br]things in the classroom

1:17:16.650,1:17:19.310
to do better in the exam.

1:17:19.310,1:17:22.990
So let me cook up[br]something ugly for you.

1:17:22.990,1:17:25.830
The same kind of disk.

1:17:25.830,1:17:28.250
And I'm changing the functions.

1:17:28.250,1:17:33.055
And I'll make it[br]more complicated.

1:17:33.055,1:17:36.140


1:17:36.140,1:17:40.478
Let's see how you[br]perform on this one.

1:17:40.478,1:17:47.240


1:17:47.240,1:17:49.710
We avoided that one,[br]probably, on finals

1:17:49.710,1:17:52.478
because I think the[br]majority of students

1:17:52.478,1:17:58.060
wouldn't have understood what[br]theorem they needed to apply.

1:17:58.060,1:17:59.690
It looks a little bit scary.

1:17:59.690,1:18:01.910
But let's say that I've[br]given you the hint,

1:18:01.910,1:18:05.003
apply Greens theorem[br]on the same path

1:18:05.003,1:18:10.380
integral, which is a circle[br]of origin 0 and radius r.

1:18:10.380,1:18:14.410
I now draw counterclockwise.

1:18:14.410,1:18:18.490
You apply Green's theorem and[br]you say, I know how to do this,

1:18:18.490,1:18:21.130
because now I know the theorem.

1:18:21.130,1:18:27.270
This is M. This is N. And I--[br]my t-shirt did not say M and N.

1:18:27.270,1:18:30.870
It said P and Q. Do you[br]want to put P and Q?

1:18:30.870,1:18:31.750
I put P and Q.

1:18:31.750,1:18:34.740
So I can-- I can have this[br]like it is on my t-shirt.

1:18:34.740,1:18:39.230
So this is going[br]to be P sub x-- no.

1:18:39.230,1:18:39.730
Q sub x.

1:18:39.730,1:18:40.730
Sorry.

1:18:40.730,1:18:44.560
M and N. So the second[br]one with respect to x.

1:18:44.560,1:18:48.560
The one that sticks to the y[br]is prime root respect to x.

1:18:48.560,1:18:54.344
The one that sticks to dx is[br]prime root with respect to y.

1:18:54.344,1:18:56.885
And I think one[br]time-- the one time

1:18:56.885,1:19:01.140
when that my friend and[br]colleague wrote that,

1:19:01.140,1:19:02.680
he did it differently.

1:19:02.680,1:19:05.595
He wrote something[br]like, just-- I'll

1:19:05.595,1:19:09.750
put-- I don't remember what.

1:19:09.750,1:19:10.800
He put this one.

1:19:10.800,1:19:13.506


1:19:13.506,1:19:17.470
Then the student[br]was used to dx/dy

1:19:17.470,1:19:19.250
and got completely confused.

1:19:19.250,1:19:25.625
So pay attention to[br]what you are saying.

1:19:25.625,1:19:29.400
Most of us write it[br]in x and y first.

1:19:29.400,1:19:32.960
And we can see that the[br]derivative with respect

1:19:32.960,1:19:38.870
to x of q, because that is[br]the one next to be the y.

1:19:38.870,1:19:42.446
When he gave it to me[br]like that, he messed up

1:19:42.446,1:19:44.760
everybody's notations.

1:19:44.760,1:19:46.000
No.

1:19:46.000,1:19:47.190
Good students steal data.

1:19:47.190,1:19:49.770
So you guys have to[br]put it in standard form

1:19:49.770,1:19:52.825
and pay attention to[br]what you are doing.

1:19:52.825,1:19:53.770
All right.

1:19:53.770,1:19:57.140
So that one form can[br]be swapped by people

1:19:57.140,1:19:58.620
who try to play games.

1:19:58.620,1:20:02.570


1:20:02.570,1:20:08.312
Now in this one-- So you[br]have q sub x minus b sub y.

1:20:08.312,1:20:15.990
You have 3x squared minus[br]minus, or just plus, 3y squared.

1:20:15.990,1:20:16.490
Good.

1:20:16.490,1:20:17.460
Wonderful.

1:20:17.460,1:20:20.370
Am I happy, do you[br]think I'm happy?

1:20:20.370,1:20:22.740
Why would I be so happy?

1:20:22.740,1:20:25.790
Why is this a happy thing?

1:20:25.790,1:20:27.740
I could have had[br]something more wild.

1:20:27.740,1:20:28.270
I don't.

1:20:28.270,1:20:30.210
I'm happy I don't.

1:20:30.210,1:20:31.760
Why am I so happy?

1:20:31.760,1:20:34.360
Let's see.

1:20:34.360,1:20:39.850
3 out over the disk.

1:20:39.850,1:20:41.710
Is this ringing a bell?

1:20:41.710,1:20:48.850


1:20:48.850,1:20:49.880
Yeah.

1:20:49.880,1:20:53.150
It's r squared if I[br]do this in former.

1:20:53.150,1:20:58.640
So if I do this in former,[br]its going to be rdr, d theta.

1:20:58.640,1:21:01.340
So life is not as[br]hard as you believe.

1:21:01.340,1:21:04.035
It can look like[br]a harder problem,

1:21:04.035,1:21:06.250
but in reality, it's not really.

1:21:06.250,1:21:11.970
So I have 3 times-- now, I[br]have r squared, I have r cubed.

1:21:11.970,1:21:16.780
r cubed dr d theta, r between.

1:21:16.780,1:21:20.560


1:21:20.560,1:21:30.710
r was between 0 and big[br]R. Theta will always

1:21:30.710,1:21:34.150
be between 0 and 2 pi.

1:21:34.150,1:21:42.316
So, I want you, without[br]me to compute the answer

1:21:42.316,1:21:44.454
and tell me what you got.

1:21:44.454,1:21:46.839
STUDENT: Just say it?

1:21:46.839,1:21:48.270
MAGDALENA TODA: Yep.

1:21:48.270,1:21:52.580
STUDENT: 3/2, pi[br]r to the fourth.

1:21:52.580,1:21:54.350
MAGDALENA TODA: So[br]how did you do that?

1:21:54.350,1:21:56.610
You said, r to the[br]4 over 4, coming

1:21:56.610,1:22:00.470
from integration times the 2 pi,[br]coming from integration times

1:22:00.470,1:22:01.966
3.

1:22:01.966,1:22:05.312
Are you guys with me?

1:22:05.312,1:22:08.690
Is everybody with me on this?

1:22:08.690,1:22:12.130
OK so, we will simplify[br]the answer, we'll do that.

1:22:12.130,1:22:16.696
What regard is the[br]radius of the disk?

1:22:16.696,1:22:18.320
STUDENT: How did he[br]solve that integral

1:22:18.320,1:22:20.008
without switching the poles?

1:22:20.008,1:22:26.456


1:22:26.456,1:22:28.936
MAGDALENA TODA: It would[br]have been a killer.

1:22:28.936,1:22:30.920
Let me write it out.

1:22:30.920,1:22:32.408
[LAUGHTER]

1:22:32.408,1:22:34.910
Because you want to[br]write it out, of course.

1:22:34.910,1:22:40.550
OK, 3 integral, integral[br]x squared plus y

1:22:40.550,1:22:45.450
squared, dy/dx, just to make[br]my life a little bit funnier,

1:22:45.450,1:22:50.280
and then y between minus[br]square root-- you're

1:22:50.280,1:22:52.060
looking for trouble, huh?

1:22:52.060,1:22:59.580
Y squared minus x squared to[br]r squared minus s squared.

1:22:59.580,1:23:01.250
And again, you could[br]do what you just

1:23:01.250,1:23:04.536
said, split into four integrals[br]over four different domains,

1:23:04.536,1:23:07.440
or two up and down.

1:23:07.440,1:23:11.950
And minus r and are[br]you guys with me?

1:23:11.950,1:23:14.890
And then, when you go[br]and integrate that,

1:23:14.890,1:23:22.960
you integrate with respect[br]to y-- [INAUDIBLE].

1:23:22.960,1:23:25.300
Well he's right,[br]so you can get x

1:23:25.300,1:23:27.835
squared y plus y cubed over 3.

1:23:27.835,1:23:30.770


1:23:30.770,1:23:34.240
Between those points, minus 12.

1:23:34.240,1:23:36.120
And from that moment,[br]that would just

1:23:36.120,1:23:39.563
leave it and go for a walk.

1:23:39.563,1:23:43.260
I will not have the[br]patience to do this.

1:23:43.260,1:23:44.800
Just a second, Matthew.

1:23:44.800,1:23:46.680
For this kind of[br]stuff, of course

1:23:46.680,1:23:50.410
I could put this in Maple.

1:23:50.410,1:23:54.150
You know Maple has these[br]little interactive fields,

1:23:54.150,1:23:55.940
like little squares?

1:23:55.940,1:23:58.940
And you go inside there[br]and add your endpoints.

1:23:58.940,1:24:02.740
And even if it looks very ugly,[br]Maple will spit you the answer.

1:24:02.740,1:24:05.668
If you know your[br]syntax and do it right,

1:24:05.668,1:24:07.620
even if you don't[br]switch to polar

1:24:07.620,1:24:10.060
coordinates or put[br]it in Cartesian.

1:24:10.060,1:24:12.980
Give it the right data, and[br]it's going to spit the answer.

1:24:12.980,1:24:13.942
Yes, Matthew?

1:24:13.942,1:24:16.106
STUDENT: I was[br]out of the room, I

1:24:16.106,1:24:19.242
was wondering why[br]it's now y cubed.

1:24:19.242,1:24:21.450
MAGDALENA TODA: Because if[br]you integrate with respect

1:24:21.450,1:24:24.140
to y first--

1:24:24.140,1:24:27.460
STUDENT: Because when I[br]walked out, it was negative y.

1:24:27.460,1:24:29.084
MAGDALENA TODA: If[br]I didn't put minus.

1:24:29.084,1:24:30.250
STUDENT: It's a new problem.

1:24:30.250,1:24:32.120
That's what he's confused about.

1:24:32.120,1:24:34.700
He walked out of the room[br]during the previous problem

1:24:34.700,1:24:36.420
and came back after this one.

1:24:36.420,1:24:37.709
And now he's confused.

1:24:37.709,1:24:40.000
MAGDALENA TODA: You don't[br]care about what I just asked?

1:24:40.000,1:24:40.380
STUDENT: Oh.

1:24:40.380,1:24:40.880
No.

1:24:40.880,1:24:44.125


1:24:44.125,1:24:45.942
I like the polar coordinates.

1:24:45.942,1:24:47.650
MAGDALENA TODA: Let[br]me ask you a question

1:24:47.650,1:24:50.100
before I talk any further.

1:24:50.100,1:24:53.250
I was about to put a plus here.

1:24:53.250,1:24:56.460
What would have been the problem[br]if I had put a plus here?

1:24:56.460,1:24:59.810


1:24:59.810,1:25:02.560
If I worked this out,[br]I would have gotten

1:25:02.560,1:25:05.930
x squared minus y squared.

1:25:05.930,1:25:08.300
Would that have been[br]the end of the world?

1:25:08.300,1:25:10.040
No.

1:25:10.040,1:25:16.156
But it would have complicated[br]my life a little bit more.

1:25:16.156,1:25:20.990
Let's do that one as well.

1:25:20.990,1:25:22.422
STUDENT: I was[br]just curious of how

1:25:22.422,1:25:25.247
you do any of these problems[br]when you can't switch to polar.

1:25:25.247,1:25:27.830
MAGDALENA TODA: Right, let's see[br]what-- because Actually, even

1:25:27.830,1:25:32.440
in this case, life is not so[br]hard, not as hard as you think.

1:25:32.440,1:25:34.750
The persistence in that matters.

1:25:34.750,1:25:37.620
You never give up on a[br]problem that freaks you out.

1:25:37.620,1:25:41.240
That's the definition[br]of a mathematician.

1:25:41.240,1:25:48.380
3x squared minus 3y[br]squared over dx/dy.

1:25:48.380,1:25:50.270
Do it slowly because[br]I'm not in a hurry.

1:25:50.270,1:25:55.670
We are almost done with 13.4.

1:25:55.670,1:25:57.407
This is OK, right?

1:25:57.407,1:25:59.355
Just the minus sign again?

1:25:59.355,1:26:01.303
STUDENT: Well not[br]the minus sign.

1:26:01.303,1:26:03.738
I was just wondering because[br]in the previous problem

1:26:03.738,1:26:07.147
you were doing the ellipse, you[br]started out with the equation

1:26:07.147,1:26:08.608
with the negative y--

1:26:08.608,1:26:10.556
MAGDALENA TODA:[br]For this one that's

1:26:10.556,1:26:14.844
just the limit that says that[br]this is the go double integral

1:26:14.844,1:26:18.920
of the area of the domain.

1:26:18.920,1:26:23.120
It's just a consequence--[br]or correlate if you want.

1:26:23.120,1:26:27.715
It's a consequence[br]of Green's theorem.

1:26:27.715,1:26:31.100
When you forget that consequence[br]of Green's theorem and we say

1:26:31.100,1:26:32.005
goodbye to that.

1:26:32.005,1:26:35.810
But while you were out,[br]this is Green's theorem.

1:26:35.810,1:26:38.635
The real Green's theorem,[br]the one that was a teacher.

1:26:38.635,1:26:41.310
There are several[br]Greens I can give you.

1:26:41.310,1:26:43.280
The famous Green[br]theorem is the one

1:26:43.280,1:26:46.850
I said when you have--[br]this is what we apply here.

1:26:46.850,1:26:50.972
The integral of M dx plus M dy.

1:26:50.972,1:27:00.820
You have a double integral of[br]M sub x minus M sub y over c.

1:27:00.820,1:27:03.450


1:27:03.450,1:27:07.690
So I'm assuming we would have[br]had this case of maybe me not

1:27:07.690,1:27:10.990
paying attention, or[br]being mean and not wanting

1:27:10.990,1:27:14.484
to give you a simple problem.

1:27:14.484,1:27:17.661
And what do you[br]do in such a case?

1:27:17.661,1:27:19.660
It's not obvious to[br]everybody, but you will see.

1:27:19.660,1:27:22.050
It's so pretty at some[br]point, if you know

1:27:22.050,1:27:24.034
how to get out of the mess.

1:27:24.034,1:27:27.010


1:27:27.010,1:27:30.820
I was already thinking, but[br]I'm using polar coordinates.

1:27:30.820,1:27:35.620
So that's arc of sine, so I[br]have to go back to the basics.

1:27:35.620,1:27:40.000
If I go back to the[br]basics, ideas come to me.

1:27:40.000,1:27:41.760
Right?

1:27:41.760,1:27:45.930
So, OK.

1:27:45.930,1:27:51.560
r-- let's put dr d theta,[br]just to get rid of it,

1:27:51.560,1:27:53.780
because it's on my nerves.

1:27:53.780,1:28:00.370
This is 0 to 2 pi,[br]this is 0 to r.

1:28:00.370,1:28:03.340
And now, you say,[br]OK, in our mind,

1:28:03.340,1:28:06.960
because we are lazy[br]people, plug in those

1:28:06.960,1:28:10.970
and pull out what you can.

1:28:10.970,1:28:14.970
One 3 out equals for what?

1:28:14.970,1:28:18.400
Inside, you have r squared.

1:28:18.400,1:28:20.720
Do you agree?

1:28:20.720,1:28:29.510
And times your favorite[br]expression, which is cosine

1:28:29.510,1:28:32.550
squared theta, minus[br]i squared theta.

1:28:32.550,1:28:34.520
And you're going to ask me why.

1:28:34.520,1:28:36.070
You shouldn't ask me why.

1:28:36.070,1:28:40.305
You just square these[br]and subtract them,

1:28:40.305,1:28:44.050
and see what in the world[br]you're going to get.

1:28:44.050,1:28:48.290
Because you get r squared[br]times cosine squared,

1:28:48.290,1:28:49.510
minus i squared.

1:28:49.510,1:28:51.434
I'm too lazy to write[br]down the argument.

1:28:51.434,1:28:52.850
But you know we[br]have trigonometry.

1:28:52.850,1:28:55.670


1:28:55.670,1:28:58.170
Yes, you see why it's[br]important for you

1:28:58.170,1:29:02.010
to learn trigonometry[br]when you are little.

1:29:02.010,1:29:05.730
You may be 50 or[br]60, in high school,

1:29:05.730,1:29:09.188
or you may be freshman year.

1:29:09.188,1:29:11.980
I don't care when, but you[br]have to learn that this is

1:29:11.980,1:29:14.306
the cosine of the double angle.

1:29:14.306,1:29:16.240
How many of you remember that?

1:29:16.240,1:29:18.540
Maybe you learned that?

1:29:18.540,1:29:19.460
Remember that?

1:29:19.460,1:29:21.400
OK.

1:29:21.400,1:29:25.800
I don't blame you at all[br]when you don't remember,

1:29:25.800,1:29:30.920
because since I've been[br]the main checker of finals

1:29:30.920,1:29:39.490
for the past five years--[br]it's a lot of finals.

1:29:39.490,1:29:40.690
Yeah, the i is there.

1:29:40.690,1:29:42.840
That's exactly what[br]I wanted to tell you,

1:29:42.840,1:29:46.450
that's why I left some room.

1:29:46.450,1:29:52.566
This data would be t.

1:29:52.566,1:29:56.370
The double angle formula did[br]not appear on many finals.

1:29:56.370,1:29:58.110
And I was thinking[br]it's a period.

1:29:58.110,1:29:59.960
When I ask the[br]instructors, generally they

1:29:59.960,1:30:06.550
say students have trouble[br]remembering or understanding

1:30:06.550,1:30:10.130
this later on, by[br]avoiding the issue,

1:30:10.130,1:30:14.012
you sort of bound to it for[br]the first time in Cal 2,

1:30:14.012,1:30:16.868
because there are any[br]geometric formulas.

1:30:16.868,1:30:21.634
And then, you bump again[br]inside it in Cal 3.

1:30:21.634,1:30:23.310
And it never leaves you.

1:30:23.310,1:30:27.560
So this, just knowing this[br]will help you so much.

1:30:27.560,1:30:30.852
Let me put the r nicely here.

1:30:30.852,1:30:34.123
And now finally, we know[br]how to solve it, because I'm

1:30:34.123,1:30:35.289
going to go ahead and erase.

1:30:35.289,1:30:44.660


1:30:44.660,1:30:49.250
So why it is good for us is[br]that-- as Matthew observed

1:30:49.250,1:30:52.810
a few moments ago,[br]whenever you have

1:30:52.810,1:30:56.730
a product of a function, you[br]not only in a function in theta

1:30:56.730,1:31:01.170
only, your life becomes easier[br]because you can separate them

1:31:01.170,1:31:03.050
between the rhos.

1:31:03.050,1:31:04.190
In two different products.

1:31:04.190,1:31:05.950
So that's would be this theorem.

1:31:05.950,1:31:11.270
And you have 3 times-- the[br]part that depends only on r,

1:31:11.270,1:31:13.990
and the part that depends[br]only on theta, let's

1:31:13.990,1:31:14.890
put them separate.

1:31:14.890,1:31:22.170
We need theta, and[br]dr. And what do you

1:31:22.170,1:31:24.520
integrate when you integrate?

1:31:24.520,1:31:25.470
r cubed.

1:31:25.470,1:31:29.100
Attention, do not do rr.

1:31:29.100,1:31:30.850
From 0 to r.

1:31:30.850,1:31:32.360
OK?

1:31:32.360,1:31:33.651
STUDENT: And cosine theta?

1:31:33.651,1:31:39.550
MAGDALENA TODA: And then you[br]have a 0 to 2 pi, cosine 2.

1:31:39.550,1:31:42.126
now, let me give[br]you-- Let me tell you

1:31:42.126,1:31:44.875
what it is, because when[br]I was young, I was naive

1:31:44.875,1:31:48.100
and I always started with that.

1:31:48.100,1:31:52.490
You should always start with the[br]part, the trig part in theta.

1:31:52.490,1:31:54.030
Because that becomes 0.

1:31:54.030,1:31:56.552
So no matter how[br]ugly this is, I've

1:31:56.552,1:31:59.790
had professors who are[br]playing games with us,

1:31:59.790,1:32:03.800
and they were giving us[br]some extremely ugly thing

1:32:03.800,1:32:06.470
that would take you forever[br]for you to integrate.

1:32:06.470,1:32:09.290
Or sometimes, it would have[br]been impossible to integrate.

1:32:09.290,1:32:11.570
But then, the whole[br]thing would have been 0

1:32:11.570,1:32:14.295
because when you[br]integrate cosine 2 theta,

1:32:14.295,1:32:16.610
it goes to sine theta.

1:32:16.610,1:32:20.600
Sine 2 theta at 2 pi and 0[br]are the same things, 0 minus 0

1:32:20.600,1:32:21.120
equals z.

1:32:21.120,1:32:23.800
So the answer is z.

1:32:23.800,1:32:27.040
I cannot tell you how many[br]professors I've had who will

1:32:27.040,1:32:28.569
play this game with us.

1:32:28.569,1:32:30.360
They give us something[br]that discouraged us.

1:32:30.360,1:32:34.010
No, it's not a piece of cake,[br]compared to what I have.

1:32:34.010,1:32:37.450
Some integral value[br]will go over two lines,

1:32:37.450,1:32:40.490
with a huge polynomial[br]or something.

1:32:40.490,1:32:44.000
But in the end, the integral[br]was 0 for such a result. Yes?

1:32:44.000,1:32:45.250
STUDENT: So I have a question.

1:32:45.250,1:32:49.700
Could we take that force and[br]prove that it was conservative?

1:32:49.700,1:32:54.940
MAGDALENA TODA: So now[br]that I'm questioning this,

1:32:54.940,1:32:59.660
I'm not questioning[br]you, but I-- is

1:32:59.660,1:33:05.630
the force, that is with you--[br]what is the original force

1:33:05.630,1:33:08.360
that Alex is talking about?

1:33:08.360,1:33:17.000
If I take y cubed i plus x cubed[br]j-- and you have to be careful.

1:33:17.000,1:33:18.915
Is this conservative?

1:33:18.915,1:33:22.670


1:33:22.670,1:33:25.130
STUDENT: Yeah.

1:33:25.130,1:33:27.740
MAGDALENA TODA: Really?

1:33:27.740,1:33:30.936
Why would we pick[br]a conservative?

1:33:30.936,1:33:33.814
STUDENT: Y squared plus[br]x squared over 2 is--

1:33:33.814,1:33:35.605
MAGDALENA TODA: Why is[br]it not conservative?

1:33:35.605,1:33:38.860


1:33:38.860,1:33:40.560
IT doesn't pass the hole test.

1:33:40.560,1:33:43.590


1:33:43.590,1:33:48.280
So p sub y is not[br]equal to q sub x.

1:33:48.280,1:33:52.090
If you primed this with respect[br]to y, you get that dy squared.

1:33:52.090,1:33:54.640
Prime this with this respect[br]to x, you get 3x squared.

1:33:54.640,1:33:57.430
So it's not concerned with him.

1:33:57.430,1:34:00.642
And still, I'm[br]getting-- it's a loop,

1:34:00.642,1:34:04.520
and I'm getting a 0, sort[br]of like I would expect it

1:34:04.520,1:34:07.530
I had any dependence of that.

1:34:07.530,1:34:09.250
What is the secret here?

1:34:09.250,1:34:14.360
STUDENT: That is conservative,[br]given a condition.

1:34:14.360,1:34:16.730
MAGDALENA TODA: Yes,[br]given a condition

1:34:16.730,1:34:21.170
that your x and y are moving[br]on the serpent's circle.

1:34:21.170,1:34:25.900
And that happens, because this[br]is a symmetric expression,

1:34:25.900,1:34:28.420
and x and y are[br]moving on a circle,

1:34:28.420,1:34:31.090
and one is the cosine theta[br]and one is sine theta.

1:34:31.090,1:34:35.160
So in the end, it[br]simplifies out.

1:34:35.160,1:34:40.280
But in general, if I would[br]have this kind of problem--

1:34:40.280,1:34:43.950
if somebody asked me is this[br]conservative, the answer is no.

1:34:43.950,1:34:46.125
Let me give you a[br]few more examples.

1:34:46.125,1:34:57.620


1:34:57.620,1:35:14.765
One example that maybe will look[br]hard to most people is here.

1:35:14.765,1:35:36.545


1:35:36.545,1:35:49.750
The vector value function[br]given by f of x, y incline,

1:35:49.750,1:35:51.950
are two values.

1:35:51.950,1:35:54.966
No, I mean define two[br]values of [INAUDIBLE].

1:35:54.966,1:36:17.040


1:36:17.040,1:36:19.170
A typical exam problem.

1:36:19.170,1:36:22.980
And I saw it at[br]Texas A&M, as well.

1:36:22.980,1:36:27.610
So maybe some people like this[br]kind of a, b, c, d problem.

1:36:27.610,1:36:28.836
Is f conservative?

1:36:28.836,1:36:37.220


1:36:37.220,1:36:38.397
STUDENT: Yep

1:36:38.397,1:36:39.855
MAGDALENA TODA:[br]You already did it?

1:36:39.855,1:36:41.160
Good for you guys.

1:36:41.160,1:36:45.335
So if I gave you one that[br]has three components what

1:36:45.335,1:36:47.480
did you have to do?

1:36:47.480,1:36:49.860
Compute the curl.

1:36:49.860,1:36:52.460
You can, of course, compute[br]the curl also on this one

1:36:52.460,1:36:55.115
and have 0 for the[br]third component.

1:36:55.115,1:37:01.820
But the simplest thing[br]is to do f1 and f2.

1:37:01.820,1:37:07.040
f1 prime with respect to y[br]equals f2 prime with respect

1:37:07.040,1:37:08.090
to x.

1:37:08.090,1:37:12.870
So I'm going to[br]make a smile here.

1:37:12.870,1:37:16.310
And you realize that the authors[br]of such a problem, whether they

1:37:16.310,1:37:21.270
are at Tech or at Texas[br]A&M. They do that on purpose

1:37:21.270,1:37:28.390
so that you can use this[br]result to the next level.

1:37:28.390,1:38:04.070
And they're saying compute[br]the happy u over the curve

1:38:04.070,1:38:23.070
x cubed and y cubed equals 8 on[br]the path that connects points

1:38:23.070,1:38:29.375
2, 1 and 1, 2 in [INAUDIBLE].

1:38:29.375,1:38:39.275


1:38:39.275,1:38:46.205
Does this integral depend on f?

1:38:46.205,1:38:51.140


1:38:51.140,1:38:52.112
State why.

1:38:52.112,1:38:58.260


1:38:58.260,1:39:04.540
And you see, they don't tell[br]you find the scalar potential.

1:39:04.540,1:39:06.920
Which is bad, and[br]many of you will

1:39:06.920,1:39:09.115
be able to see it[br]because you have

1:39:09.115,1:39:14.000
good mathematical intuition,[br]and a computer process

1:39:14.000,1:39:17.290
planning in the background[br]over all the other processes.

1:39:17.290,1:39:18.900
We are very visual people.

1:39:18.900,1:39:22.310
If you realize that every time[br]just there with each other

1:39:22.310,1:39:25.720
through the classroom, there[br]are hundreds of distractions.

1:39:25.720,1:39:28.165
There's the screen,[br]there is somebody

1:39:28.165,1:39:31.210
who's next to you[br]who's sneezing,

1:39:31.210,1:39:34.568
all sorts of distractions.

1:39:34.568,1:39:37.860
Still, your computer[br]unit can still

1:39:37.860,1:39:41.200
function, trying to[br]integrate and find the scalar

1:39:41.200,1:39:42.450
potential, which is a miracle.

1:39:42.450,1:39:46.250
I don't know how we managed[br]to do that after all.

1:39:46.250,1:39:49.840
If you don't manage to do that,[br]what do you have to set up?

1:39:49.840,1:39:54.870
You have to say, find is[br]there-- well, you know there is.

1:39:54.870,1:39:59.900
So you're not going to question[br]the existence of the scalar

1:39:59.900,1:40:03.680
potential You know it exists,[br]but you don't know what it is.

1:40:03.680,1:40:09.750
What is f such that f sub[br]x would be 6xy plus 1,

1:40:09.750,1:40:14.486
and m sub y will be 3x squared?

1:40:14.486,1:40:18.315
And normally, you would[br]have to integrate backwards.

1:40:18.315,1:40:21.350
Now, I'll give you 10 seconds.

1:40:21.350,1:40:25.190
If in 10 seconds, you don't[br]find me a scalar potential,

1:40:25.190,1:40:27.120
I'm going to make you[br]integrate backwards.

1:40:27.120,1:40:31.360
So this is finding the scalar[br]potential by integration.

1:40:31.360,1:40:34.455
The way you should, if[br]you weren't very smart.

1:40:34.455,1:40:37.970
But I think you're[br]smart enough to smell

1:40:37.970,1:40:42.410
the potential-- Very good.

1:40:42.410,1:40:44.030
But what if you don't?

1:40:44.030,1:40:46.050
OK I'm asking.

1:40:46.050,1:40:50.475
So we had one or two[br]student who figured it out.

1:40:50.475,1:40:51.225
What if you don't?

1:40:51.225,1:40:55.240
If you don't, you can still do[br]perfectly fine on this problem.

1:40:55.240,1:41:01.120
Let's see how we do it[br]without seeing or guessing.

1:41:01.120,1:41:03.310
His brain was running[br]in the background.

1:41:03.310,1:41:05.060
He came up with the answer.

1:41:05.060,1:41:05.655
He's happy.

1:41:05.655,1:41:09.640
He can move on to[br]the next level.

1:41:09.640,1:41:12.270
STUDENT: Integrate both[br]sides with respect to r.

1:41:12.270,1:41:15.970
MAGDALENA TODA: Right, and[br]then mix and match them.

1:41:15.970,1:41:17.665
Make them in work.

1:41:17.665,1:41:20.550
So try to integrate[br]with respect to x.

1:41:20.550,1:41:25.970
6y-- or plus 1, I'm sorry guys.

1:41:25.970,1:41:28.970
And once you get it,[br]you're going to get--

1:41:28.970,1:41:32.130
STUDENT: 3x squared y plus x.

1:41:32.130,1:41:34.690
MAGDALENA TODA: And[br]plus a c of what?

1:41:34.690,1:41:37.970
And then take this fellow and[br]prime it with respect to y.

1:41:37.970,1:41:41.480
And you're going to[br]get-- it's not hard.

1:41:41.480,1:41:44.260
You're going to get dx[br]squared plus nothing,

1:41:44.260,1:41:50.170
plus c from the y, and it's[br]good because I gave you

1:41:50.170,1:41:51.330
a simple one.

1:41:51.330,1:41:54.390
So sometimes you can[br]have something here,

1:41:54.390,1:41:57.260
but in this case, it was just 0.

1:41:57.260,1:42:00.400
So c is kappa as a constant.

1:42:00.400,1:42:04.771
So instead of why we teach[br]found with a plus kappa here,

1:42:04.771,1:42:07.937
and it still does it.

1:42:07.937,1:42:12.810
So on such a problem,[br]I don't know,

1:42:12.810,1:42:18.066
but I think I would give equal[br]weights to it, B and C. Compute

1:42:18.066,1:42:20.874
the path integral[br]over the curve.

1:42:20.874,1:42:24.200
This is horrible, as[br]an increasing curve.

1:42:24.200,1:42:26.620
But I know that[br]there is a path that

1:42:26.620,1:42:28.680
connects the points 2, 1 and 1.

1:42:28.680,1:42:30.430
What I have to pay[br]attention to in my mind

1:42:30.430,1:42:33.390
is that these points[br]actually are on the curve.

1:42:33.390,1:42:36.170
And they are, because I[br]have 8 times 1 equals 8,

1:42:36.170,1:42:37.810
1 times 8 equals 8.

1:42:37.810,1:42:41.355
So while I was writing it,[br]I had to think a little bit

1:42:41.355,1:42:42.580
on the problem.

1:42:42.580,1:42:45.440
If you were to[br]draw-- well that's

1:42:45.440,1:42:48.450
for you have to find[br]out when you go home.

1:42:48.450,1:42:52.162
What do you think[br]this is going to be?

1:42:52.162,1:42:55.080


1:42:55.080,1:42:58.002
Actually, we have to[br]do it now, because it's

1:42:58.002,1:43:01.770
a lot simpler than[br]you think it is.

1:43:01.770,1:43:06.510
x and y will be positive,[br]I can also restrict that.

1:43:06.510,1:43:09.060
It looks horrible, but[br]it's actually much easier

1:43:09.060,1:43:09.990
than you think.

1:43:09.990,1:43:15.770
So how do I compute that path[br]integral that makes the points?

1:43:15.770,1:43:19.263
I'm going to have[br]fundamental there.

1:43:19.263,1:43:22.756


1:43:22.756,1:43:27.350
Which has f of x at q[br]minus f, with p, which

1:43:27.350,1:43:31.280
says that little f is here.

1:43:31.280,1:43:41.120
3x squared y plus[br]x at 2, 1 minus 3x

1:43:41.120,1:43:47.300
squared y plus x at 1, 2.

1:43:47.300,1:43:53.450
So all I have to do is[br]go ahead and-- do you

1:43:53.450,1:43:56.980
see what I'm actually doing?

1:43:56.980,1:43:57.790
It's funny.

1:43:57.790,1:44:02.460
Which one is the origin, and[br]which one is the endpoint?

1:44:02.460,1:44:03.830
The problem doesn't tell you.

1:44:03.830,1:44:07.302
It tells you only you are[br]connecting the two points.

1:44:07.302,1:44:10.164
But which one is the alpha,[br]and which one is the omega?

1:44:10.164,1:44:10.955
Where do you start?

1:44:10.955,1:44:12.550
You start here or[br]you start here?

1:44:12.550,1:44:16.460


1:44:16.460,1:44:17.070
OK.

1:44:17.070,1:44:19.150
Sort of arbitrary.

1:44:19.150,1:44:22.230
How do you handle this problem?

1:44:22.230,1:44:26.150
Depending on the direction--[br]pick one direction you move on

1:44:26.150,1:44:28.530
along the r, it's up to you.

1:44:28.530,1:44:31.690
And then you get an answer, and[br]if you change the direction,

1:44:31.690,1:44:34.250
what's going to happen[br]to the integral?

1:44:34.250,1:44:37.970
It's just change the[br]sign and that's all.

1:44:37.970,1:44:43.270
3 times 4, times 1, plus 2--[br]guys, keep an eye on my algebra

1:44:43.270,1:44:48.104
please, because I[br]don't want to mess up.

1:44:48.104,1:44:49.879
Am I right, here?

1:44:49.879,1:44:50.420
STUDENT: Yes.

1:44:50.420,1:44:52.300
MAGDALENA TODA: So how much?

1:44:52.300,1:44:53.990
14, is it?

1:44:53.990,1:44:56.315
STUDENT: It's 7.

1:44:56.315,1:44:57.315
MAGDALENA TODA: Minus 7.

1:44:57.315,1:45:06.491


1:45:06.491,1:45:06.990
Good.

1:45:06.990,1:45:07.880
Wonderful.

1:45:07.880,1:45:11.670
So we know what to get,[br]and we know this does not

1:45:11.670,1:45:13.180
depend on the fact.

1:45:13.180,1:45:16.770
How much blah, blah,[br]blah does the instructor

1:45:16.770,1:45:20.887
expect for you to get full[br]credit on the problem?

1:45:20.887,1:45:22.220
STUDENT: Just enough to explain.

1:45:22.220,1:45:23.845
MAGDALENA TODA: Just[br]enough to explain.

1:45:23.845,1:45:29.810
About 2 lines or 1 line saying[br]you can say anything really.

1:45:29.810,1:45:34.470
You can say this is the theorem[br]that either shows independence

1:45:34.470,1:45:35.861
of that integral.

1:45:35.861,1:45:43.310
If the force F vector value[br]function is conservative,

1:45:43.310,1:45:46.746
then this is what[br]you have to write.

1:45:46.746,1:45:49.252
This doesn't depend[br]on the path c.

1:45:49.252,1:45:51.450
And you apply the[br]fundamental theorem

1:45:51.450,1:45:54.290
of path integrals for[br]the scalar potential.

1:45:54.290,1:45:57.760
And that scalar potential[br]depends on the endpoints

1:45:57.760,1:45:59.660
that you're taking.

1:45:59.660,1:46:02.150
And the value of[br]the work depends--

1:46:02.150,1:46:06.290
the work depends only on the[br]scalar potential and the two

1:46:06.290,1:46:07.510
points.

1:46:07.510,1:46:08.320
That's enough.

1:46:08.320,1:46:09.780
That's more than enough.

1:46:09.780,1:46:13.816
What if somebody's[br]not good with wording?

1:46:13.816,1:46:17.170
I'm not going to write[br]her all that explanation.

1:46:17.170,1:46:21.650
I'm just going to say whatever.

1:46:21.650,1:46:25.100
I'm going to give[br]her the theorem

1:46:25.100,1:46:27.210
in mathematical compressed way.

1:46:27.210,1:46:30.960
And I don't care if she[br]understands it or not.

1:46:30.960,1:46:34.620
Even if you write this[br]formula with not much wording,

1:46:34.620,1:46:36.870
I still give you credit.

1:46:36.870,1:46:38.560
But I would prefer[br]that you give me

1:46:38.560,1:46:41.640
some sort of-- some[br]sort of explanation.

1:46:41.640,1:46:42.802
Yes, sir.

1:46:42.802,1:46:44.093
STUDENT: You said answer was 0.

1:46:44.093,1:46:45.801
Then it would have[br]been path independent?

1:46:45.801,1:46:50.860


1:46:50.860,1:46:53.630
MAGDALENA TODA: No, the[br]answer would not be for sure 0

1:46:53.630,1:46:56.770
if it was a longer loop.

1:46:56.770,1:46:58.870
If it were a longer[br]closed curve,

1:46:58.870,1:47:03.570
that way where it[br]starts, it ends.

1:47:03.570,1:47:07.470
Even if I take a weekly[br]road between the two points,

1:47:07.470,1:47:09.170
I still get 7, right?

1:47:09.170,1:47:11.290
That's the whole idea.

1:47:11.290,1:47:12.530
Am I clear about that?

1:47:12.530,1:47:14.710
Are we clear about that?

1:47:14.710,1:47:21.360
Let me ask you though,[br]how do you find out?

1:47:21.360,1:47:25.760
Because I don't know how[br]many of you figured out

1:47:25.760,1:47:28.526
what kind of curve that is.

1:47:28.526,1:47:32.600
And it looks like an enemy[br]to you, but there is a catch.

1:47:32.600,1:47:38.520
It's an old friend of[br]yours and you don't see it.

1:47:38.520,1:47:40.379
So what is the curve?

1:47:40.379,1:47:41.265
What is the curve?

1:47:41.265,1:47:46.650
And what is this arc of a[br]curve between 2, 1 and 1, 2?

1:47:46.650,1:47:48.230
Can we find out what that is?

1:47:48.230,1:47:49.350
Of course, or cubic.

1:47:49.350,1:47:50.450
It's a fake cubic.

1:47:50.450,1:47:53.780
It's a fake cubic--

1:47:53.780,1:47:56.224
STUDENT: To function together?

1:47:56.224,1:47:58.150
MAGDALENA TODA: Let's[br]see what this is.

1:47:58.150,1:48:03.060
xy cubed minus 2 cubed equals 0.

1:48:03.060,1:48:06.315
We were in fourth grade--[br]well, our teachers--

1:48:06.315,1:48:13.640
I think our teachers teach us[br]when we were little that this,

1:48:13.640,1:48:16.660
if you divided by a[br]minus- I wasn't little.

1:48:16.660,1:48:19.236
I was in high school.

1:48:19.236,1:48:21.132
Well, 14-year-old.

1:48:21.132,1:48:22.080
STUDENT: A cubed.

1:48:22.080,1:48:23.280
STUDENT: A squared.

1:48:23.280,1:48:24.370
MAGDALENA TODA: A squared.

1:48:24.370,1:48:26.010
STUDENT: Minus 2AB.

1:48:26.010,1:48:27.080
Plus 2AB.

1:48:27.080,1:48:28.960
MAGDALENA TODA: Very good.

1:48:28.960,1:48:30.870
Plus AB, not 2AB.

1:48:30.870,1:48:31.630
STUDENT: Oh, darn.

1:48:31.630,1:48:33.750
MAGDALENA TODA: Plus B squared.

1:48:33.750,1:48:34.890
Suppose you don't believe.

1:48:34.890,1:48:36.730
That proves this.

1:48:36.730,1:48:38.060
Let's multiply.

1:48:38.060,1:48:41.660
A cubed plus A squared[br]B plus AB squared.

1:48:41.660,1:48:44.020
I'm done with the[br]first multiplication.

1:48:44.020,1:48:50.366
Minus BA squared minus[br]AB squared minus B cubed.

1:48:50.366,1:48:52.290
Do they cancel out?

1:48:52.290,1:48:53.733
Yes.

1:48:53.733,1:48:55.180
Good.

1:48:55.180,1:48:57.605
Cancel out.

1:48:57.605,1:49:00.040
And cancel out.

1:49:00.040,1:49:01.560
Out, poof.

1:49:01.560,1:49:02.850
We've proved it, why?

1:49:02.850,1:49:08.700
Because maybe some of you--[br]nobody gave it to proof before.

1:49:08.700,1:49:11.590


1:49:11.590,1:49:17.510
So as an application,[br]what is this?

1:49:17.510,1:49:18.010
There.

1:49:18.010,1:49:19.100
Who is A and who is B?

1:49:19.100,1:49:23.790
A is xy, B is 2.

1:49:23.790,1:49:34.440
So you have xy minus 2 times[br]all this fluffy guy, xy

1:49:34.440,1:49:42.381
squared plus 2xy plus--

1:49:42.381,1:49:44.510
STUDENT: 4.

1:49:44.510,1:49:45.260
MAGDALENA TODA: 4.

1:49:45.260,1:49:49.292
And I also said, because[br]I was sneaky, that's why.

1:49:49.292,1:49:54.510
To make your life easier[br]or harder. xy is positive.

1:49:54.510,1:49:57.783
When I said xy was positive,[br]what was I intending?

1:49:57.783,1:50:03.320
I was intending for you to see[br]that this cannot be 0 ever.

1:50:03.320,1:50:07.678
So the only possible[br]for you to have 0 here

1:50:07.678,1:50:10.070
is when xy equals 2.

1:50:10.070,1:50:14.200
And xy equals 2 is a[br]much simpler curve.

1:50:14.200,1:50:17.720
And I want to know[br]if you realize

1:50:17.720,1:50:22.160
that this will have the points[br]2,1 and 1, 2 staring at you.

1:50:22.160,1:50:23.370
Have a nice day today.

1:50:23.370,1:50:25.201
Take care.

1:50:25.201,1:50:26.632
And good luck.

1:50:26.632,1:50:31.880


1:50:31.880,1:50:34.075
What is it?

1:50:34.075,1:50:34.950
STUDENT: [INAUDIBLE].

1:50:34.950,1:50:36.760
MAGDALENA TODA:[br]Some sort of animal.

1:50:36.760,1:50:38.380
It's a curve, a linear curve.

1:50:38.380,1:50:42.276
It's not a line.

1:50:42.276,1:50:43.250
What is it?

1:50:43.250,1:50:47.633
Talking about conics because[br]I was talking a little bit

1:50:47.633,1:50:49.581
with Casey about conics.

1:50:49.581,1:50:52.503
Is this a conic?

1:50:52.503,1:50:53.477
Yeah.

1:50:53.477,1:50:55.430
What is a conic?

1:50:55.430,1:50:59.970
A conic is any kind of[br]curve that looks like this.

1:50:59.970,1:51:04.530
In general form--[br]oh my god, ABCD.

1:51:04.530,1:51:08.190
Now I got my ABC[br]plus f equals 0.

1:51:08.190,1:51:10.030
This is a conic in plane.

1:51:10.030,1:51:13.650
My conic is missing[br]everything else.

1:51:13.650,1:51:16.020
And B is 0.

1:51:16.020,1:51:18.830
And there is a way where[br]you-- I showed you how you

1:51:18.830,1:51:21.730
know what kind of conic it is.

1:51:21.730,1:51:28.130
A, A, B, B, C. A is[br]positive is-- no, A is 0,

1:51:28.130,1:51:31.740
B is-- it should be 2 here.

1:51:31.740,1:51:34.158
So you split this in half.

1:51:34.158,1:51:37.110
1/2, 1/2, and 0.

1:51:37.110,1:51:41.086
The determinant of this is[br]negative, the discriminant.

1:51:41.086,1:51:43.653
That's why we call it[br]discriminant about the conic.

1:51:43.653,1:51:44.900
So it cannot be an ellipse.

1:51:44.900,1:51:46.491
So what the heck is it?

1:51:46.491,1:51:47.365
STUDENT: [INAUDIBLE].

1:51:47.365,1:51:48.698
MAGDALENA TODA: Well, I'm silly.

1:51:48.698,1:51:50.180
I should have pulled out for y.

1:51:50.180,1:51:53.040


1:51:53.040,1:51:56.725
And I knew that it[br]goes down like 1/x.

1:51:56.725,1:52:00.590
But I'm asking you, why in[br]the world is that a conic?

1:52:00.590,1:52:01.670
Because you say, wait.

1:52:01.670,1:52:03.230
Wait a minute.

1:52:03.230,1:52:09.900
I know this curve since I was[br]five year old in kindergarten.

1:52:09.900,1:52:13.150
And this is the point 2, 1.

1:52:13.150,1:52:16.510


1:52:16.510,1:52:17.240
It's on it.

1:52:17.240,1:52:22.930
And there is a symmetric[br]point for your pleasure here.

1:52:22.930,1:52:25.120
1, 2.

1:52:25.120,1:52:26.640
And between the[br]two points, there

1:52:26.640,1:52:31.530
is just one arc of a curve.

1:52:31.530,1:52:34.120
And this is the path that[br]you are dragging some object

1:52:34.120,1:52:35.390
with force f.

1:52:35.390,1:52:37.980
You are computing[br]the work of a-- maybe

1:52:37.980,1:52:40.990
you're computing the work of[br]a neutron between those two

1:52:40.990,1:52:42.930
locations.

1:52:42.930,1:52:43.786
It's a--

1:52:43.786,1:52:44.642
STUDENT: Hyperbola?

1:52:44.642,1:52:46.000
MAGDALENA TODA: Hyperbola.

1:52:46.000,1:52:47.030
Why Nitish?

1:52:47.030,1:52:47.768
Yes, sir.

1:52:47.768,1:52:49.518
STUDENT: I was just[br]wondering, couldn't we

1:52:49.518,1:52:51.696
have gone to xy equals 2 plane?

1:52:51.696,1:52:52.820
STUDENT: Yeah, way quicker.

1:52:52.820,1:52:55.259
STUDENT: x cubed, y[br]cubed equals 2 cubed.

1:52:55.259,1:52:56.550
Then you'd just do both sides--

1:52:56.550,1:52:57.250
MAGDALENA TODA:[br]That's what I did.

1:52:57.250,1:52:57.865
STUDENT: The cubed root.

1:52:57.865,1:52:59.400
MAGDALENA TODA:[br]Didn't I do that?

1:52:59.400,1:53:02.840
No, because in[br]general, it's not--

1:53:02.840,1:53:07.482
you cannot say if and only[br]if xy equals 2 in general.

1:53:07.482,1:53:10.600
You have to write to[br]decompose the polynomial.

1:53:10.600,1:53:12.450
You were lucky[br]this was positive.

1:53:12.450,1:53:15.049
STUDENT: Well, because[br]we divided by x cubed.

1:53:15.049,1:53:16.590
We could have just[br]divided everything

1:53:16.590,1:53:18.610
by x cubed, and then taken[br]the cube root of both sides.

1:53:18.610,1:53:20.401
MAGDALENA TODA: He's[br]saying the same thing.

1:53:20.401,1:53:23.931
But in mathematics, we don't--[br]let me show you something.

1:53:23.931,1:53:25.472
STUDENT: It would[br]work for this case,

1:53:25.472,1:53:26.860
but not necessarily[br]for all cases?

1:53:26.860,1:53:27.735
MAGDALENA TODA: Yeah.

1:53:27.735,1:53:39.140
Let me show you some other[br]example where you just-- how

1:53:39.140,1:53:41.130
do you solve this equation?

1:53:41.130,1:53:46.020
By the way, a math[br]field test is coming.

1:53:46.020,1:53:48.270
No, only if you're a math major.

1:53:48.270,1:53:50.510
Sorry, junior or senior.

1:53:50.510,1:53:53.360
In one math field test,[br]you don't have to take it.

1:53:53.360,1:53:56.880
But some people who[br]go to graduate school,

1:53:56.880,1:54:01.495
if they take the math field[br]test, that replaces the GRE,

1:54:01.495,1:54:03.475
if the school agrees.

1:54:03.475,1:54:06.450
So there was this questions,[br]how many roots does it have

1:54:06.450,1:54:07.770
and what kind?

1:54:07.770,1:54:11.380
Two are imaginary[br]and one is real.

1:54:11.380,1:54:14.741
But everybody said[br]it only had one root.

1:54:14.741,1:54:17.030
How can it have one root[br]if it's a cubic equation?

1:54:17.030,1:54:18.580
So one root.

1:54:18.580,1:54:20.596
x1 is 1.

1:54:20.596,1:54:23.081
The other two are imaginary.

1:54:23.081,1:54:24.330
This is the case in this also.

1:54:24.330,1:54:26.450
You have some imaginary roots.

1:54:26.450,1:54:31.440
So those roots[br]are funny, but you

1:54:31.440,1:54:35.740
would have to[br]solve this equation

1:54:35.740,1:54:42.100
because this is x minus 1[br]times x squared plus x plus 1.

1:54:42.100,1:54:45.580
So the roots are minus[br]1, plus minus square root

1:54:45.580,1:54:52.230
of b squared minus 4ac[br]over 2, which are minus 1

1:54:52.230,1:54:57.290
plus minus square[br]root of 3i over 2.

1:54:57.290,1:55:01.230
Do you guys know[br]how they are called?

1:55:01.230,1:55:05.600
You know them because in[br]some countries we learn them.

1:55:05.600,1:55:07.982
But do you know the notations?

1:55:07.982,1:55:09.190
STUDENT: What they call them?

1:55:09.190,1:55:10.065
MAGDALENA TODA: Yeah.

1:55:10.065,1:55:12.190


1:55:12.190,1:55:14.150
There is a Greek letter.

1:55:14.150,1:55:15.872
STUDENT: Iota.

1:55:15.872,1:55:17.330
MAGDALENA TODA: In[br]India, probably.

1:55:17.330,1:55:19.060
In my country, it was omega.

1:55:19.060,1:55:19.874
But I don't think--

1:55:19.874,1:55:20.874
STUDENT: In India, iota.

1:55:20.874,1:55:22.875


1:55:22.875,1:55:25.810
MAGDALENA TODA: But we call[br]them omega and omega squared.

1:55:25.810,1:55:28.640
Because one is the[br]square of the other.

1:55:28.640,1:55:30.140
They are, of course,[br]both imaginary.

1:55:30.140,1:55:35.662
And we call this the[br]cubic roots of unity.

1:55:35.662,1:55:39.110


1:55:39.110,1:55:41.970
You say Magdalena, why would[br]you talk about imaginary numbers

1:55:41.970,1:55:43.880
when everything is real?

1:55:43.880,1:55:44.500
OK.

1:55:44.500,1:55:48.140
It's real for the time being[br]while you are still with me.

1:55:48.140,1:55:50.330
The moment you're going[br]to say goodbye to me

1:55:50.330,1:55:55.120
and you know in 3350 your[br]life is going to change.

1:55:55.120,1:55:57.340
In that course,[br]they will ask you

1:55:57.340,1:56:02.571
to solve this equation just like[br]we asked all our 3350 students.

1:56:02.571,1:56:05.050
To our surprise,[br]the students don't

1:56:05.050,1:56:06.880
know what imaginary roots are.

1:56:06.880,1:56:07.940
Many, you know.

1:56:07.940,1:56:10.376
You will refresh your memory.

1:56:10.376,1:56:12.000
But the majority of[br]the students didn't

1:56:12.000,1:56:15.140
know how to get to[br]those imaginary numbers.

1:56:15.140,1:56:20.360
You're going to need to not[br]only use them, but also express

1:56:20.360,1:56:22.557
these in terms of trigonometry.

1:56:22.557,1:56:25.540


1:56:25.540,1:56:31.730
So just out of curiosity, since[br]I am already talking to you,

1:56:31.730,1:56:34.885
and since I've preparing you a[br]little bit for the differential

1:56:34.885,1:56:38.662
equations class where you[br]have lots of electric circuits

1:56:38.662,1:56:41.430
and applications[br]of trigonometry,

1:56:41.430,1:56:45.780
these imaginary numbers[br]can also be put-- they

1:56:45.780,1:56:50.726
are in general of the form[br]a plus ib. a plus minus ib.

1:56:50.726,1:56:54.868
And we agree that in[br]3350 you have to do that.

1:56:54.868,1:56:56.945
Out of curiosity,[br]is there anybody

1:56:56.945,1:57:02.926
who knows the trigonometric[br]form of these complex numbers?

1:57:02.926,1:57:06.300
STUDENT: Isn't it r e to the j--

1:57:06.300,1:57:10.170


1:57:10.170,1:57:14.240
MAGDALENA TODA: So you would[br]have exactly what he says here.

1:57:14.240,1:57:18.485
This number will[br]be-- if it's plus.

1:57:18.485,1:57:20.949
r e to the i theta.

1:57:20.949,1:57:26.220
He knows a little bit[br]more than most students.

1:57:26.220,1:57:34.280
And that is cosine[br]theta plus i sine theta.

1:57:34.280,1:57:36.810
Can you find me the[br]angle theta if I

1:57:36.810,1:57:42.870
want to write cosine theta[br]plus i sine theta or cosine

1:57:42.870,1:57:46.390
theta minus i sine theta?

1:57:46.390,1:57:50.210
Can you find me[br]the angle of theta?

1:57:50.210,1:57:50.965
Is it hard?

1:57:50.965,1:57:53.290
Is it easy?

1:57:53.290,1:57:54.990
What in the world is it?

1:57:54.990,1:57:59.810


1:57:59.810,1:58:01.400
Think like this.

1:58:01.400,1:58:04.440
We are done with this[br]example, but I'm just

1:58:04.440,1:58:08.170
saying some things that[br]will help you in 3350.

1:58:08.170,1:58:11.860
If you want cosine[br]theta to be minus 1/2

1:58:11.860,1:58:20.553
and you want sine theta to be[br]root 3 over 2, which quadrant?

1:58:20.553,1:58:22.900
Which quadrant are you in?

1:58:22.900,1:58:24.120
STUDENT: Second.

1:58:24.120,1:58:25.620
MAGDALENA TODA: The[br]second quadrant.

1:58:25.620,1:58:27.030
Very good.

1:58:27.030,1:58:28.030
All right.

1:58:28.030,1:58:31.465
So think cosine.

1:58:31.465,1:58:36.420
If cosine would be a half and[br]sine would be root 3 over 2,

1:58:36.420,1:58:38.390
it would be in first quadrant.

1:58:38.390,1:58:40.420
And what angle would that be?

1:58:40.420,1:58:40.922
STUDENT: 60.

1:58:40.922,1:58:41.755
STUDENT: That's 60--

1:58:41.755,1:58:45.720
MAGDALENA TODA: 60 degrees,[br]which is pi over 3, right?

1:58:45.720,1:58:52.900
But pi over 3 is your[br]friend, so he's happy.

1:58:52.900,1:58:54.900
Well, he is there somewhere.

1:58:54.900,1:58:58.900


1:58:58.900,1:59:00.770
STUDENT: 120.

1:59:00.770,1:59:05.300
MAGDALENA TODA: Where you[br]are here, you are at what?

1:59:05.300,1:59:07.200
How much is 120-- very good.

1:59:07.200,1:59:09.210
How much is 120 pi?

1:59:09.210,1:59:10.769
STUDENT: 4 pi?

1:59:10.769,1:59:11.560
MAGDALENA TODA: No.

1:59:11.560,1:59:11.850
STUDENT: 2 pi over 3.

1:59:11.850,1:59:13.155
MAGDALENA TODA: 2 pi over 3.

1:59:13.155,1:59:14.460
Excellent.

1:59:14.460,1:59:16.840
So 2 pi over 3.

1:59:16.840,1:59:21.130
This would be if you[br]were to think about it--

1:59:21.130,1:59:22.390
this is in radians.

1:59:22.390,1:59:24.320
Let me write radians.

1:59:24.320,1:59:28.470
In degrees, that's 120 degrees.

1:59:28.470,1:59:38.820
So to conclude my detour[br]to introduction to 3350.

1:59:38.820,1:59:47.104
When they will ask you to solve[br]this equation, x cubed minus 1,

1:59:47.104,1:59:49.580
you have to tell them like that.

1:59:49.580,1:59:53.142
They will ask you to put[br]it in trigonometric form.

1:59:53.142,2:00:04.140
x1 is 1, x2 is cosine of 2 pi[br]over 3 plus i sine 2 pi over 3.

2:00:04.140,2:00:07.330
And the other one[br]is x3 equals cosine

2:00:07.330,2:00:15.682
of 2 pi over 3 minus[br]i sine of 2 pi over 3.

2:00:15.682,2:00:16.840
The last thing.

2:00:16.840,2:00:18.660
Because I should let you go.

2:00:18.660,2:00:20.015
There was no break.

2:00:20.015,2:00:23.440
I squeezed your brains[br]really bad today.

2:00:23.440,2:00:26.370
We still have like 150 minutes.

2:00:26.370,2:00:29.190
I stole from you--[br]no, I stole really big

2:00:29.190,2:00:33.330
because we would have-- yeah,[br]we still have 15 minutes.

2:00:33.330,2:00:37.294
But the break was 10 minutes,[br]so I didn't give you a break.

2:00:37.294,2:00:40.030
What would this be if[br]you wanted to express it

2:00:40.030,2:00:43.030
in terms of another angle?

2:00:43.030,2:00:47.142
That's the last thing[br]I'm asking of you.

2:00:47.142,2:00:48.603
STUDENT: [INAUDIBLE].

2:00:48.603,2:00:50.551
MAGDALENA TODA: Not minus.

2:00:50.551,2:00:53.170
Like cosine of an angle[br]plus i sine of an angle.

2:00:53.170,2:00:55.942
You would need to go to[br]another quadrant, right?

2:00:55.942,2:00:57.570
And which quadrant?

2:00:57.570,2:00:58.407
STUDENT: 4.

2:00:58.407,2:00:59.990
MAGDALENA TODA:[br]You've said it before.

2:00:59.990,2:01:02.570
That would be 4 pi over 3.

2:01:02.570,2:01:05.230
And 4 pi over 3.

2:01:05.230,2:01:10.160
Keep in mind these things[br]with imaginary numbers because

2:01:10.160,2:01:13.540
in 3350, they will rely on[br]you knowing these things.

2:01:13.540,2:01:15.709


2:01:15.709,2:01:17.750
STUDENT: Then you apply[br]Euler's formula up there.

2:01:17.750,2:01:21.429


2:01:21.429,2:01:22.470
MAGDALENA TODA: Oh, yeah.

2:01:22.470,2:01:24.261
By the way, this is[br]called Euler's formula.

2:01:24.261,2:01:27.535


2:01:27.535,2:01:30.930
STUDENT: In middle[br]school, they teach you,

2:01:30.930,2:01:33.840
and they tell you when[br]discriminant is small,

2:01:33.840,2:01:35.885
there's no solutions.

2:01:35.885,2:01:36.760
MAGDALENA TODA: Yeah.

2:01:36.760,2:01:37.890
STUDENT: And you[br]go to [INAUDIBLE].

2:01:37.890,2:01:40.393
MAGDALENA TODA: When the[br]discriminant is less than 0,

2:01:40.393,2:01:42.357
there are no real solutions.

2:01:42.357,2:01:44.321
But you have in pairs[br]imaginary solutions.

2:01:44.321,2:01:46.285
They always come in pairs.

2:01:46.285,2:01:50.230


2:01:50.230,2:01:52.496
Do you want me to[br]show you probably

2:01:52.496,2:01:55.240
the most important problem[br]in 3350 in 2 minutes,

2:01:55.240,2:01:58.568
and then I'll let you go?

2:01:58.568,2:02:01.003
STUDENT: Sure.

2:02:01.003,2:02:07.690
MAGDALENA TODA: So somebody[br]gives you the equation

2:02:07.690,2:02:10.390
of the harmonic oscillator.

2:02:10.390,2:02:12.010
And you say, what[br]the heck is that?

2:02:12.010,2:02:17.420
You have a little spring[br]and you pull that spring.

2:02:17.420,2:02:19.360
And it's going to come back.

2:02:19.360,2:02:21.860
You displace it, it comes back.

2:02:21.860,2:02:24.360
It oscillates back and forth,[br]oscillates back and forth.

2:02:24.360,2:02:27.950
If you were to write the[br]solutions of the harmonic

2:02:27.950,2:02:29.800
oscillator in electric[br]circuits, there

2:02:29.800,2:02:31.300
would be oscillating functions.

2:02:31.300,2:02:36.530
So it has to do with sine and[br]cosine, so they must be trig.

2:02:36.530,2:02:38.910
If somebody gives[br]you this equation,

2:02:38.910,2:02:57.056
let's say ax squared-- y[br]double prime of x minus b.

2:02:57.056,2:02:59.040
Plus.

2:02:59.040,2:03:04.000
Equals to 0.

2:03:04.000,2:03:09.470
And here is a y equals 0.

2:03:09.470,2:03:12.500
Why would that[br]show up like that?

2:03:12.500,2:03:19.370
Well, Hooke's law tells[br]you that there is a force.

2:03:19.370,2:03:21.680
And there is a[br]force and the force

2:03:21.680,2:03:23.430
is mass times acceleration.

2:03:23.430,2:03:27.485
And acceleration is like this[br]type of second derivative

2:03:27.485,2:03:30.830
of the displacement.

2:03:30.830,2:03:37.580
And F and the displacement[br]are proportional,

2:03:37.580,2:03:41.230
when you write F[br]equals displacement,

2:03:41.230,2:03:43.980
let's call it y of x.

2:03:43.980,2:03:47.535
When you have y of x, x is time.

2:03:47.535,2:03:49.297
That's the displacement.

2:03:49.297,2:03:50.005
That's the force.

2:03:50.005,2:03:50.630
That's the k.

2:03:50.630,2:03:53.060
So you have a certain[br]Hooke's constant.

2:03:53.060,2:03:54.880
Hooke's law constant.

2:03:54.880,2:03:56.870
So when you write[br]this, Hooke's law

2:03:56.870,2:03:58.590
is going to become like that.

2:03:58.590,2:04:04.930
Mass times y double prime of[br]x equals-- this is the force.

2:04:04.930,2:04:06.725
k times y of x.

2:04:06.725,2:04:09.990


2:04:09.990,2:04:16.540
But it depends because[br]you can have plus minus.

2:04:16.540,2:04:18.580
So you can have plus or minus.

2:04:18.580,2:04:20.146
And these are[br]positive functions.

2:04:20.146,2:04:27.500


2:04:27.500,2:04:31.920
You have two equations[br]in that case.

2:04:31.920,2:04:39.642
One equation is the form y[br]double prime plus-- give me

2:04:39.642,2:04:40.470
a number.

2:04:40.470,2:04:43.405
Cy equals 0.

2:04:43.405,2:04:49.570
And the other one would be y[br]double prime minus cy equals 0.

2:04:49.570,2:04:51.371
All right.

2:04:51.371,2:04:54.976
Now, how hard is to[br]guess your solutions?

2:04:54.976,2:04:59.916


2:04:59.916,2:05:02.170
Can you guess the[br]solutions with naked eyes?

2:05:02.170,2:05:04.760


2:05:04.760,2:05:05.635
STUDENT: e to the x--

2:05:05.635,2:05:09.080


2:05:09.080,2:05:13.920
MAGDALENA TODA: So if you have--[br]you have e to the something.

2:05:13.920,2:05:17.780
If you didn't have a c, it[br]would make your life easier.

2:05:17.780,2:05:18.700
Forget about the c.

2:05:18.700,2:05:20.910
The c will act the[br]same in the end.

2:05:20.910,2:05:25.796
So here, what are the[br]possible solutions?

2:05:25.796,2:05:27.012
STUDENT: e to the--

2:05:27.012,2:05:29.220
MAGDALENA TODA: e to the t[br]is one of them. e to the x

2:05:29.220,2:05:32.750
is one of them, right?

2:05:32.750,2:05:35.540
So in the end, to[br]solve such a problem

2:05:35.540,2:05:37.300
they teach you the method.

2:05:37.300,2:05:39.660
You take the equation.

2:05:39.660,2:05:42.080
And for that, you associate[br]the so-called characteristic

2:05:42.080,2:05:44.480
equation.

2:05:44.480,2:05:47.250
For power 2, you put r squared.

2:05:47.250,2:05:51.474
Then you minus n for-- this[br]is how many times is it prime?

2:05:51.474,2:05:52.250
No times.

2:05:52.250,2:05:53.080
0 times.

2:05:53.080,2:05:55.250
So you put a 1.

2:05:55.250,2:05:58.010
If it's prime one times,[br]y prime is missing.

2:05:58.010,2:06:01.770
It's prime 1 time,[br]you would put minus r.

2:06:01.770,2:06:02.880
Equals 0.

2:06:02.880,2:06:06.950
And then you look at[br]the two roots of that.

2:06:06.950,2:06:07.820
And what are they?

2:06:07.820,2:06:08.710
Plus minus 1.

2:06:08.710,2:06:11.490
So r1 is 1, r2 is 2.

2:06:11.490,2:06:13.370
And there is a[br]theorem that says--

2:06:13.370,2:06:15.260
STUDENT: r2 is minus 1.

2:06:15.260,2:06:17.125
MAGDALENA TODA: r2 is minus 1.

2:06:17.125,2:06:19.620
Excuse me.

2:06:19.620,2:06:23.970
OK, there's a theorem that[br]says all the solutions

2:06:23.970,2:06:28.160
of this equation come as[br]linear combinations of e

2:06:28.160,2:06:31.230
to the r1t and e to the r2t.

2:06:31.230,2:06:33.106
So linear combination[br]means you can

2:06:33.106,2:06:39.150
take any number a and any[br]number b, or c1 and c2, anything

2:06:39.150,2:06:40.140
like that.

2:06:40.140,2:06:44.650
So all the solutions of[br]this will look like e

2:06:44.650,2:06:48.230
to the t with an a in[br]front plus e to the minus

2:06:48.230,2:06:50.055
t with a b in front.

2:06:50.055,2:06:53.300
Could you have seen[br]that with naked eye?

2:06:53.300,2:06:54.350
Well, yeah.

2:06:54.350,2:06:57.270
I mean, you are smart[br]and you guessed one.

2:06:57.270,2:06:59.250
An you said e to[br]the t satisfied.

2:06:59.250,2:07:02.320
Because if you put e to the[br]p and prime it as many times

2:07:02.320,2:07:04.870
as you want, you[br]still get e to the t.

2:07:04.870,2:07:06.050
So you get 0.

2:07:06.050,2:07:09.180
But nobody thought of-- or maybe[br]some people thought about e

2:07:09.180,2:07:10.483
to the minus t.

2:07:10.483,2:07:11.066
STUDENT: Yeah.

2:07:11.066,2:07:11.880
I was about to go[br]through that one.

2:07:11.880,2:07:13.030
MAGDALENA TODA: You were about.

2:07:13.030,2:07:13.800
STUDENT: That's for a selection.

2:07:13.800,2:07:16.008
MAGDALENA TODA: So even if[br]you take e to the minus t,

2:07:16.008,2:07:17.240
you get the same answer.

2:07:17.240,2:07:19.790
And you get this thing.

2:07:19.790,2:07:24.040
All right, all the combinations[br]will satisfy the same equation

2:07:24.040,2:07:24.660
as well.

2:07:24.660,2:07:26.630
This is a superposition[br]principle.

2:07:26.630,2:07:28.622
With this, it was easy.

2:07:28.622,2:07:31.610
But this is the so-called[br]harmonic oscillator equation.

2:07:31.610,2:07:36.100


2:07:36.100,2:07:40.290
So either you have it simplified[br]y double prime plus y equals 0,

2:07:40.290,2:07:46.250
or you have some constant c.

2:07:46.250,2:07:48.695
Well, what do you[br]do in that case?

2:07:48.695,2:07:50.766
Let's assume you have 1.

2:07:50.766,2:07:53.946
Who can guess the solutions?

2:07:53.946,2:07:55.920
STUDENT: 0 and cosine--

2:07:55.920,2:07:57.982
MAGDALENA TODA: No, 0[br]is the trivial solution

2:07:57.982,2:07:59.160
and it's not going to count.

2:07:59.160,2:08:03.860
You can get it from the[br]combination of the--

2:08:03.860,2:08:05.350
STUDENT: y equals sine t.

2:08:05.350,2:08:06.850
MAGDALENA TODA:[br]Sine t is a solution

2:08:06.850,2:08:11.665
because sine t prime is cosine.

2:08:11.665,2:08:13.770
When you prime it[br]again, it's minus sine.

2:08:13.770,2:08:16.680
When you add sine and[br]minus sine, you get 0.

2:08:16.680,2:08:19.410
So you just guessed[br]1 and you're right.

2:08:19.410,2:08:20.670
Make a face.

2:08:20.670,2:08:21.790
Do you see another one?

2:08:21.790,2:08:22.770
STUDENT: Cosine t.

2:08:22.770,2:08:23.728
MAGDALENA TODA: Cosine.

2:08:23.728,2:08:26.155


2:08:26.155,2:08:28.600
They are independent,[br]linear independent.

2:08:28.600,2:08:31.220
And so the multitude[br]of solutions

2:08:31.220,2:08:34.270
for that-- I taught you[br]a whole chapter in 3350.

2:08:34.270,2:08:37.140
Now you don't have[br]to take it anymore--

2:08:37.140,2:08:39.717
is going to be a equals sine t--

2:08:39.717,2:08:41.050
STUDENT: How about e to the i t?

2:08:41.050,2:08:42.300
MAGDALENA TODA: Plus b sine t.

2:08:42.300,2:08:43.440
I tell you in a second.

2:08:43.440,2:08:46.530
All right, we have to[br]do an e to the i t.

2:08:46.530,2:08:47.520
OK.

2:08:47.520,2:08:51.350
So you guessed that all the[br]solutions will be combinations

2:08:51.350,2:08:56.020
like-- on the monitor when you[br]have cosine and sine, if you

2:08:56.020,2:08:58.870
add them up-- multiply[br]and add them up,

2:08:58.870,2:09:02.390
you get something like the[br]monitor thing at the hospital.

2:09:02.390,2:09:04.480
So any kind of[br]oscillation like that

2:09:04.480,2:09:07.686
is a combination of this kind.

2:09:07.686,2:09:13.050
Maybe with some different[br]phases and amplitudes.

2:09:13.050,2:09:16.830
You have cosine of 70 or[br]cosine of 5t or something.

2:09:16.830,2:09:18.650
But let me show[br]you what they are

2:09:18.650,2:09:24.640
going to show you [INAUDIBLE][br]for the harmonic oscillator

2:09:24.640,2:09:27.270
equation how the method goes.

2:09:27.270,2:09:29.250
You solve for the[br]characteristic equation.

2:09:29.250,2:09:34.600
So you have r squared[br]plus 1 equals 0.

2:09:34.600,2:09:38.860
Now, here's where most of[br]the students in 3350 fail.

2:09:38.860,2:09:40.468
They understand that.

2:09:40.468,2:09:43.880
And some of them say, OK,[br]this has no solutions.

2:09:43.880,2:09:46.990
Some of them even say this[br]has solutions plus minus 1.

2:09:46.990,2:09:48.870
I mean, crazy stuff.

2:09:48.870,2:09:51.490
Now, what are the[br]solutions of that?

2:09:51.490,2:09:53.455
Because the theory[br]in this case says

2:09:53.455,2:09:57.330
if your solutions are[br]imaginary, then y1

2:09:57.330,2:10:00.980
would be e to the ax cosine bx.

2:10:00.980,2:10:05.450
And y2 will be e[br]to the ax sine bx

2:10:05.450,2:10:09.446
where your imaginary[br]solutions are a plus minus ib.

2:10:09.446,2:10:13.990
It has a lot to do with[br]Euler's formula in a way.

2:10:13.990,2:10:21.000
So if you knew the theory in[br]3350 and not be just very smart

2:10:21.000,2:10:24.130
and get these by yourselves[br]by guessing them,

2:10:24.130,2:10:26.520
how are you supposed[br]to know that?

2:10:26.520,2:10:30.580
Well, r squared[br]equals minus 1, right?

2:10:30.580,2:10:34.036
The square root of minus 1 is i.

2:10:34.036,2:10:34.910
STUDENT: Or negative.

2:10:34.910,2:10:36.159
MAGDALENA TODA: Or negative i.

2:10:36.159,2:10:40.600
So r1 is 0 plus minus i.

2:10:40.600,2:10:42.460
So who is a?

2:10:42.460,2:10:44.181
a is 0.

2:10:44.181,2:10:45.710
Who is b?

2:10:45.710,2:10:46.720
b is 1.

2:10:46.720,2:10:53.010
So the solutions are e to[br]the 0x equal cosine 1x and e

2:10:53.010,2:10:59.040
to the 0x sine 1x, which[br]is cosine x, sine x.

2:10:59.040,2:11:03.210
Now you know why you can[br]do everything formalized

2:11:03.210,2:11:06.170
and you get all these[br]solutions from a method.

2:11:06.170,2:11:10.286
This method is an[br]entire chapter.

2:11:10.286,2:11:12.400
It's so much easier than in 350.

2:11:12.400,2:11:14.670
So much easier than Calculus 3.

2:11:14.670,2:11:16.420
You will say this is easy.

2:11:16.420,2:11:17.720
It's a pleasure.

2:11:17.720,2:11:22.790
You spend about one[br]fourth of the semester

2:11:22.790,2:11:24.562
just on this method.

2:11:24.562,2:11:26.270
So now you don't have[br]to take it anymore.

2:11:26.270,2:11:29.080
You can learn it all by[br]yourself and you're going

2:11:29.080,2:11:33.060
to be ready for the next thing.

2:11:33.060,2:11:34.760
So I'm just giving you courage.

2:11:34.760,2:11:40.350
If you do really, really well in[br]Calc 3, 3350 will be a breeze.

2:11:40.350,2:11:42.290
You can breeze through that.

2:11:42.290,2:11:46.170
You only have the probability[br]in stats for most engineers

2:11:46.170,2:11:48.595
to take.

2:11:48.595,2:11:54.210
Math is not so complicated.