0:00:00.000,0:00:03.190
PROFESSOR TODA: And Calc II.

0:00:03.190,0:00:08.189
And I will go ahead and[br]solve some problems today out

0:00:08.189,0:00:10.780
of chapter 10 as a review.

0:00:10.780,0:00:14.494


0:00:14.494,0:00:15.478
Meaning what?

0:00:15.478,0:00:22.540
Meaning, that you have[br]section 10.1 followed by 10.2

0:00:22.540,0:00:24.740
followed by 10.4.

0:00:24.740,0:00:26.920
These ones are[br]required sections,

0:00:26.920,0:00:34.520
but I'm putting the material[br]all together as a compact set.

0:00:34.520,0:00:38.816
So, if we cannot officially[br]cut between, as I told you,

0:00:38.816,0:00:41.720
cut between the sections.

0:00:41.720,0:00:47.270
One thing that I did[br]not work examples on,

0:00:47.270,0:00:50.400
trusting that you'd[br]remember it was integration.

0:00:50.400,0:00:53.270
In particular, I didn't[br]cover integration

0:00:53.270,0:00:56.220
of vector valued functions[br]and examples that

0:00:56.220,0:00:58.070
are very very important.

0:00:58.070,0:01:02.820
Now, do you need to learn[br]something special for that?

0:01:02.820,0:01:03.320
No.

0:01:03.320,0:01:07.820
But just like you cannot learn[br]organic chemistry without

0:01:07.820,0:01:11.540
knowing inorganic chemistry,[br]then you could not know how

0:01:11.540,0:01:17.032
to integrate a vector value[br]function r prime of d to get r

0:01:17.032,0:01:21.220
of d unless you know calculus[br]one and caluculus two, right?

0:01:21.220,0:01:25.680
So let's say first[br]a bunch of formulas

0:01:25.680,0:01:30.570
that you use going back[br]to last week's knowledge

0:01:30.570,0:01:32.100
what have we learned?

0:01:32.100,0:01:38.900
We work with regular[br]curves in r3.

0:01:38.900,0:01:42.200
And in particular if[br]they are part of R2,

0:01:42.200,0:01:45.170
they are plain curves.

0:01:45.170,0:01:47.690
I want to encourage[br]you to ask questions

0:01:47.690,0:01:50.090
about the example[br][INAUDIBLE] now.

0:01:50.090,0:01:57.110
In the review session we[br]have applications [INAUDIBLE]

0:01:57.110,0:01:58.870
from 2 2 3.

0:01:58.870,0:02:00.870
What was a regular curve?

0:02:00.870,0:02:04.185
Is anybody willing to tell[br]me what a regular curve was?

0:02:04.185,0:02:08.092
Was it vector value function?

0:02:08.092,0:02:09.699
Do you like big r or little r?

0:02:09.699,0:02:10.699
STUDENT: Doesn't matter.

0:02:10.699,0:02:11.980
PROFESSOR TODA: Big r of t.

0:02:11.980,0:02:14.120
Vector value function.

0:02:14.120,0:02:17.940
x of t [INAUDIBLE] You know,[br]I told you that sometimes we

0:02:17.940,0:02:19.290
use brackets here.

0:02:19.290,0:02:25.040
Sometimes we use round[br]parentheses depending

0:02:25.040,0:02:31.130
how you represent a vector in r3[br]in our book they use brackets,

0:02:31.130,0:02:37.070
but in other calculus books,[br]they use round parentheses

0:02:37.070,0:02:38.510
around it.

0:02:38.510,0:02:44.270
So these are the coordinates[br]of the moving particle in time.

0:02:44.270,0:02:47.600
Doesn't have to be a specific[br]object, could be a fly,

0:02:47.600,0:02:50.780
could be just a[br]particle, anything

0:02:50.780,0:02:58.000
in physical motion between this[br]point a of b equals a and b

0:02:58.000,0:03:00.140
of t equals b.

0:03:00.140,0:03:02.336
So at time a and[br]time b you are there.

0:03:02.336,0:03:03.210
What have we learned?

0:03:03.210,0:03:11.280
We've learned that a regular[br]curve means its differentiable

0:03:11.280,0:03:15.120
and the derivative is[br]continuous, it's a c1 function.

0:03:15.120,0:03:16.290
And what else?

0:03:16.290,0:03:19.934
The derivative of[br]the position vector

0:03:19.934,0:03:22.850
called velocity never vanishes.

0:03:22.850,0:03:26.670
So it's different from 0[br]for every t in the interval

0:03:26.670,0:03:30.050
that you take, like ab.

0:03:30.050,0:03:31.840
That's a regular curve.

0:03:31.840,0:03:38.720
Regular curve was something we[br]talked about at least 5 times.

0:03:38.720,0:03:44.170
The point is how do we[br]see the backwards process?

0:03:44.170,0:03:52.160
That means if somebody gives you[br]the velocity of a vector curve,

0:03:52.160,0:03:55.186
they ask you for[br]the position vector.

0:03:55.186,0:03:57.320
So let's see an example.

0:03:57.320,0:04:02.990
Integration example[br]1 says I gave you

0:04:02.990,0:04:07.820
the veclocity vector or[br]a certain law of motion

0:04:07.820,0:04:09.030
that I don't know.

0:04:09.030,0:04:13.175
I just know the velocity[br]vector is being 1 over 1

0:04:13.175,0:04:15.600
plus t squared.

0:04:15.600,0:04:17.055
Should I put the brace here?

0:04:17.055,0:04:19.000
An angular bracket?

0:04:19.000,0:04:20.720
One over one plus t squared.

0:04:20.720,0:04:32.863
And I'm gonna put a cosign[br]on 2t, and t squared

0:04:32.863,0:04:37.500
plus equal to minus t.

0:04:37.500,0:04:43.290
And somebody says,[br]that's all I know for P

0:04:43.290,0:04:47.210
on an arbitrary real integral.

0:04:47.210,0:04:54.980
And we know via the[br]0 as being even.

0:04:54.980,0:05:03.091
Let's say it's even[br]as 0 0 and that

0:05:03.091,0:05:06.610
takes a little bit of thinking.

0:05:06.610,0:05:09.000
I don't know.

0:05:09.000,0:05:20.260
How about a 1, which[br]would be just k.

0:05:20.260,0:05:24.450
Using this velocity vector[br]find me being normal,

0:05:24.450,0:05:26.750
which means find[br]the position vector

0:05:26.750,0:05:29.690
corresponding to this velocity.

0:05:29.690,0:05:31.370
What is this?

0:05:31.370,0:05:34.340
It's actually initial value

0:05:34.340,0:05:40.620
STUDENT: [INAUDIBLE][br]1, 1, and 1?

0:05:40.620,0:05:42.400
PROFESSOR TODA: 0, what is it?

0:05:42.400,0:05:44.061
When place 0 in?

0:05:44.061,0:05:45.432
STUDENT: Yeah.

0:05:45.432,0:05:47.670
[INTERPOSING VOICES]

0:05:47.670,0:05:49.530
STUDENT: Are these[br]the initial conditions

0:05:49.530,0:05:50.802
for the location, or--

0:05:50.802,0:05:51.885
PROFESSOR TODA: I'm sorry.

0:05:51.885,0:05:58.780
I wrote r the intial[br]condition for the location.

0:05:58.780,0:06:01.040
Thank you so much, OK?

0:06:01.040,0:06:04.520
I probably would've realized[br]it as soon as possible.

0:06:04.520,0:06:07.030
Not the initial velocity[br]I wanted to give you,

0:06:07.030,0:06:11.120
but the initial position.

0:06:11.120,0:06:17.730
All right, so how do[br]I get to the r of d?

0:06:17.730,0:06:20.000
I would say integrate,[br]and when I integrate,

0:06:20.000,0:06:25.680
I have to keep in mind that[br]I have to add the constants.

0:06:25.680,0:06:26.320
Right?

0:06:26.320,0:06:27.000
OK.

0:06:27.000,0:06:29.442
So from v, v is our priority.

0:06:29.442,0:06:32.400


0:06:32.400,0:06:37.876
It follows that r will[br]be-- who tells me?

0:06:37.876,0:06:42.514
Do you guys remember the[br]integral of 1 plus t squared?

0:06:42.514,0:06:43.462
STUDENT: [INAUDIBLE]

0:06:43.462,0:06:45.358
PROFESSOR TODA: So[br]that's the inverse.

0:06:45.358,0:06:49.012
Or, I'll write it [? arc tan, ?][br]and I'm very happy that you

0:06:49.012,0:06:51.220
remember that, but there[br]are many students who don't.

0:06:51.220,0:06:54.530
If you feel you don't, that[br]means that you have to open

0:06:54.530,0:06:59.740
the -- where? -- Between[br]chapters 5 and chapter 7.

0:06:59.740,0:07:03.680
You have all these[br]integration chapters--

0:07:03.680,0:07:05.974
the main ones over there.

0:07:05.974,0:07:08.140
It's a function definted[br]on the whole real interval,

0:07:08.140,0:07:11.860
so I don't care[br]to worry about it.

0:07:11.860,0:07:14.700
This what we call an IVP,[br]initial value problem.

0:07:14.700,0:07:18.650


0:07:18.650,0:07:20.945
So what kind of problem is that?

0:07:20.945,0:07:23.030
It's a problem[br]like somebody would

0:07:23.030,0:07:29.570
give you knowing that f[br]prime of t is the little f,

0:07:29.570,0:07:32.820
and knowing that big f[br]of 0 is the initial value

0:07:32.820,0:07:37.340
for your function of find f.

0:07:37.340,0:07:42.860
So you have actually an initial[br]value problem of the calc

0:07:42.860,0:07:47.160
that you've seen[br]in previous class.

0:07:47.160,0:07:54.680
arctangent of t plus c1 and then[br]if you miss the c1 in general,

0:07:54.680,0:07:59.690
this can mess up the whole thing[br]because-- see, in your case,

0:07:59.690,0:08:02.280
you're really lucky.

0:08:02.280,0:08:06.650
If you plug in the 0 here,[br]what are you gonna have?

0:08:06.650,0:08:10.500
You're gonna have arctangent[br]of 0, and that is 0.

0:08:10.500,0:08:12.732
So in that case c1 is just 0.

0:08:12.732,0:08:15.190
And [? three ?] [? not ?] and[br]if you forgot it would not be

0:08:15.190,0:08:19.520
the end of the world, but[br]if you forgot it in general,

0:08:19.520,0:08:20.870
it would be a big problem.

0:08:20.870,0:08:22.990
So don't forget[br]about the constant.

0:08:22.990,0:08:25.260
When you integrate-- the[br]familiar of antiderivatives

0:08:25.260,0:08:26.450
is cosine 2t.

0:08:26.450,0:08:29.980


0:08:29.980,0:08:32.510
I know you know it.

0:08:32.510,0:08:35.870
1/2 sine of t.

0:08:35.870,0:08:37.240
Am I done?

0:08:37.240,0:08:40.390
No, I should say plus C2.

0:08:40.390,0:08:43.039
And finally the familiar[br]of antiderivatives of t

0:08:43.039,0:08:45.700
squared plus e to minus t.

0:08:45.700,0:08:48.230
STUDENT: 2t minus e[br]to the negative t.

0:08:48.230,0:08:50.120
PROFESSOR TODA: No, integral of.

0:08:50.120,0:08:52.570
So what's the integral of--

0:08:52.570,0:08:53.700
STUDENT: t 2 squared.

0:08:53.700,0:08:57.860
PROFESSOR TODA: t cubed[br]over 3-- minus, excellent.

0:08:57.860,0:09:01.310
Now, do you want one[br]of you guys almost

0:09:01.310,0:09:03.427
kill me during the weekend.

0:09:03.427,0:09:04.010
But that's OK.

0:09:04.010,0:09:06.340
I mean, this problem[br]had something

0:09:06.340,0:09:08.100
to do with integral minus.

0:09:08.100,0:09:12.130
He put that integral of e to the[br]minus t was equal to minus t.

0:09:12.130,0:09:14.450
So pay attention to the sign.

0:09:14.450,0:09:16.925
Remember that integral[br]of e to the at,

0:09:16.925,0:09:22.220
the t is to the at over a plus.

0:09:22.220,0:09:23.020
Right?

0:09:23.020,0:09:26.810
OK, so this is what you[br]have, a minus plus C3.

0:09:26.810,0:09:28.770
Pay attention also to the exam.

0:09:28.770,0:09:30.708
Because in the[br]exams, when you rush,

0:09:30.708,0:09:33.030
you make lots of[br]mistakes like that.

0:09:33.030,0:09:36.810
R of 0 is even.

0:09:36.810,0:09:43.090
So the initial position[br]is given as C1.

0:09:43.090,0:09:44.820
I'm replacing in my formula.

0:09:44.820,0:09:49.430
It's going to be[br]C1, C2, and what?

0:09:49.430,0:09:52.264
When I replace the 0 here,[br]what am I going to get?

0:09:52.264,0:09:53.930
STUDENT: You're going[br]to get negative 1.

0:09:53.930,0:09:59.060
PROFESSOR TODA: Minus 1 plus C3.

0:09:59.060,0:10:03.040
Note that I fabricated this[br]example, so that C3 is not

0:10:03.040,0:10:04.440
going to be 0.

0:10:04.440,0:10:06.790
I wanted some customs to[br]be zero and some customs

0:10:06.790,0:10:10.235
to not be 0, just for[br]you to realize it's

0:10:10.235,0:10:12.560
important to pay attention.

0:10:12.560,0:10:14.720
OK, minus 1 plus C3.

0:10:14.720,0:10:22.490
And then I have 0, 0, 1 as[br]given as initial position.

0:10:22.490,0:10:28.130
So what do you get by solving[br]this linear system that's

0:10:28.130,0:10:29.230
very simple?

0:10:29.230,0:10:32.300
In general, you can get[br]more complicated stuff.

0:10:32.300,0:10:35.740
C1 is 0, C2 is 0, C3 is a--

0:10:35.740,0:10:36.360
STUDENT: 2.

0:10:36.360,0:10:37.110
PROFESSOR TODA: 2.

0:10:37.110,0:10:38.860
And so it was a piece of cake.

0:10:38.860,0:10:40.880
What is my formula?

0:10:40.880,0:10:43.590
If you leave it like[br]that, generally you're

0:10:43.590,0:10:44.670
going to get full credit.

0:10:44.670,0:10:47.400
What would you need to[br]do to get full credit?

0:10:47.400,0:10:53.424
STUDENT: Rt is equal to R10[br]plus 1/2 sine of 2t plus tq--

0:10:53.424,0:10:55.424
PROFESSOR TODA: Precisely,[br]and thank you so much

0:10:55.424,0:10:56.810
for your help.

0:10:56.810,0:11:01.550
So you have R10 of[br]t, 1/2 sine of 2t

0:11:01.550,0:11:09.905
and t cubed over 3 minus[br]e to the minus e plus 2.

0:11:09.905,0:11:11.650
And close, and that's it.

0:11:11.650,0:11:14.240
And box your answer.

0:11:14.240,0:11:16.260
So I got the long motion back.

0:11:16.260,0:11:22.000
Similarly, you could find,[br]if somebody gives you

0:11:22.000,0:11:27.840
the acceleration of a[br]long motion and asks you

0:11:27.840,0:11:29.830
this is the acceleration.

0:11:29.830,0:11:31.910
And I give you some[br]initial values.

0:11:31.910,0:11:34.520
And you have to find[br]first the velocity,

0:11:34.520,0:11:36.270
going backwards one step.

0:11:36.270,0:11:39.680
And from the velocity,[br]backwards a second step,

0:11:39.680,0:11:42.050
get the position vector.

0:11:42.050,0:11:44.050
And that sounds a little[br]bit more elaborate.

0:11:44.050,0:11:47.110
But it doesn't have to[br]be a long computation.

0:11:47.110,0:11:50.960
In general, we do not[br]focus on giving you

0:11:50.960,0:11:54.100
an awfully long computation.

0:11:54.100,0:11:58.870
We just want to test your[br]understanding of the concepts.

0:11:58.870,0:12:04.490
And having this in mind,[br]I picked another example.

0:12:04.490,0:12:08.570
I would like to[br]see what that is.

0:12:08.570,0:12:14.377
And the initial velocity[br]will be given in this case.

0:12:14.377,0:12:16.812
This is what I was thinking[br]a little bit ahead of that.

0:12:16.812,0:12:22.890
So somebody gives you the[br]acceleration in the velocity

0:12:22.890,0:12:31.130
vector at 0 and is asking you[br]to find the velocity vector So

0:12:31.130,0:12:36.300
let me give it to you[br]for t between 0 and 2 pi.

0:12:36.300,0:12:38.050
I give you the[br]acceleration vector,

0:12:38.050,0:12:40.370
it will be nice and sassy.

0:12:40.370,0:12:45.690
Let's see, that's going to be[br]cosine of t, sine of t and 0.

0:12:45.690,0:12:48.340
And you'll say, oh, I[br]know how to do those.

0:12:48.340,0:12:49.770
Of course you know.

0:12:49.770,0:12:52.370
But I want you to pay[br]attention to the constraints

0:12:52.370,0:12:53.140
of integration.

0:12:53.140,0:12:58.250
This is why I do this[br]kind of exercise again.

0:12:58.250,0:13:02.810
So what do we have for V of t.

0:13:02.810,0:13:09.980
V of 0 is-- somebody will say,[br]let's give something nice,

0:13:09.980,0:13:15.530
and let's say this would be--[br]I have no idea what I want.

0:13:15.530,0:13:21.770
Let's say i, j, and that's it.

0:13:21.770,0:13:24.590


0:13:24.590,0:13:26.950
How do you do that?

0:13:26.950,0:13:27.640
V of t.

0:13:27.640,0:13:30.006
Let's integrate together.

0:13:30.006,0:13:31.530
You don't like this?

0:13:31.530,0:13:35.270
I hope that by now,[br]you've got used to it.

0:13:35.270,0:13:38.560
A bracket, I'm doing a[br]bracket, like in the book.

0:13:38.560,0:13:42.700
So sine t plus a constant.

0:13:42.700,0:13:45.090
What's the integral[br]of sine, class?

0:13:45.090,0:13:48.200
V equals sine t plus a constant.

0:13:48.200,0:13:51.380
And C3 is a constant.

0:13:51.380,0:13:52.790
And there I go.

0:13:52.790,0:13:54.670
You say, oh my god,[br]what am I having?

0:13:54.670,0:13:58.360
V of 0-- is as a[br]vector, I presented it

0:13:58.360,0:14:03.810
in the canonical standard[br]basis as 1, 1, and 0.

0:14:03.810,0:14:07.350
So from that one, you[br]can jump to this one

0:14:07.350,0:14:10.740
and say, yes, I'm going to[br]plug in 0, see what I get.

0:14:10.740,0:14:13.376
In the general formula,[br]when you plug in 0,

0:14:13.376,0:14:19.070
you get C1-- what[br]is cosine of 0?

0:14:19.070,0:14:22.470
Minus 1, I have here, plus C2.

0:14:22.470,0:14:27.900
And C3, that is always there.

0:14:27.900,0:14:34.530
And then V of 0 is[br]what I got here.

0:14:34.530,0:14:39.730
V of 0 has to be compared to[br]what your initial data was.

0:14:39.730,0:14:48.380
So C1 is 1, C2 is 2, and C3 is--

0:14:48.380,0:14:51.460
So let me replace it.

0:14:51.460,0:15:05.427
I say the answer will be--[br]cosine t plus 1, sine t plus 2,

0:15:05.427,0:15:13.315
and the constants.

0:15:13.315,0:15:19.240


0:15:19.240,0:15:24.740
But then somebody, who is[br]really an experimental guy,

0:15:24.740,0:15:25.539
says well--

0:15:25.539,0:15:26.830
STUDENT: You have it backwards.

0:15:26.830,0:15:28.971
It's sine of t plus[br]1, and then you

0:15:28.971,0:15:31.054
have the cosine of t plus 2.

0:15:31.054,0:15:32.095
PROFESSOR TODA: Oh, yeah.

0:15:32.095,0:15:35.470


0:15:35.470,0:15:36.490
Wait a minute.

0:15:36.490,0:15:41.550
This is-- I[br]miscopied looking up.

0:15:41.550,0:15:52.208
So I have sine t, I was[br]supposed to-- minus cosine t

0:15:52.208,0:15:56.550
and I'm done.

0:15:56.550,0:15:58.380
So thank you for telling me.

0:15:58.380,0:16:04.030
So sum t plus 1 minus[br]cosine t plus 2 and 0

0:16:04.030,0:16:13.070
are the functions that I put[br]here by replacing C1, C2, C3.

0:16:13.070,0:16:15.260
And then, somebody[br]says, wait a minute,

0:16:15.260,0:16:18.500
now let me give you V of 0.

0:16:18.500,0:16:20.860
Let me give you R of 0.

0:16:20.860,0:16:22.495
We have zeroes already there.

0:16:22.495,0:16:25.290


0:16:25.290,0:16:28.800
And you were supposed[br]to get R from here.

0:16:28.800,0:16:36.320
So what is R of t, the[br]position vector, find it.

0:16:36.320,0:16:38.000
V of t is given.

0:16:38.000,0:16:40.230
Actually, it's given by[br]you, because you found it

0:16:40.230,0:16:41.940
at the previous step.

0:16:41.940,0:16:46.300
And R of 0 is given as well.

0:16:46.300,0:16:59.218
And let's say that would[br]be-- let's say 1, 1, and 1.

0:16:59.218,0:17:02.630


0:17:02.630,0:17:05.220
So what do you need to do next?

0:17:05.220,0:17:14.390


0:17:14.390,0:17:18.069
You have R prime given.

0:17:18.069,0:17:22.140
That leaves you to[br]integrate to get R t.

0:17:22.140,0:17:24.520
And R of t is going to be what?

0:17:24.520,0:17:29.030
Who is going to tell me[br]what I have to write down?

0:17:29.030,0:17:39.475
Minus cosine t plus t plus--[br]let's use the constant K1

0:17:39.475,0:17:40.948
integration.

0:17:40.948,0:17:42.421
And then what?

0:17:42.421,0:17:43.410
STUDENT: Sine of t.

0:17:43.410,0:17:45.380
PROFESSOR TODA: I think[br]it's minus sine, right?

0:17:45.380,0:17:56.120
Minus sine of t plus 2t[br]plus K2 and K3, right?

0:17:56.120,0:18:04.200
So R of 0 is going to be what?

0:18:04.200,0:18:07.930
First of all, we use this[br]piece of information.

0:18:07.930,0:18:12.480
Second of all, we identify[br]from the formula we got.

0:18:12.480,0:18:16.305
So from the formula I[br]got, just plugging in 0,

0:18:16.305,0:18:22.690
it should come out straight[br]as minus 1 plus K1.

0:18:22.690,0:18:28.070
0 for this guy, 0 for the[br]second term, K2 and K3.

0:18:28.070,0:18:31.830


0:18:31.830,0:18:36.510
So who is helping me solve[br]the system really quickly?

0:18:36.510,0:18:39.750
K1 is 2.

0:18:39.750,0:18:41.030
K2 is--

0:18:41.030,0:18:41.720
STUDENT: 1.

0:18:41.720,0:18:44.010
PROFESSOR TODA: K3 is 1.

0:18:44.010,0:18:50.610
And I'm going back[br]to R and replace it.

0:18:50.610,0:18:55.470
And that's my final answer[br]for this two-step problem.

0:18:55.470,0:18:57.870
So I have a two-step integration[br]from the acceleration

0:18:57.870,0:19:00.270
to the velocity,[br]from the velocity

0:19:00.270,0:19:05.080
to the position vector.

0:19:05.080,0:19:08.250
Minus cosine t plus t plus 2.

0:19:08.250,0:19:11.630
Remind me, because I have[br]a tendency to miscopy,

0:19:11.630,0:19:13.280
an I looking in the right place?

0:19:13.280,0:19:14.340
Yes.

0:19:14.340,0:19:24.930
So I have minus sine t plus[br]2t plus 1 and K3 is one.

0:19:24.930,0:19:29.751
So this is the process you[br]are supposed to remember

0:19:29.751,0:19:32.160
for the rest of the semester.

0:19:32.160,0:19:33.420
It's not a hard one.

0:19:33.420,0:19:36.960
It's something that[br]everybody should master.

0:19:36.960,0:19:38.060
Is it hard?

0:19:38.060,0:19:39.810
How many of you understood this?

0:19:39.810,0:19:41.820
Please raise hands.

0:19:41.820,0:19:45.110
Oh, no problem, good.

0:19:45.110,0:19:52.160
Now would you tell me--[br]I'm not going to ask you

0:19:52.160,0:19:53.800
what kind of motion this is.

0:19:53.800,0:19:57.030
It's a little bit close to[br]a circular motion but not

0:19:57.030,0:19:58.376
a circular motion.

0:19:58.376,0:20:00.670
However, can you tell[br]me anything interesting

0:20:00.670,0:20:04.800
about the type of trajectory[br]that I have, in terms

0:20:04.800,0:20:06.380
of the acceleration vector?

0:20:06.380,0:20:10.820
The acceleration[br]vector is beautiful,

0:20:10.820,0:20:13.840
just like in the[br]case of the washer.

0:20:13.840,0:20:18.870
That was a vector[br]that-- like this

0:20:18.870,0:20:20.860
would be the circular motion.

0:20:20.860,0:20:23.040
The acceleration would[br]be this unique vector

0:20:23.040,0:20:25.050
that comes inside.

0:20:25.050,0:20:26.920
Is this going outside[br]or coming inside?

0:20:26.920,0:20:29.600
Is it a unit vector?

0:20:29.600,0:20:32.720
Yes, it is a unit vector.

0:20:32.720,0:20:37.430
So suppose that I'm[br]looking at the trajectory,

0:20:37.430,0:20:40.290
if it were more or[br]less a motion that has

0:20:40.290,0:20:44.760
to do with mixing into a bowl.

0:20:44.760,0:20:48.830
Would this go inside or outside?

0:20:48.830,0:20:51.960
Towards the outside[br]or towards the inside?

0:20:51.960,0:20:57.650
I plugged j-- depends on[br]what I'm looking at, in terms

0:20:57.650,0:21:00.150
of surface that I'm on, right?

0:21:00.150,0:21:01.560
Do you remember[br]from last time we

0:21:01.560,0:21:04.055
had that helix that[br]was on a cylinder.

0:21:04.055,0:21:07.920
And we asked ourselves, how[br]is that [INAUDIBLE] pointing?

0:21:07.920,0:21:11.780
And it was pointing[br]outside of the cylinder,

0:21:11.780,0:21:16.052
in the direction[br]towards the outside.

0:21:16.052,0:21:26.930
Coming back to the[br]review, there are

0:21:26.930,0:21:31.440
several things I'd like to[br]review but not all of them.

0:21:31.440,0:21:34.439
Because some of the[br]examples we have there,

0:21:34.439,0:21:38.150
you understood them really well.

0:21:38.150,0:21:40.300
I was very proud[br]of you, and I saw

0:21:40.300,0:21:43.758
that you finished--[br]almost all of you

0:21:43.758,0:21:45.746
finished the[br]homework number one.

0:21:45.746,0:21:49.100
So I was looking outside[br]at homework number

0:21:49.100,0:21:53.180
two that is over[br]these three sections.

0:21:53.180,0:21:58.471
So I was hoping you would ask[br]me today, between two and three,

0:21:58.471,0:22:00.856
if you have any difficulties[br]with homework two.

0:22:00.856,0:22:03.730
That's due February 11.

0:22:03.730,0:22:12.730
And then the latest homework[br]that I posted yesterday, I

0:22:12.730,0:22:14.980
don't know how many[br]of you logged in.

0:22:14.980,0:22:18.620
But last night I[br]posted a homework

0:22:18.620,0:22:21.800
that is getting a huge[br]extended deadline, which

0:22:21.800,0:22:23.370
is the 28th of February.

0:22:23.370,0:22:29.010
Because somebody's[br]birthday is February 29.

0:22:29.010,0:22:34.740
I was just thinking why would[br]somebody need be a whole month?

0:22:34.740,0:22:37.300
You would need the whole[br]month to have a good view

0:22:37.300,0:22:39.020
of the whole chapter 11.

0:22:39.020,0:22:40.970
I sent you the videos[br]for chapter 11.

0:22:40.970,0:22:43.540
And for chapter 11, you[br]have this huge homework

0:22:43.540,0:22:46.770
which is 49 problems.

0:22:46.770,0:22:50.430
So please do not,[br]do not leave it

0:22:50.430,0:22:52.260
to the last five[br]days or six days,

0:22:52.260,0:22:55.710
because it's going to kill you.

0:22:55.710,0:22:57.495
There are people who[br]say, I can finish

0:22:57.495,0:22:58.620
this in the next five days.

0:22:58.620,0:23:00.320
I know you can.

0:23:00.320,0:23:01.950
I know you can,[br]I don't doubt it.

0:23:01.950,0:23:04.420
That's why I left[br]you so much freedom.

0:23:04.420,0:23:07.610
But you have-- today is[br]the second or the third?

0:23:07.610,0:23:10.860
So practically you have[br]25 days to work on this.

0:23:10.860,0:23:15.200
On the 28th at 11 PM[br]it's going to close.

0:23:15.200,0:23:18.820
I would work a few[br]problems every other day.

0:23:18.820,0:23:22.050
Because I need a break,[br]so I would alternate.

0:23:22.050,0:23:25.030
But don't leave it--[br]even if you have help,

0:23:25.030,0:23:27.690
especially if you have help,[br]like a tutor or tutoring

0:23:27.690,0:23:30.280
services here that are[br]free in the department.

0:23:30.280,0:23:32.220
Do not leave it[br]to the last days.

0:23:32.220,0:23:35.160
Because you're putting pressure[br]on yourself, on your brain,

0:23:35.160,0:23:37.221
on your tutor, on everybody.

0:23:37.221,0:23:37.720
Yes sir.

0:23:37.720,0:23:38.640
STUDENT: So that's[br]homework three?

0:23:38.640,0:23:40.223
PROFESSOR TODA:[br]That's homework three,

0:23:40.223,0:23:43.305
and it's a huge homework[br]over chapter 11.

0:23:43.305,0:23:45.610
STUDENT: You said[br]there are 49 problems?

0:23:45.610,0:23:49.157
PROFESSOR TODA: I don't[br]remember exactly but 47, 49.

0:23:49.157,0:23:50.240
I don't remember how many.

0:23:50.240,0:23:52.850
STUDENT: Between 45 and 50.

0:23:52.850,0:23:55.910
PROFESSOR TODA:[br]Between 45 and 50, yes.

0:23:55.910,0:23:59.210
If you encounter any bug--[br]although there shouldn't

0:23:59.210,0:24:02.020
be bugs, maybe 1 in 1,000.

0:24:02.020,0:24:04.530
If you encounter any[br]bug that the programmer

0:24:04.530,0:24:09.710
of those problems may[br]have accidentally put in,

0:24:09.710,0:24:11.370
you let me know.

0:24:11.370,0:24:13.580
So I can contact them.

0:24:13.580,0:24:17.390
If there is a problem that I[br]consider shouldn't be there,

0:24:17.390,0:24:19.790
I will eliminate that later on.

0:24:19.790,0:24:23.450
But hopefully, everything[br]will be doable,

0:24:23.450,0:24:28.096
everything will be fair and[br]you will be able to solve it.

0:24:28.096,0:24:32.140


0:24:32.140,0:24:34.590
Any questions?

0:24:34.590,0:24:37.015
Particular questions[br]from the homework?

0:24:37.015,0:24:39.925


0:24:39.925,0:24:43.805
STUDENT: [INAUDIBLE] is it to[br]parametrize a circle of a set,

0:24:43.805,0:24:47.860
like of a certain[br]radius on the xy-plane?

0:24:47.860,0:24:49.260
PROFESSOR TODA:[br]Shall we do that?

0:24:49.260,0:24:53.228
Do you want me to do that[br]in general, in xy-plane, OK.

0:24:53.228,0:24:55.224
STUDENT: [INAUDIBLE][br]in the xy-plane.

0:24:55.224,0:24:59.220


0:24:59.220,0:25:04.660
PROFESSOR TODA: xy-plane and[br]then what was the equation?

0:25:04.660,0:25:10.162
Was it like a equals sine[br]of t or a equals sine of bt?

0:25:10.162,0:25:12.432
Because it's a[br]little bit different,

0:25:12.432,0:25:15.790
depending on how the[br]parametrization was given.

0:25:15.790,0:25:17.165
What's your name[br]again, I forgot.

0:25:17.165,0:25:18.915
I don't know what to refer you.

0:25:18.915,0:25:19.540
STUDENT: Ryder.

0:25:19.540,0:25:22.397


0:25:22.397,0:25:24.730
PROFESSOR TODA: Was that part[br]of what's due on the 11th?

0:25:24.730,0:25:27.890
STUDENT: It doesn't-- yes, it[br]doesn't give a revision set.

0:25:27.890,0:25:29.050
It says--

0:25:29.050,0:25:33.000
PROFESSOR TODA: Let me quickly[br]read-- find parametrization

0:25:33.000,0:25:38.440
of the circle of radius 7 in[br]the xy-plane, centered at 3, 1,

0:25:38.440,0:25:40.615
oriented counterclockwise.

0:25:40.615,0:25:43.238
The point 10, 1[br]should be connected--

0:25:43.238,0:25:44.675
STUDENT: Just one more second.

0:25:44.675,0:25:45.633
PROFESSOR TODA: Do[br]you mind if I put it.

0:25:45.633,0:25:47.070
I'll take good care of it.

0:25:47.070,0:25:48.028
I won't drop it.

0:25:48.028,0:25:51.860


0:25:51.860,0:25:57.880
So the point-- parametrization[br]of the circle of radius

0:25:57.880,0:26:01.980
7 in the xy-plane,[br]centered at 3, 1.

0:26:01.980,0:26:12.090
So circle centered at-- and[br]I'll say it x0, 1 0, being 3, 1.

0:26:12.090,0:26:16.340


0:26:16.340,0:26:19.040
No, because then I'm[br]solving your problem.

0:26:19.040,0:26:20.710
But I'm solving[br]your problem anyway,

0:26:20.710,0:26:23.480
even if I change[br]change the numbers.

0:26:23.480,0:26:26.060


0:26:26.060,0:26:27.960
Why don't I change[br]the numbers, and then

0:26:27.960,0:26:30.850
you do it for the given numbers.

0:26:30.850,0:26:33.880
Let's say 1, 0.

0:26:33.880,0:26:39.610
And it's the same type[br]of problem, right?

0:26:39.610,0:26:42.662
Oriented counterclockwise.

0:26:42.662,0:26:43.370
That's important.

0:26:43.370,0:26:51.720


0:26:51.720,0:26:54.290
So you have circle radius 7.

0:26:54.290,0:26:56.850
I think people could[br]have any other,

0:26:56.850,0:27:01.370
because problems are-- sometimes[br]you get a random assignment.

0:27:01.370,0:27:05.472
So you have R[br]equals 2, let's say.

0:27:05.472,0:27:08.330


0:27:08.330,0:27:14.230
And you have the point,[br]how to make up something.

0:27:14.230,0:27:21.292
The point corresponding[br]to t equals

0:27:21.292,0:27:29.794
0 will be given as you have[br][INAUDIBLE], 1, 0, whatever.

0:27:29.794,0:27:32.000
OK?

0:27:32.000,0:27:36.850
Use the t as the parameter[br]for all your answers.

0:27:36.850,0:27:39.180
So use t as a parameter[br]for all your answers,

0:27:39.180,0:27:42.915
and the answers are written in[br]the interactive field as x of t

0:27:42.915,0:27:45.310
equals what and y[br]of t equals what,

0:27:45.310,0:27:47.414
and it's waiting for[br]you to fill them in.

0:27:47.414,0:27:49.230
You know.

0:27:49.230,0:27:54.470
OK, now I was talking[br]to [INAUDIBLE].

0:27:54.470,0:27:56.890
I'm going to give[br]this back to you.

0:27:56.890,0:27:57.680
Thank you, Ryan.

0:27:57.680,0:28:02.760
So when you said it's a[br]little bit frustrating,

0:28:02.760,0:28:07.800
and I agree wit you, that[br]in this variant of webwork

0:28:07.800,0:28:10.760
problems you have to enter[br]both of them correctly

0:28:10.760,0:28:14.640
in order to say yes, correct.

0:28:14.640,0:28:18.070
I was used to another library--[br]the library was outdated

0:28:18.070,0:28:22.480
[INAUDIBLE]-- where if I[br]enter this correctly I get 50%

0:28:22.480,0:28:25.665
credit, and if I enter this[br]incorrectly it's not going

0:28:25.665,0:28:26.810
to penalize me.

0:28:26.810,0:28:29.915
So I a little bit[br]complained about it,

0:28:29.915,0:28:32.310
and I was shown the[br]old library where

0:28:32.310,0:28:35.550
I can go ahead and go[br]back and assign problems

0:28:35.550,0:28:38.870
where you get the answer[br]correct for this one

0:28:38.870,0:28:42.130
and incorrect for this one,[br]and you get partial credit.

0:28:42.130,0:28:46.590
So I'm probably going[br]to switch to that.

0:28:46.590,0:28:47.310
Let's do that.

0:28:47.310,0:28:48.530
This is a very good problem.

0:28:48.530,0:28:51.755
I'm glad you brought it up.

0:28:51.755,0:28:56.850
What have you learned about[br]conics in high school?

0:28:56.850,0:28:59.770
You've learned about--[br]well, it depends.

0:28:59.770,0:29:01.380
You've learned about ellipse.

0:29:01.380,0:29:03.000
You've learned about hyperbola.

0:29:03.000,0:29:04.500
You've learned about parabola.

0:29:04.500,0:29:07.190
Some of you put them down[br]for me for extra credit.

0:29:07.190,0:29:08.980
I was very happy you did that.

0:29:08.980,0:29:10.490
It's a good exercise.

0:29:10.490,0:29:12.169
If you have-- Alex, yes?

0:29:12.169,0:29:14.210
STUDENT: I was just[br]thinking, does that say 1, 0?

0:29:14.210,0:29:17.600


0:29:17.600,0:29:19.410
The point corresponding[br]to t0 [INAUDIBLE]?

0:29:19.410,0:29:20.450
PROFESSOR TODA: I think[br]that's what I meant.

0:29:20.450,0:29:22.060
I don't know, I just[br]came up with it.

0:29:22.060,0:29:22.620
I made it.

0:29:22.620,0:29:23.227
1, 0.

0:29:23.227,0:29:24.310
I make up all my problems.

0:29:24.310,0:29:26.120
STUDENT: But the center[br]of the circle isn't 1, 0.

0:29:26.120,0:29:27.190
PROFESSOR TODA: Oh, oops.

0:29:27.190,0:29:29.742
Yes.

0:29:29.742,0:29:31.590
Sorry.

0:29:31.590,0:29:33.330
So 2, 0.

0:29:33.330,0:29:34.100
No--

0:29:34.100,0:29:35.290
[INTERPOSING VOICES]

0:29:35.290,0:29:38.304
PROFESSOR TODA:[br]--because the radius.

0:29:38.304,0:29:40.720
This is the problem when you[br]don't think very [INAUDIBLE].

0:29:40.720,0:29:44.240
I always like to make[br]up my own problems.

0:29:44.240,0:29:48.400
When an author, when we came up[br]with the problems in the book,

0:29:48.400,0:29:51.700
of course we had to think, draw,[br]and make sure they made sense.

0:29:51.700,0:29:55.185
But when you just come up with[br]a problem out of the middle

0:29:55.185,0:29:57.475
of nowhere-- thank you so much.

0:29:57.475,0:29:58.850
Of course, we[br]would have realized

0:29:58.850,0:30:01.090
that was nonsense[br]in just a minute.

0:30:01.090,0:30:04.470
But it's good that you told me.

0:30:04.470,0:30:06.526
So x of t, y of t.

0:30:06.526,0:30:10.810


0:30:10.810,0:30:11.900
Let's find it.

0:30:11.900,0:30:13.353
Based on what?

0:30:13.353,0:30:15.560
What is the general[br]equation of a circle?

0:30:15.560,0:30:22.340
x minus x0 squared plus y minus[br]y0 squared equals R squared.

0:30:22.340,0:30:24.530
And you have learned[br]that in high school.

0:30:24.530,0:30:26.590
Am I right or not?

0:30:26.590,0:30:27.390
You have.

0:30:27.390,0:30:28.240
OK.

0:30:28.240,0:30:29.130
Good.

0:30:29.130,0:30:36.190
Now, in our case what[br]is x0 and what is y0?

0:30:36.190,0:30:40.000
x0 is 1 and y0 is 0.

0:30:40.000,0:30:42.860
Because that's[br]why-- I don't know.

0:30:42.860,0:30:44.060
I just made it up.

0:30:44.060,0:30:47.070
And I said that's the center.

0:30:47.070,0:30:49.144
I'll draw.

0:30:49.144,0:30:50.810
I should have drawn[br]it in the beginning,

0:30:50.810,0:30:54.400
and that would have[br]helped me not come up

0:30:54.400,0:31:00.430
with some nonsensical data.

0:31:00.430,0:31:01.610
c is 1, 0.

0:31:01.610,0:31:03.010
Radius is 2.

0:31:03.010,0:31:04.640
So I'm going this way.

0:31:04.640,0:31:06.790
What point is this way, guys?

0:31:06.790,0:31:08.710
Just by the way.

0:31:08.710,0:31:10.240
Because [INAUDIBLE][br]is 1, 0, right?

0:31:10.240,0:31:16.240
And this way the other[br]extreme, the antipode is 3, 0.

0:31:16.240,0:31:20.088
So that's exactly what[br]Alexander was saying.

0:31:20.088,0:31:22.020
And now it makes sense.

0:31:22.020,0:31:25.401


0:31:25.401,0:31:26.500
Well, I cannot draw today.

0:31:26.500,0:31:27.333
STUDENT: [INAUDIBLE]

0:31:27.333,0:31:30.354


0:31:30.354,0:31:31.770
PROFESSOR TODA:[br]It looks horrible.

0:31:31.770,0:31:37.065
It looks like an egg that[br]is shaped-- disabled egg.

0:31:37.065,0:31:41.470


0:31:41.470,0:31:42.480
OK.

0:31:42.480,0:31:43.060
All right.

0:31:43.060,0:31:49.640
So the motion of-- the[br]motion will come like that.

0:31:49.640,0:31:53.736
From t equals 0, when I'm[br]here, counterclockwise,

0:31:53.736,0:31:57.312
I have to draw-- any kind of[br]circle you have in the homework

0:31:57.312,0:32:00.610
should be drawn on the board.

0:32:00.610,0:32:06.210
If you have a general, you[br]don't know what the data is.

0:32:06.210,0:32:08.800
I want to help you solve[br]the general problem.

0:32:08.800,0:32:10.775
For the original problem,[br]which is a circle

0:32:10.775,0:32:15.470
of center x, 0, y, 0 and[br]radius R, generic one,

0:32:15.470,0:32:19.574
what is the parametrization[br]without data?

0:32:19.574,0:32:20.490
Without specific data.

0:32:20.490,0:32:23.240
What is the parametrization?

0:32:23.240,0:32:25.644
And I want you to pay[br]attention very well.

0:32:25.644,0:32:26.930
You are paying attention.

0:32:26.930,0:32:29.730
You are very careful today.

0:32:29.730,0:32:31.190
[INAUDIBLE]

0:32:31.190,0:32:33.940
So what do you have?

0:32:33.940,0:32:35.740
STUDENT: Cosine.

0:32:35.740,0:32:37.520
PROFESSOR TODA:[br]Before that cosine

0:32:37.520,0:32:39.580
there is an R, excellent.

0:32:39.580,0:32:43.980
So [INAUDIBLE][br]there R cosine of t.

0:32:43.980,0:32:46.550
I'm not done.

0:32:46.550,0:32:47.475
What do I put here?

0:32:47.475,0:32:48.342
STUDENT: Over d.

0:32:48.342,0:32:49.300
PROFESSOR TODA: No, no.

0:32:49.300,0:32:51.276
I'm continuing.

0:32:51.276,0:32:52.240
STUDENT: Plus x0.

0:32:52.240,0:32:53.690
PROFESSOR TODA: Plus x0.

0:32:53.690,0:32:57.460
And R sine t plus y0.

0:32:57.460,0:32:59.560
Who taught me that?

0:32:59.560,0:33:02.640
First of all, this[br]is not unique.

0:33:02.640,0:33:03.650
It's not unique.

0:33:03.650,0:33:05.920
I could put sine t[br]here and cosine t here

0:33:05.920,0:33:08.650
and it would be the same[br]type of parametrization.

0:33:08.650,0:33:11.020
But we usually put[br]the cosine first

0:33:11.020,0:33:13.500
because we look at the[br]x-axis corresponding

0:33:13.500,0:33:17.766
to the cosine and the y-axis[br]corresponding to the sine.

0:33:17.766,0:33:20.845
If I don't know that,[br]because I happen to know that

0:33:20.845,0:33:24.170
from when I was 16 in high[br]school, if I don't know that,

0:33:24.170,0:33:25.380
what do I know?

0:33:25.380,0:33:27.550
I cook up my own[br]parametrization.

0:33:27.550,0:33:29.130
And that's a very good thing.

0:33:29.130,0:33:31.160
And I'm glad Ryan[br]asked about that.

0:33:31.160,0:33:33.360
How does one come up with this?

0:33:33.360,0:33:34.360
Do we have to memorize?

0:33:34.360,0:33:38.190
In mathematics, thank god,[br]we don't memorize much.

0:33:38.190,0:33:41.730
The way we cook up things[br]is just from, in this case,

0:33:41.730,0:33:44.930
from the Pythagorean[br]theorem of-- no.

0:33:44.930,0:33:47.140
Pythagorean theorem[br]of trigonometry?

0:33:47.140,0:33:49.170
The fundamental identity[br]of trigonometry,

0:33:49.170,0:33:52.840
which is the same thing as[br]the Pythagorean theorem.

0:33:52.840,0:33:55.300
What's the fundamental[br]identity of trigonometry?

0:33:55.300,0:33:58.210
Cosine squared plus[br]sin squared equals 1.

0:33:58.210,0:34:03.681
If I have a problem[br]like that, I must

0:34:03.681,0:34:08.810
have that this is R cosine[br]t and this is R sine t.

0:34:08.810,0:34:11.310
Because when I take[br]the red guys and I

0:34:11.310,0:34:13.818
square them and I[br]add them together,

0:34:13.818,0:34:17.650
I'm going to have R squared.

0:34:17.650,0:34:18.940
All righty, good.

0:34:18.940,0:34:23.370
So no matter what[br]kind of data you have,

0:34:23.370,0:34:27.774
you should be able to come[br]up with this on your own.

0:34:27.774,0:34:33.510
And what else is[br]going to be happening?

0:34:33.510,0:34:38.139
When I solve for x of-- the[br]point corresponding to t

0:34:38.139,0:34:39.389
equals 0.

0:34:39.389,0:34:43.920
x of 0 and y of 0 will[br]therefore be what?

0:34:43.920,0:34:49.350
It will be R plus x0.

0:34:49.350,0:34:51.429
This is going to be what?

0:34:51.429,0:34:53.710
Just y0.

0:34:53.710,0:34:56.194
Does anybody give them to me?

0:34:56.194,0:34:59.110
STUDENT: 3, 0.

0:34:59.110,0:35:01.630
PROFESSOR TODA: Alexander[br]gave me the correct ones.

0:35:01.630,0:35:05.670
They will be 3 and 0.

0:35:05.670,0:35:06.670
Are you guys with me?

0:35:06.670,0:35:10.920
They could be anything,[br]anything that makes sense.

0:35:10.920,0:35:15.100
All right, for example somebody[br]would say, I'm starting here.

0:35:15.100,0:35:16.580
I give you other points.

0:35:16.580,0:35:19.910
Then you put them in, you[br]plug in that initial point,

0:35:19.910,0:35:22.980
meaning that you're[br]starting your motion here.

0:35:22.980,0:35:26.120
And you do go around[br]the circle one

0:35:26.120,0:35:31.688
because, you take [INAUDIBLE][br]only between 0 and 2 pi.

0:35:31.688,0:35:32.852
Alexander.

0:35:32.852,0:35:34.018
STUDENT: I have [INAUDIBLE].

0:35:34.018,0:35:34.950
PROFESSOR TODA: OK.

0:35:34.950,0:35:35.882
STUDENT: [INAUDIBLE]

0:35:35.882,0:35:38.310
PROFESSOR TODA: No, I thought[br]that I misprinted something

0:35:38.310,0:35:38.610
again.

0:35:38.610,0:35:40.784
STUDENT: No, I was about to[br]say something really dumb.

0:35:40.784,0:35:41.575
PROFESSOR TODA: OK.

0:35:41.575,0:35:43.840


0:35:43.840,0:35:48.692
So how do we make sense[br]of what we have here?

0:35:48.692,0:35:52.280
Well, y0 corresponds[br]to what I said.

0:35:52.280,0:35:55.680
So this is a[br]superfluous equation.

0:35:55.680,0:35:57.620
I don't need that.

0:35:57.620,0:36:01.200
What do I know from that?

0:36:01.200,0:36:05.824
R will be 2.

0:36:05.824,0:36:07.690
x1 is 1.

0:36:07.690,0:36:10.000
I have a superfluous equation.

0:36:10.000,0:36:13.890
I have to get identities[br]in that case, right?

0:36:13.890,0:36:14.700
OK, now.

0:36:14.700,0:36:19.640


0:36:19.640,0:36:26.640
What is going to be my--[br]my bunch of equations

0:36:26.640,0:36:47.640
will be x of t equals 2[br]cosine t plus 1 and y of t

0:36:47.640,0:36:49.230
equals-- I don't[br]like this marker.

0:36:49.230,0:36:49.920
I hate it.

0:36:49.920,0:36:50.772
Where did I get it?

0:36:50.772,0:36:51.730
In the math department.

0:36:51.730,0:36:53.460
And it's a new one.

0:36:53.460,0:36:54.820
I got it as a new one.

0:36:54.820,0:36:56.740
It's not working.

0:36:56.740,0:36:58.180
OK, y of t.

0:36:58.180,0:37:01.150


0:37:01.150,0:37:03.830
The blue contrast is invisible.

0:37:03.830,0:37:07.897
I have 2 sine t.

0:37:07.897,0:37:08.396
Okey dokey.

0:37:08.396,0:37:11.590
When you finish a[br]problem, always quickly

0:37:11.590,0:37:15.742
verify if what you[br]got makes sense.

0:37:15.742,0:37:19.500
And obviously if I[br]look at everything,

0:37:19.500,0:37:21.130
it's matching the whole point.

0:37:21.130,0:37:21.906
Right?

0:37:21.906,0:37:22.680
OK.

0:37:22.680,0:37:29.843
Now going back to-- this is[br]reminding me of something in 3d

0:37:29.843,0:37:34.673
that I wanted to talk[br]to you today about.

0:37:34.673,0:37:36.605
This is [INAUDIBLE].

0:37:36.605,0:37:42.900


0:37:42.900,0:37:45.120
I'm going to give[br]you, in a similar way

0:37:45.120,0:37:47.501
with this simple[br]problem, I'm going

0:37:47.501,0:37:49.886
to give you something[br]more complicated

0:37:49.886,0:38:16.700
and say find the[br]parametrization of a helix.

0:38:16.700,0:38:19.610
And you say, well,[br]I'm happy that this

0:38:19.610,0:38:21.824
isn't a made-up problem again.

0:38:21.824,0:38:23.918
I have to be a little[br]bit more careful

0:38:23.918,0:38:27.430
with these made-up problems[br]so that they make sense.

0:38:27.430,0:38:44.380
Of a helix R of t such that[br]it is contained or it lies,

0:38:44.380,0:38:59.770
it lies on the circular[br]cylinder x squared

0:38:59.770,0:39:03.822
plus y squared equals 4.

0:39:03.822,0:39:04.780
Why is that a cylinder?

0:39:04.780,0:39:07.910
The z's missing, so it's[br]going to be a cylinder whose

0:39:07.910,0:39:09.470
main axis is the z axis.

0:39:09.470,0:39:09.970
Right?

0:39:09.970,0:39:11.450
Are you guys with me?

0:39:11.450,0:39:14.750
I think we are on the same page.

0:39:14.750,0:39:19.449
And you cannot solve the[br]problem just with this data.

0:39:19.449,0:39:22.160
Do you agree with me?

0:39:22.160,0:39:47.190
And knowing that, the[br]curvature of the helix is k

0:39:47.190,0:40:04.030
equals 2/5 at every point.

0:40:04.030,0:40:06.080
And of course it's an oxymoron.

0:40:06.080,0:40:08.240
Because what I[br]proved last time is

0:40:08.240,0:40:12.530
that the curvature of[br]a helix is a constant.

0:40:12.530,0:40:27.220
So remember, we got the[br]curvature of a helix

0:40:27.220,0:40:30.056
as being a constant.

0:40:30.056,0:40:34.455


0:40:34.455,0:40:36.413
STUDENT: What's that last[br]word of the sentence?

0:40:36.413,0:40:38.589
It's "the curvature[br]is at every" what?

0:40:38.589,0:40:39.880
PROFESSOR TODA: At every point.

0:40:39.880,0:40:45.070
I'm sorry I said, it very--[br]I abbreviated [INAUDIBLE].

0:40:45.070,0:40:48.100
So at every point you[br]have the same curvature.

0:40:48.100,0:40:50.920
When you draw a[br]helix you say, wait,

0:40:50.920,0:40:53.820
the helix is bent uniformly.

0:40:53.820,0:40:58.760
If you were to play with a[br]spring taken from am old bed,

0:40:58.760,0:41:01.910
you would go with your[br]hands along the spring.

0:41:01.910,0:41:04.700
And then you say, oh,[br]it bends about the same.

0:41:04.700,0:41:06.010
Yes, it does.

0:41:06.010,0:41:08.800
And that means the[br]curvature is the same.

0:41:08.800,0:41:11.720
How would you[br]solve this problem?

0:41:11.720,0:41:16.380
This problem is hard,[br]because you cannot integrate

0:41:16.380,0:41:17.450
the curvature.

0:41:17.450,0:41:19.110
Well, what is the curvature?

0:41:19.110,0:41:21.040
The curvature would be--

0:41:21.040,0:41:22.040
STUDENT: Absolute value.

0:41:22.040,0:41:23.910
PROFESSOR TODA: Just[br]absolute value of R

0:41:23.910,0:41:28.410
double prime if it were in s.

0:41:28.410,0:41:31.030
And you cannot integrate.

0:41:31.030,0:41:34.000
If somebody gave you[br]the vector equation

0:41:34.000,0:41:36.570
of double prime of[br]this, them you say,

0:41:36.570,0:41:38.730
yes, I can integrate[br]one step going back.

0:41:38.730,0:41:40.330
I get R prime of s.

0:41:40.330,0:41:41.729
Then I go back to R of s.

0:41:41.729,0:41:43.270
But this is a little[br]bit complicated.

0:41:43.270,0:41:45.492
I'm giving you a scalar.

0:41:45.492,0:41:50.760
You have to be a little bit[br]aware of what you did last time

0:41:50.760,0:41:54.596
and try to remember[br]what we did last time.

0:41:54.596,0:41:56.330
What did we do last time?

0:41:56.330,0:41:58.110
I would not give you[br]a problem like that

0:41:58.110,0:42:03.290
on the final, because it would[br]assume that you have solved

0:42:03.290,0:42:06.282
the problem we did last[br]time in terms of R of t

0:42:06.282,0:42:09.950
equals A equals sine t.

0:42:09.950,0:42:11.400
A sine t and [? vt. ?]

0:42:11.400,0:42:15.770
And we said, this is the[br]standard parametrized helix

0:42:15.770,0:42:22.330
that sits on a cylinder of[br]radius A and has the phb.

0:42:22.330,0:42:27.510
So the distance between[br]consecutive spirals

0:42:27.510,0:42:28.770
really matters.

0:42:28.770,0:42:30.160
That really makes[br]the difference.

0:42:30.160,0:42:30.620
STUDENT: I have a question.

0:42:30.620,0:42:32.578
PROFESSOR TODA: You wanted[br]to ask me something.

0:42:32.578,0:42:34.350
STUDENT: Is s always[br]the reciprocal of t?

0:42:34.350,0:42:35.952
Are they always--

0:42:35.952,0:42:37.410
PROFESSOR TODA:[br]No, not reciprocal.

0:42:37.410,0:42:45.800
You mean s of t is a function[br]is from t0 to t of the speed.

0:42:45.800,0:42:50.430
R prime and t-- d tau, right?

0:42:50.430,0:42:51.630
Tau not t. [INAUDIBLE].

0:42:51.630,0:42:54.220


0:42:54.220,0:43:00.650
t and s are[br]different parameters.

0:43:00.650,0:43:01.993
Different times.

0:43:01.993,0:43:04.369
Different parameter times.

0:43:04.369,0:43:04.910
And you say--

0:43:04.910,0:43:06.701
STUDENT: Isn't s[br]the parameter time

0:43:06.701,0:43:08.807
when [INAUDIBLE] parametrized?

0:43:08.807,0:43:09.890
PROFESSOR TODA: Very good.

0:43:09.890,0:43:12.365
So what is the magic s?

0:43:12.365,0:43:13.850
I'm proud of you.

0:43:13.850,0:43:15.940
This is the important[br]thing to remember.

0:43:15.940,0:43:17.530
t could be any time.

0:43:17.530,0:43:19.960
I start measuring[br]wherever I want.

0:43:19.960,0:43:23.690
I can set my watch to start now.

0:43:23.690,0:43:24.950
It could be crazy.

0:43:24.950,0:43:26.640
Doesn't have to be uniform.

0:43:26.640,0:43:27.603
Motion, I don't care.

0:43:27.603,0:43:30.760


0:43:30.760,0:43:33.330
s is a friend of[br]yours that says,

0:43:33.330,0:43:38.290
I am that special time[br]so that according to me

0:43:38.290,0:43:40.590
the speed will become one.

0:43:40.590,0:43:45.650
So for a physicist to measure[br]the speed with respect to this,

0:43:45.650,0:43:49.410
parameter s time, the speed[br]will always become one.

0:43:49.410,0:43:51.660
That is the arclength[br]time and position.

0:43:51.660,0:43:54.480
How you get from one[br]another, I told you last time

0:43:54.480,0:43:57.376
that for both of them[br]you have-- this is R of t

0:43:57.376,0:43:59.230
and this is little r of s.

0:43:59.230,0:44:01.240
And there is a composition.

0:44:01.240,0:44:03.460
s can be viewed as[br]a function of t,

0:44:03.460,0:44:06.340
and t can be viewed[br]as a function of s.

0:44:06.340,0:44:09.580
As functions they are[br]inverse to one another.

0:44:09.580,0:44:12.650
So going back to who they[br]are, a very good question,

0:44:12.650,0:44:15.580
because this is a review[br]anyway, [? who wants ?]

0:44:15.580,0:44:19.180
s as a function of t for[br]this particular problem?

0:44:19.180,0:44:23.949
I hope you remember, we were[br]like-- have you seen this movie

0:44:23.949,0:44:28.395
with Mickey Mouse going[br]on a mountain that

0:44:28.395,0:44:32.100
was more like a cylinder.

0:44:32.100,0:44:35.320
And this is the train[br]going at a constant slope.

0:44:35.320,0:44:42.590
And one of my colleagues,[br]actually, he's at Stanford,

0:44:42.590,0:44:47.300
was telling me that he[br]gave his students in Calc 1

0:44:47.300,0:44:51.860
to prove, formally prove,[br]that the angle formed

0:44:51.860,0:44:56.590
by the law of motion[br]by the velocity vector,

0:44:56.590,0:45:01.990
with the horizontal plane[br]passing through the particle,

0:45:01.990,0:45:04.050
is always a constant.

0:45:04.050,0:45:07.165
I didn't think about doing[br]in now, but of course we can.

0:45:07.165,0:45:08.520
We could do that.

0:45:08.520,0:45:10.964
So maybe the next[br]thing would be, like,

0:45:10.964,0:45:12.630
if you [INAUDIBLE][br]an extra problem, can

0:45:12.630,0:45:17.280
we show that angle between the[br]velocity vector on the helix

0:45:17.280,0:45:20.702
and the horizontal plane through[br]that point is a constant.

0:45:20.702,0:45:22.535
STUDENT: Wouldn't it[br]just be, because B of t

0:45:22.535,0:45:23.935
is just a constant times t?

0:45:23.935,0:45:24.810
PROFESSOR TODA: Yeah.

0:45:24.810,0:45:25.590
We'll get to that.

0:45:25.590,0:45:27.170
We'll get to that in a second.

0:45:27.170,0:45:32.487
So he reminded me of an old[br]movie from like 70 years ago,

0:45:32.487,0:45:33.820
with Mickey Mouse and the train.

0:45:33.820,0:45:38.650
And the train going[br]up at the same speed.

0:45:38.650,0:45:41.160
You have to maintain[br]the same speed.

0:45:41.160,0:45:44.810
Because if you risk it[br]not, then you sort of

0:45:44.810,0:45:46.160
are getting trouble.

0:45:46.160,0:45:47.760
So you never stop.

0:45:47.760,0:45:49.290
If you stop you go back.

0:45:49.290,0:45:50.740
So it's a regular curve.

0:45:50.740,0:45:52.875
What I have here is[br]that such a curve.

0:45:52.875,0:45:54.786
Regular curve, never stop.

0:45:54.786,0:45:56.800
Get up with a constant speed.

0:45:56.800,0:45:58.826
Do you guys remember the[br]speed from last time?

0:45:58.826,0:46:01.077
We'll square root the a[br]squared plus b squared.

0:46:01.077,0:46:04.430
When we did the[br]velocity thingie.

0:46:04.430,0:46:10.730
And I get square root a[br]squared plus b squared times t.

0:46:10.730,0:46:19.040
Now, today I would like[br]to ask you one question.

0:46:19.040,0:46:21.520
What if-- Ryan brought this up.

0:46:21.520,0:46:22.460
It's very good.

0:46:22.460,0:46:23.660
b is a constant.

0:46:23.660,0:46:26.550
What if b would[br]not be a constant,

0:46:26.550,0:46:28.610
or maybe could be worse?

0:46:28.610,0:46:32.710
For example, instead of having[br]another linear function with t,

0:46:32.710,0:46:36.178
but something that contains[br]higher powers of t.

0:46:36.178,0:46:39.360


0:46:39.360,0:46:43.410
Then you don't go at the[br]constant speed anymore.

0:46:43.410,0:46:45.370
You can say goodbye[br]to the cartoon.

0:46:45.370,0:46:45.880
Yes, sir?

0:46:45.880,0:46:49.017
STUDENT: And then[br]it's [INAUDIBLE].

0:46:49.017,0:46:50.100
One that goes [INAUDIBLE].

0:46:50.100,0:46:51.600
PROFESSOR TODA: I[br]mean, it's still--

0:46:51.600,0:46:54.826
STUDENT: s is not[br]multiplied by a constant.

0:46:54.826,0:46:57.017
The function between t and[br]s is not a constant one.

0:46:57.017,0:46:59.600
PROFESSOR TODA: It's going to[br]be a different parameterization,

0:46:59.600,0:47:00.580
different speed.

0:47:00.580,0:47:03.770
Sometimes-- OK, you[br]have to understand.

0:47:03.770,0:47:06.740
Let's say I have a cone.

0:47:06.740,0:47:10.230
And I'm going slow[br]at first, and I

0:47:10.230,0:47:11.980
go faster and faster[br]and faster and faster

0:47:11.980,0:47:13.900
to the end of the cone.

0:47:13.900,0:47:18.390
But then I have the[br]same physical curve,

0:47:18.390,0:47:21.040
and I parameterized[br][INAUDIBLE] the length.

0:47:21.040,0:47:24.310
And I say, no, I'm a mechanic.

0:47:24.310,0:47:26.840
Or I'm the engineer[br]of the strain.

0:47:26.840,0:47:29.420
I can make the motion[br]have a constant speed.

0:47:29.420,0:47:33.130
So even if the helix[br]is no longer circular,

0:47:33.130,0:47:36.560
and it's sort of a crazy helix[br]going on top of the mountain,

0:47:36.560,0:47:39.330
as an engineer I[br]can just say, oh no,

0:47:39.330,0:47:42.150
I want cruise control[br]for my little train.

0:47:42.150,0:47:45.500
And I will go at the same speed.

0:47:45.500,0:47:48.940
See, the problem is[br]the slope a constant.

0:47:48.940,0:47:51.270
And thinking of[br]what they did that

0:47:51.270,0:47:53.067
stand for, because[br]it didn't stand

0:47:53.067,0:47:54.880
for [INAUDIBLE] in honors.

0:47:54.880,0:47:57.280
We can do it in honors as well.

0:47:57.280,0:47:58.700
We'll do it in a second.

0:47:58.700,0:48:04.950
Now, k obviously is what?

0:48:04.950,0:48:08.460
Some of you have[br]very good memory,

0:48:08.460,0:48:13.250
and like the memory of a[br]medical doctor, which is great.

0:48:13.250,0:48:14.560
Some of you don't.

0:48:14.560,0:48:18.772
But if you don't you just go[br]back and look at the notes.

0:48:18.772,0:48:20.756
What I'm trying to[br]do, but I don't know,

0:48:20.756,0:48:22.960
it's also a matter[br]of money-- I don't

0:48:22.960,0:48:26.040
want to use the math[br]department copier-- I'd

0:48:26.040,0:48:29.850
like to make a stack of notes.

0:48:29.850,0:48:33.090
So that's why I'm collecting[br]these notes, to bring them back

0:48:33.090,0:48:33.971
to you.

0:48:33.971,0:48:34.470
For free!

0:48:34.470,0:48:36.470
I'm not going to[br]sell them to you.

0:48:36.470,0:48:38.060
I'm [INAUDIBLE].

0:48:38.060,0:48:41.515
So that you can have those[br]with you whenever you want,

0:48:41.515,0:48:45.475
or put them in a spiral,[br]punch holes in them,

0:48:45.475,0:48:48.680
and have them for[br]review at any time.

0:48:48.680,0:48:51.450
Reminds me of what that[br]was-- that was in the notes.

0:48:51.450,0:48:54.694
a over a squared plus b squared.

0:48:54.694,0:48:57.300
So who can tell me, a[br]and b really quickly,

0:48:57.300,0:49:00.870
so we don't waste too[br]much time, Mr. a is--?

0:49:00.870,0:49:05.729


0:49:05.729,0:49:07.020
STUDENT: So this is another way

0:49:07.020,0:49:07.601
STUDENT: 2.

0:49:07.601,0:49:08.350
PROFESSOR TODA: 2.

0:49:08.350,0:49:13.037
STUDENT: So is this another[br]way of defining k in k of s?

0:49:13.037,0:49:14.120
PROFESSOR TODA: Actually--

0:49:14.120,0:49:16.895
STUDENT: That's the general[br]curvature for [INAUDIBLE].

0:49:16.895,0:49:21.040
PROFESSOR TODA: You know[br]what is the magic thing?

0:49:21.040,0:49:23.095
Even if-- the curvature[br]is an invariant.

0:49:23.095,0:49:26.526
It doesn't depend the[br]reparametrization.

0:49:26.526,0:49:29.830
There is a way maybe I'm going[br]to teach you, although this

0:49:29.830,0:49:32.160
is not in the book.

0:49:32.160,0:49:35.870
What are the formulas[br]corresponding

0:49:35.870,0:49:41.580
to the [INAUDIBLE] t and v that[br]depend on curvature and torsion

0:49:41.580,0:49:44.000
and the speed along the curve.

0:49:44.000,0:49:48.750
And if you analyze the notion[br]of curvature, [INAUDIBLE],

0:49:48.750,0:49:52.230
no matter what your[br]parameter will be, t, s, tau,

0:49:52.230,0:49:56.690
God knows what, k will[br]still be the same number.

0:49:56.690,0:49:59.300
So k is viewed as an[br]invariant with respect

0:49:59.300,0:50:01.425
to the parametrization.

0:50:01.425,0:50:04.120
STUDENT: So then that a over[br]a squared plus b squared,

0:50:04.120,0:50:05.912
that's another way of finding k?

0:50:05.912,0:50:07.120
PROFESSOR TODA: Say it again?

0:50:07.120,0:50:09.285
STUDENT: So using a over[br]a squared plus b squared

0:50:09.285,0:50:10.829
is another way of finding k?

0:50:10.829,0:50:11.620
PROFESSOR TODA: No.

0:50:11.620,0:50:13.916
Somebody gave you k.

0:50:13.916,0:50:17.440
And then you say, if it's[br]a standard parametrization,

0:50:17.440,0:50:25.290
and then I get 2/5,[br]can I be sure a is 2?

0:50:25.290,0:50:28.220
I'm sure a is 2 from nothing.

0:50:28.220,0:50:32.860
This is what makes me aware[br]that a is 2 the first place.

0:50:32.860,0:50:36.640
Because its the radius[br]of the cylinder.

0:50:36.640,0:50:39.290
This is x squared, x and y.

0:50:39.290,0:50:41.860
You see, x squared plus[br]y squared is a squared.

0:50:41.860,0:50:43.650
This is where I get a from.

0:50:43.650,0:50:44.620
a is 2.

0:50:44.620,0:50:47.140
I replace it in here[br]and I say, all righty,

0:50:47.140,0:50:51.777
so I only have one[br]choice. a is 2 and b is?

0:50:51.777,0:50:52.610
STUDENT: [INAUDIBLE]

0:50:52.610,0:50:57.260


0:50:57.260,0:51:00.470
PROFESSOR TODA: But can b[br]plus-- So what I'm saying,

0:51:00.470,0:51:01.380
a is 2, right?

0:51:01.380,0:51:04.390
We know that from this.

0:51:04.390,0:51:08.440
If I block in here I have 4[br]and somebody says plus minus 1.

0:51:08.440,0:51:09.520
No.

0:51:09.520,0:51:11.000
b is always positive.

0:51:11.000,0:51:13.380
So you remember the[br]last time we discussed

0:51:13.380,0:51:16.640
about the standard[br]parametrization.

0:51:16.640,0:51:20.280
But somebody will say,[br]but what if I put a minus?

0:51:20.280,0:51:22.840
What if I'm going[br]to put a minus?

0:51:22.840,0:51:24.150
That's an excellent question.

0:51:24.150,0:51:27.072
What's going to happen[br]if you put minus t?

0:51:27.072,0:51:28.010
[INTERPOSING VOICES]

0:51:28.010,0:51:29.010
PROFESSOR TODA: Exactly.

0:51:29.010,0:51:31.260
In the opposite direction.

0:51:31.260,0:51:35.550
Instead of going[br]up, you go down.

0:51:35.550,0:51:37.430
All right.

0:51:37.430,0:51:41.095
Now, I'm gonna-- what else?

0:51:41.095,0:51:43.270
Ah, I said, let's do this.

0:51:43.270,0:51:47.986
Let's prove that the[br]angle is a constant,

0:51:47.986,0:51:51.080
the angle that's[br]made by the velocity

0:51:51.080,0:51:56.220
vector of the train with the[br]horizontal plane is a constant.

0:51:56.220,0:51:57.840
Is this hard?

0:51:57.840,0:51:58.340
Nah.

0:51:58.340,0:51:58.840
Yes, sir?

0:51:58.840,0:52:03.930
STUDENT: Are we still going[br]to find R of t given only k?

0:52:03.930,0:52:05.550
PROFESSOR TODA: But didn't we?

0:52:05.550,0:52:07.300
We did.

0:52:07.300,0:52:13.750
R of t was 2 cosine[br]t, 2 sine t, and t.

0:52:13.750,0:52:16.280
All right?

0:52:16.280,0:52:17.470
OK, so we are done.

0:52:17.470,0:52:18.640
What did I say?

0:52:18.640,0:52:22.410
I said that let's[br]prove-- it's a proof.

0:52:22.410,0:52:27.305
Let's prove that the angle made[br]by the velocity to the train--

0:52:27.305,0:52:30.635
to the train?-- to the direction[br]of motion, which is the helix.

0:52:30.635,0:52:37.438
And the horizontal[br]plane is a constant.

0:52:37.438,0:52:38.426
Is this hard?

0:52:38.426,0:52:39.908
How are we going to do that?

0:52:39.908,0:52:42.872
Now I start waking up,[br]because I was very tired.

0:52:42.872,0:52:44.259
STUDENT: [INAUDIBLE]

0:52:44.259,0:52:45.342
PROFESSOR TODA: Excuse me.

0:52:45.342,0:52:46.840
STUDENT: [INAUDIBLE]

0:52:46.840,0:53:01.242
PROFESSOR TODA: So you see,[br]the helix contains this point.

0:53:01.242,0:53:03.920
And I'm looking at[br]the velocity vector

0:53:03.920,0:53:06.310
that is standard to the helix.

0:53:06.310,0:53:09.320
And I'll call that R prime.

0:53:09.320,0:53:10.980
And then you say,[br]yea, but how am I

0:53:10.980,0:53:13.990
going to compute that angle?

0:53:13.990,0:53:15.632
What is that angle?

0:53:15.632,0:53:17.987
STUDENT: It's a function of b.

0:53:17.987,0:53:20.820


0:53:20.820,0:53:21.980
PROFESSOR TODA: It will be.

0:53:21.980,0:53:24.820
But we have to do it rigorously.

0:53:24.820,0:53:27.925
So what's going to happen[br]for me to draw that angle?

0:53:27.925,0:53:30.094
First of all, I should[br]take-- from the tip

0:53:30.094,0:53:33.240
of the vector I should[br]draw perpendicular

0:53:33.240,0:53:36.235
to the horizontal plane[br]passing through the point.

0:53:36.235,0:53:37.110
And I'll get P prime.

0:53:37.110,0:53:37.693
God knows why.

0:53:37.693,0:53:41.490
I don't know why, I don't know[br]why. [? Q. ?] And this is PR,

0:53:41.490,0:53:42.910
and P-- not PR.

0:53:42.910,0:53:46.990
PR is too much[br][INAUDIBLE] radius, M.

0:53:46.990,0:53:50.835
OK, so then you would[br]take PQ and then

0:53:50.835,0:53:52.604
you would measure this angle.

0:53:52.604,0:53:54.770
Well, you have to be a[br]little bit smarter than that,

0:53:54.770,0:53:58.390
because you can[br]prove something else.

0:53:58.390,0:54:02.930
This is the complement of[br]another angle that you love.

0:54:02.930,0:54:07.095
And using chapter 9 you can[br]do that angle in no time.

0:54:07.095,0:54:15.840


0:54:15.840,0:54:20.800
So this is the[br]complement of the angle

0:54:20.800,0:54:23.500
formed by the velocity vector[br]of prime with the normal.

0:54:23.500,0:54:26.680


0:54:26.680,0:54:29.720
But not the normal principle[br]normal to the curve,

0:54:29.720,0:54:32.340
but the normal to the plane.

0:54:32.340,0:54:34.510
And what is the[br]normal to the plane?

0:54:34.510,0:54:38.960
Let's call the principal normal[br]n to the curve big N bar.

0:54:38.960,0:54:42.110
So in order to avoid confusion,[br]I'll write this little n.

0:54:42.110,0:54:42.945
How about that?

0:54:42.945,0:54:45.240
Do you guys know-- like[br]they do in mechanics.

0:54:45.240,0:54:48.360
If you have two normals,[br]they call that 1n.

0:54:48.360,0:54:51.200
1 is little n, and[br]stuff like that.

0:54:51.200,0:54:52.980
So this is the complement.

0:54:52.980,0:54:55.430
If I were able to prove[br]that that complement

0:54:55.430,0:54:59.600
is a constant-- this is the[br]Stanford [? property-- ?] then

0:54:59.600,0:55:00.990
I will be happy.

0:55:00.990,0:55:03.100
Is it hard?

0:55:03.100,0:55:04.286
No, for god's sake.

0:55:04.286,0:55:07.034
Who is little n?

0:55:07.034,0:55:11.350
Little n would be-- is[br]that the normal to a plane

0:55:11.350,0:55:12.330
that you love?

0:55:12.330,0:55:13.340
What is your plane?

0:55:13.340,0:55:14.090
STUDENT: xy plane.

0:55:14.090,0:55:16.520
PROFESSOR TODA: Your[br]plane is horizontal plane.

0:55:16.520,0:55:17.320
STUDENT: xy.

0:55:17.320,0:55:18.570
PROFESSOR TODA: Yes, xy plane.

0:55:18.570,0:55:22.120
Or xy plane shifted,[br]shifted, shifted, shifted.

0:55:22.120,0:55:23.180
That's the normal change?

0:55:23.180,0:55:23.679
No.

0:55:23.679,0:55:24.886
Who is the normal?

0:55:24.886,0:55:26.174
STUDENT: [INAUDIBLE]

0:55:26.174,0:55:27.340
PROFESSOR TODA: [INAUDIBLE].

0:55:27.340,0:55:28.308
STUDENT: 0, 0, 1.

0:55:28.308,0:55:29.308
PROFESSOR TODA: 0, 0, 1.

0:55:29.308,0:55:29.808
OK.

0:55:29.808,0:55:32.110
When I put 0 I was [INAUDIBLE].

0:55:32.110,0:55:33.720
So this is k.

0:55:33.720,0:55:36.420


0:55:36.420,0:55:37.730
All right.

0:55:37.730,0:55:39.620
And what is our prime?

0:55:39.620,0:55:42.300
I was-- that was[br]a piece of cake.

0:55:42.300,0:55:47.360
We did it last time minus a[br]sine t, a equals sine t and b.

0:55:47.360,0:55:50.720


0:55:50.720,0:55:53.620
Let's find that angle.

0:55:53.620,0:55:54.690
Well, I don't know.

0:55:54.690,0:55:58.320
You have to teach me, because[br]you have chapter 9 fresher

0:55:58.320,0:56:01.590
in your memory than I have it.

0:56:01.590,0:56:03.920
Are you taking attendance also?

0:56:03.920,0:56:07.177
Are you writing your name down?

0:56:07.177,0:56:08.260
Oh, no problem whatsoever.

0:56:08.260,0:56:09.391
STUDENT: We didn't get it.

0:56:09.391,0:56:10.807
PROFESSOR TODA:[br]You didn't get it.

0:56:10.807,0:56:11.750
Circulate it.

0:56:11.750,0:56:16.660
All right, so who is going[br]to help me with the angle?

0:56:16.660,0:56:19.870
What is the angle between[br]two vectors, guys?

0:56:19.870,0:56:24.070
That should be review from what[br]we just covered in chapter 9.

0:56:24.070,0:56:27.980
Let me call them[br]u and v. And who's

0:56:27.980,0:56:29.916
going to tell me how[br]I get that angle?

0:56:29.916,0:56:31.960
STUDENT: [INAUDIBLE] is equal[br]to the inverse cosine of the dot

0:56:31.960,0:56:33.290
product of [? the magnitude. ?]

0:56:33.290,0:56:35.081
PROFESSOR TODA: Do you[br]like me to write arc

0:56:35.081,0:56:36.450
cosine or cosine [INAUDIBLE].

0:56:36.450,0:56:37.760
Doesn't matter.

0:56:37.760,0:56:39.850
Arc cosine of--

0:56:39.850,0:56:40.960
STUDENT: The dot products.

0:56:40.960,0:56:47.476
PROFESSOR TODA: The dot[br]product between u and v.

0:56:47.476,0:56:48.940
STUDENT: Over magnitude.

0:56:48.940,0:56:52.466
PROFESSOR TODA: Divided by the[br]product of their magnitudes.

0:56:52.466,0:56:54.692
Look, I will change the[br]order, because you're not

0:56:54.692,0:56:56.140
going to like it.

0:56:56.140,0:56:56.910
Doesn't matter.

0:56:56.910,0:56:57.680
OK?

0:56:57.680,0:57:03.450
So the angle phi between[br]my favorite vectors

0:57:03.450,0:57:08.460
here is going to be[br]simply the dot product.

0:57:08.460,0:57:09.570
That's a blessing.

0:57:09.570,0:57:10.242
It's a constant.

0:57:10.242,0:57:11.908
STUDENT: So you're[br]doing the dot product

0:57:11.908,0:57:13.416
between the normal [INAUDIBLE]?

0:57:13.416,0:57:14.999
PROFESSOR TODA:[br]Between this and that.

0:57:14.999,0:57:18.065
So this is u and this[br]is v. So the dot product

0:57:18.065,0:57:22.340
would be 0 plus v.[br]So the dot product

0:57:22.340,0:57:28.520
is arc cosine of v, which,[br]thank god, is a constant.

0:57:28.520,0:57:30.310
I don't have to do[br]anything anymore.

0:57:30.310,0:57:33.154
I'm done with the proof[br]bit, because arc cosine

0:57:33.154,0:57:36.000
of a constant will[br]be a constant.

0:57:36.000,0:57:36.720
OK?

0:57:36.720,0:57:37.600
All right.

0:57:37.600,0:57:40.850
So I have v over what?

0:57:40.850,0:57:45.090
What is the length[br]of this vector?

0:57:45.090,0:57:46.750
1. [INAUDIBLE].

0:57:46.750,0:57:50.510
What's the length[br]of that vector?

0:57:50.510,0:57:55.900
Square root of a[br]squared plus b squared.

0:57:55.900,0:57:56.430
All right?

0:57:56.430,0:58:01.831


0:58:01.831,0:58:05.280
STUDENT: How did[br]you [INAUDIBLE].

0:58:05.280,0:58:07.647
PROFESSOR TODA: So now[br]let me ask you one thing.

0:58:07.647,0:58:11.362


0:58:11.362,0:58:13.910
What kind of function[br]is arc cosine?

0:58:13.910,0:58:16.430
Of course I said arc cosine[br]of a constant is a constant.

0:58:16.430,0:58:18.390
What kind of a[br]function is arc cosine?

0:58:18.390,0:58:21.740
I'm doing review with you[br]because I think it's useful.

0:58:21.740,0:58:26.068
Arc cosine is defined on[br]what with values in what?

0:58:26.068,0:58:30.289


0:58:30.289,0:58:32.640
STUDENT: Repeat the question?

0:58:32.640,0:58:33.890
PROFESSOR TODA: Arc cosine.

0:58:33.890,0:58:36.100
Or cosine inverse,[br]like Ryan prefers.

0:58:36.100,0:58:38.130
Cosine inverse is[br]the same thing.

0:58:38.130,0:58:40.440
It's a function defined[br]by where to where?

0:58:40.440,0:58:43.190
Cosine is defined[br]from where to where?

0:58:43.190,0:58:46.136
From R to minus 1.

0:58:46.136,0:58:47.740
It's a cosine of t.

0:58:47.740,0:58:49.690
t could be any real number.

0:58:49.690,0:58:51.800
The range is minus 1, 1.

0:58:51.800,0:58:53.242
Close the interval.

0:58:53.242,0:58:54.950
STUDENT: So it's-- so[br]I just wonder why--

0:58:54.950,0:58:57.014
PROFESSOR TODA: Minus[br]1 to 1, close interval.

0:58:57.014,0:58:58.315
But pay attention, please.

0:58:58.315,0:59:03.040
Because it cannot go back to R.[br]It has to be a 1 to 1 function.

0:59:03.040,0:59:05.960
You cannot have an inverse[br]function if you don't take

0:59:05.960,0:59:09.242
a restriction of a[br]function to be 1 to 1.

0:59:09.242,0:59:11.597
And we took that[br]restriction of a function.

0:59:11.597,0:59:14.894
And do you remember what it was?

0:59:14.894,0:59:15.840
[INTERPOSING VOICES]

0:59:15.840,0:59:17.640
PROFESSOR TODA: 0 to pi.

0:59:17.640,0:59:19.710
Now, on this one[br]I'm really happy.

0:59:19.710,0:59:23.160
Because I asked[br]several people-- people

0:59:23.160,0:59:27.133
come to my office to get[br]all sorts of transcripts,

0:59:27.133,0:59:27.633
[INAUDIBLE].

0:59:27.633,0:59:30.600
And in trigonometry[br]I asked one student,

0:59:30.600,0:59:31.910
so you took trigonometry.

0:59:31.910,0:59:32.910
So do you remember that?

0:59:32.910,0:59:34.360
He didn't remember that.

0:59:34.360,0:59:35.300
So I'm glad you do.

0:59:35.300,0:59:40.130
How about when I had[br]the sine inverse?

0:59:40.130,0:59:44.760
How was my restriction so that[br]would be a 1 to 1 function?

0:59:44.760,0:59:46.900
It's got to go[br]from minus 1 to 1.

0:59:46.900,0:59:48.180
What is the range?

0:59:48.180,0:59:49.016
[INTERPOSING VOICES]

0:59:49.016,0:59:51.300
PROFESSOR TODA: Minus pi over 2.

0:59:51.300,0:59:53.253
You guys know your trig.

0:59:53.253,0:59:53.752
Good.

0:59:53.752,0:59:55.630
That's a very good thing.

0:59:55.630,0:59:59.440
You were in high school[br]when you learned that?

0:59:59.440,1:00:00.430
Here at Lubbock High?

1:00:00.430,1:00:01.150
STUDENT: Yes.

1:00:01.150,1:00:02.066
PROFESSOR TODA: Great.

1:00:02.066,1:00:03.560
Good job, Lubbock High.

1:00:03.560,1:00:06.230
But many students, I caught[br]them, who wanted credit

1:00:06.230,1:00:08.350
for trig who didn't know that.

1:00:08.350,1:00:09.680
Good.

1:00:09.680,1:00:19.870
So since arc cosine is a[br]function that is of 0, pi,

1:00:19.870,1:00:25.230
for example, what if my--[br]let me give you an example.

1:00:25.230,1:00:26.910
What was last time, guys?

1:00:26.910,1:00:30.800
a was 1. b was 1.

1:00:30.800,1:00:32.010
For one example.

1:00:32.010,1:00:33.810
In that case, 1 with 5b.

1:00:33.810,1:00:36.327
STUDENT: [INAUDIBLE] ask you[br]for the example you just did?

1:00:36.327,1:00:37.535
PROFESSOR TODA: No last time.

1:00:37.535,1:00:39.560
STUDENT: A was 3 and b was--

1:00:39.560,1:00:44.255
PROFESSOR TODA: So what would[br]that be, in this case 5?

1:00:44.255,1:00:46.680
STUDENT: That would be[br]b over the square root--

1:00:46.680,1:00:47.560
STUDENT: 3 over pi.

1:00:47.560,1:00:49.985


1:00:49.985,1:00:52.360
PROFESSOR TODA: a is 1 and b[br]is 1, like we did last time.

1:00:52.360,1:00:55.050
STUDENT: [INAUDIBLE][br]2, which is--

1:00:55.050,1:00:57.059
PROFESSOR TODA: Plug[br]in 1 is a, b is 1.

1:00:57.059,1:00:57.600
What is this?

1:00:57.600,1:00:59.417
STUDENT: It's just pi over 4.

1:00:59.417,1:01:00.500
PROFESSOR TODA: Pi over 4.

1:01:00.500,1:01:06.800
So pi will be our cosine, of[br]1 over square root 2, which

1:01:06.800,1:01:12.090
is 45 degree angle, which is--[br]you said pi over 4, right?

1:01:12.090,1:01:14.540
[INAUDIBLE].

1:01:14.540,1:01:19.800
So exactly, you would[br]have that over here.

1:01:19.800,1:01:22.580
This is where the[br]cosine [INAUDIBLE].

1:01:22.580,1:01:28.220
Now you see, guys, the way we[br]have, the way I assume a and b,

1:01:28.220,1:01:30.980
the way anybody-- the[br]book also introduces

1:01:30.980,1:01:33.350
a and b to be positive numbers.

1:01:33.350,1:01:37.230
Can you tell me what kind[br]of angle phi will be,

1:01:37.230,1:01:39.900
not only restricted to 0 pi?

1:01:39.900,1:01:41.360
Well, a is positive.

1:01:41.360,1:01:42.480
b is positive.

1:01:42.480,1:01:44.360
a doesn't matter.

1:01:44.360,1:01:46.670
The whole thing[br]will be positive.

1:01:46.670,1:01:50.510
Arc cosine of a[br]positive number--

1:01:50.510,1:01:52.010
STUDENT: Between[br]0 and pi over 2.

1:01:52.010,1:01:53.010
PROFESSOR TODA: That is.

1:01:53.010,1:01:56.326
Yeah, so it has to be[br]between 0 and pi over 2.

1:01:56.326,1:01:57.950
So it's going to be[br]only this quadrant.

1:01:57.950,1:01:59.640
Does that make sense?

1:01:59.640,1:02:03.388
Yes, think with the[br]imagination of your eyes,

1:02:03.388,1:02:05.220
or the eyes of your imagination.

1:02:05.220,1:02:06.430
OK.

1:02:06.430,1:02:08.360
You have a cylinder.

1:02:08.360,1:02:10.270
And you are moving[br]along that cylinder.

1:02:10.270,1:02:12.160
And this is how you turn.

1:02:12.160,1:02:14.400
You turn with that little train.

1:02:14.400,1:02:16.580
Du-du-du-du-du, you go up.

1:02:16.580,1:02:19.910
When you turn the[br]velocity vector and you

1:02:19.910,1:02:23.123
look at the-- mm.

1:02:23.123,1:02:23.956
STUDENT: The normal.

1:02:23.956,1:02:24.860
PROFESSOR TODA: The normal!

1:02:24.860,1:02:25.360
Thank you.

1:02:25.360,1:02:30.245
The z axis, you always have an[br]angle between 0 and pi over 2.

1:02:30.245,1:02:31.712
So it makes sense.

1:02:31.712,1:02:34.157
I'm going to go ahead and[br]erase the whole thing.

1:02:34.157,1:02:41.020


1:02:41.020,1:02:47.720
So we reviewed, more or less, s[br]of t, integration, derivation,

1:02:47.720,1:02:52.020
moving from position vector[br]to velocity to acceleration

1:02:52.020,1:02:56.260
and back, acceleration to[br]velocity to position vector,

1:02:56.260,1:02:58.500
the meaning of arclength.

1:02:58.500,1:03:00.890
There are some things I[br]would like to tell you,

1:03:00.890,1:03:07.829
because Ryan asked me a few more[br]questions about the curvature.

1:03:07.829,1:03:11.560
The curvature[br]formula depends very

1:03:11.560,1:03:16.830
much on the type of formula[br]you used for the curve.

1:03:16.830,1:03:18.800
So you say, wait,[br]wait, wait, Magdelena,

1:03:18.800,1:03:21.290
you told us-- you[br]are confusing us.

1:03:21.290,1:03:23.710
You told us that the[br]curvature is uniquely

1:03:23.710,1:03:33.740
defined as the magnitude[br]of the acceleration vector

1:03:33.740,1:03:36.800
when the law of motion[br]is an arclength.

1:03:36.800,1:03:38.850
And that is correct.

1:03:38.850,1:03:43.190
So suppose my original law of[br]motion was R of t [INAUDIBLE]

1:03:43.190,1:03:47.750
time, any time, t,[br]any time parameter.

1:03:47.750,1:03:49.370
I'm making a face.

1:03:49.370,1:03:53.290
But then from that we switch[br]to something beautiful,

1:03:53.290,1:03:56.445
which is called the[br]arclength parametrization.

1:03:56.445,1:03:58.280
Why am I so happy?

1:03:58.280,1:04:04.970
Because in this parametrization[br]the magnitude of the speed

1:04:04.970,1:04:07.120
is 1.

1:04:07.120,1:04:17.700
And I define k to[br]be the magnitude

1:04:17.700,1:04:19.870
of R double prime of s, right?

1:04:19.870,1:04:22.430
The acceleration only in[br]the arclength [? time ?]

1:04:22.430,1:04:23.426
parameterization.

1:04:23.426,1:04:24.920
And then this was[br]the definition.

1:04:24.920,1:04:30.410


1:04:30.410,1:04:36.550
A. Can you prove-- what?

1:04:36.550,1:04:40.190
Can you prove the[br]following formula?

1:04:40.190,1:04:52.200


1:04:52.200,1:04:58.514
T prime of s equals[br]k times N of s.

1:04:58.514,1:05:02.550
This is famous for people[br]who do-- not for everybody.

1:05:02.550,1:05:05.530
But imagine you have[br]an engineer who does

1:05:05.530,1:05:08.430
research of the laws of motion.

1:05:08.430,1:05:13.130
Maybe he works for[br]the railways and he's

1:05:13.130,1:05:17.170
looking at skew[br]curves, or he is one

1:05:17.170,1:05:20.480
of those people who[br]project the ski slopes,

1:05:20.480,1:05:25.360
or all sorts of winter sports[br]slope or something, that

1:05:25.360,1:05:29.150
involve a lot of[br]curvatures and torsions.

1:05:29.150,1:05:31.240
That guy has to know[br]the Frenet formula.

1:05:31.240,1:05:34.260
So this is the famous[br]first Frenet formula.

1:05:34.260,1:05:40.140


1:05:40.140,1:05:46.690
Frenet was a mathematician[br]who gave the name to the TNB

1:05:46.690,1:05:48.476
vectors, the trihedron.

1:05:48.476,1:05:49.970
You have the T was what?

1:05:49.970,1:05:52.958
The T was the tangent[br][INAUDIBLE] vector.

1:05:52.958,1:05:58.180
The N was the[br]principal unit normal.

1:05:58.180,1:06:00.930
In those videos that I'm[br]watching that I also sent you--

1:06:00.930,1:06:02.400
I like most of them.

1:06:02.400,1:06:05.660
I like the Khan Academy[br]more than everything.

1:06:05.660,1:06:09.100
Also I like the one that[br]was made by Dr. [? Gock ?]

1:06:09.100,1:06:12.540
But Dr. [? Gock ?] made a[br]little bit of a mistake.

1:06:12.540,1:06:13.920
A conceptual mistake.

1:06:13.920,1:06:17.013
We all make mistakes by[br]misprinting or misreading

1:06:17.013,1:06:18.366
or goofy mistake.

1:06:18.366,1:06:20.690
But he said this is[br]the normal vector.

1:06:20.690,1:06:22.930
This is not-- it's the[br]principle normal vectors.

1:06:22.930,1:06:24.726
There are many normals.

1:06:24.726,1:06:26.630
There is only one[br]tangent direction,

1:06:26.630,1:06:29.010
but in terms of normals[br]there are many that

1:06:29.010,1:06:30.914
are-- all of these are normals.

1:06:30.914,1:06:34.940
All the perpendicular in[br]the plane-- [INAUDIBLE]

1:06:34.940,1:06:39.780
so this is my law of motion,[br]T. All this plane is normal.

1:06:39.780,1:06:41.960
So any of these[br]vectors is a normal.

1:06:41.960,1:06:44.990
The one we choose and[br]defined as T prime

1:06:44.990,1:06:47.220
over T prime [INAUDIBLE][br]absolute values

1:06:47.220,1:06:48.990
called the principal normal.

1:06:48.990,1:06:51.350
It's like the principal[br]of a high school.

1:06:51.350,1:06:53.230
He is important.

1:06:53.230,1:06:58.352
So T and B-- B goes[br]down, or goes-- down.

1:06:58.352,1:07:04.540
Well, yeah, because B is T cross[br]N. So when you find the Frenet

1:07:04.540,1:07:10.440
Trihedron, TNB, it's like that.

1:07:10.440,1:07:15.685
T, N, and B. What's special,[br]why do we call it the frame,

1:07:15.685,1:07:18.460
is that every[br][? payer ?] of vectors

1:07:18.460,1:07:20.090
are mutually orthogonal.

1:07:20.090,1:07:22.270
And they are all unit vectors.

1:07:22.270,1:07:25.910
This is the famous Frenet frame.

1:07:25.910,1:07:27.580
Now, Mr. Frenet was a smart guy.

1:07:27.580,1:07:32.330
He found-- I don't know whether[br]he was adopting mathematics

1:07:32.330,1:07:33.050
or not.

1:07:33.050,1:07:34.290
Doesn't matter.

1:07:34.290,1:07:37.970
He found a bunch of formulas,[br]of which this is the first one.

1:07:37.970,1:07:42.265
And it's called a[br]first Frenet formula.

1:07:42.265,1:07:44.230
That's one thing[br]I want to ask you.

1:07:44.230,1:07:47.170
And then I'm going to give you[br]more formulas for curvatures,

1:07:47.170,1:07:50.460
depending on how you[br]define your curve.

1:07:50.460,1:08:08.832
So for example, base B[br]based on the definition one

1:08:08.832,1:08:18.870
can prove that for a curve[br]that is not parametrizing

1:08:18.870,1:08:22.870
arclength-- you say, ugh,[br]forget about parametrization

1:08:22.870,1:08:23.590
in arclength.

1:08:23.590,1:08:26.840
This time you're[br]assuming, I want to know!

1:08:26.840,1:08:29.410
I'm coming to this[br]because Ryan asked.

1:08:29.410,1:08:32.381
I want to know, what is[br]the formula directly?

1:08:32.381,1:08:34.439
Is there a direct[br]formula that comes

1:08:34.439,1:08:38.529
from here for the curvature?

1:08:38.529,1:08:41.310
Yeah, but it's a lot[br]more complicated.

1:08:41.310,1:08:45.265
When I was a freshman, maybe[br]a freshman or a sophomore,

1:08:45.265,1:08:48.090
I don't remember, when[br]I was asked to memorize

1:08:48.090,1:08:52.689
that, I did not memorize it.

1:08:52.689,1:08:56.649
Then when I started working[br]as a faculty member,

1:08:56.649,1:09:01.810
I saw that I am supposed[br]to ask it from my students.

1:09:01.810,1:09:05.578
So this is going to be[br]R prime plus product

1:09:05.578,1:09:12.057
R double prime in magnitude[br]over R prime cubed.

1:09:12.057,1:09:14.550
So how am I supposed[br]to remember that?

1:09:14.550,1:09:15.658
It's not so easy.

1:09:15.658,1:09:17.800
Are you cold there?

1:09:17.800,1:09:18.649
It's cold there.

1:09:18.649,1:09:22.590
I don't know how[br]these roofs are made.

1:09:22.590,1:09:24.670
Velocity times acceleration.

1:09:24.670,1:09:26.620
This is what I try[br]to teach myself.

1:09:26.620,1:09:29.810
I was old already, 26 or 27.

1:09:29.810,1:09:32.979
Velocity times[br]acceleration, cross product,

1:09:32.979,1:09:35.760
take the magnitude,[br]divide by the speed, cube.

1:09:35.760,1:09:36.810
Oh my god.

1:09:36.810,1:09:41.340
So I was supposed to know[br]that when I was 18 or 19.

1:09:41.340,1:09:44.510
Now, I was teaching majors[br]of mechanical engineering.

1:09:44.510,1:09:45.840
They knew that by heart.

1:09:45.840,1:09:48.210
I didn't, so I had to learn it.

1:09:48.210,1:09:51.475
So if one is too[br]lazy or it's simply

1:09:51.475,1:09:54.915
inconvenient to try to[br]reparametrize from R of T

1:09:54.915,1:10:00.790
being arclength parametrization[br]R of s and do that thing here,

1:10:00.790,1:10:05.300
one can just plug in and[br]find the curvature like that.

1:10:05.300,1:10:08.450
For example, guys,[br]as Ryan asked,

1:10:08.450,1:10:13.290
if I have A cosine, [INAUDIBLE],[br]and I do this with respect

1:10:13.290,1:10:16.740
to T, can I get k[br]without-- k will not

1:10:16.740,1:10:18.950
depend on T or s or tau.

1:10:18.950,1:10:20.860
It will always be the same.

1:10:20.860,1:10:23.400
I will still get A[br]over A squared plus B

1:10:23.400,1:10:25.260
squared, no matter what.

1:10:25.260,1:10:28.930
So even if I use this[br]formula for my helix,

1:10:28.930,1:10:30.950
I'm going to get the same thing.

1:10:30.950,1:10:33.060
I'll get A over A[br]squared plus B squared,

1:10:33.060,1:10:35.390
because curvature[br]is an invariant.

1:10:35.390,1:10:38.510
There is another invariant[br]that's-- the other invariant,

1:10:38.510,1:10:40.550
of course, in space[br]is called torsion.

1:10:40.550,1:10:43.680
We want to talk a little[br]bit about that later.

1:10:43.680,1:10:48.780
So is this hard?

1:10:48.780,1:10:49.280
No.

1:10:49.280,1:10:50.450
It shouldn't be hard.

1:10:50.450,1:10:54.930
And you guys should be able[br]to help me on that, hopefully.

1:10:54.930,1:10:56.600
How do we prove that?

1:10:56.600,1:10:58.480
STUDENT: N is G[br]prime [INAUDIBLE].

1:10:58.480,1:11:01.950


1:11:01.950,1:11:03.450
PROFESSOR TODA:[br]That's right, proof.

1:11:03.450,1:11:06.080
And that's a very good[br]start, wouldn't you say?

1:11:06.080,1:11:09.090
So what were the definitions?

1:11:09.090,1:11:14.350
Let me start from[br]the definition of T.

1:11:14.350,1:11:17.180
That's going to be-- I[br]am in hard planes, right?

1:11:17.180,1:11:21.050
So you say, wait, why do[br]you write it as a quotient?

1:11:21.050,1:11:22.430
You're being silly.

1:11:22.430,1:11:24.530
You are in arclength, Magdalena.

1:11:24.530,1:11:25.470
I am.

1:11:25.470,1:11:26.340
I am.

1:11:26.340,1:11:29.860
I just pretend that[br]I cannot see that.

1:11:29.860,1:11:32.160
So if I'm in[br]arclength, that means

1:11:32.160,1:11:35.870
that the denominator is 1.

1:11:35.870,1:11:37.320
So I'm being silly.

1:11:37.320,1:11:44.380
So R prime of s is[br]T. Say it again.

1:11:44.380,1:11:49.280
R prime of s is T. OK.

1:11:49.280,1:11:53.720
Now, did we know that[br]T and N are orthogonal?

1:11:53.720,1:12:00.530


1:12:00.530,1:12:04.350
How did we know that T[br]and N were orthogonal?

1:12:04.350,1:12:07.524
We proved that last[br]time, actually.

1:12:07.524,1:12:11.003
T and N are orthogonal.

1:12:11.003,1:12:12.991
How do I write[br]that? [INAUDIBLE].

1:12:12.991,1:12:15.973


1:12:15.973,1:12:21.990
Meaning that T is[br]perpendicular to N, right?

1:12:21.990,1:12:24.110
From the definition.

1:12:24.110,1:12:26.000
You said it right, Sandra.

1:12:26.000,1:12:28.000
But why is it from[br]the definition

1:12:28.000,1:12:30.900
that I can jump to[br]conclusions and say, oh,

1:12:30.900,1:12:36.120
since I have T prime here, then[br]this is perpendicular to T?

1:12:36.120,1:12:37.435
Well, we did that last time.

1:12:37.435,1:12:39.236
STUDENT: Two parallel vectors.

1:12:39.236,1:12:41.110
PROFESSOR TODA: We did[br]it-- how did we do it?

1:12:41.110,1:12:42.250
We did this last.

1:12:42.250,1:12:45.800
We said T dot T equals 1.

1:12:45.800,1:12:47.960
Prime the whole thing.

1:12:47.960,1:12:54.270
T prime times T plus T times T[br]prime, T dot T prime will be 0.

1:12:54.270,1:12:57.030
So T and T prime are[br]perpendicular always.

1:12:57.030,1:12:58.150
Right?

1:12:58.150,1:13:03.030
OK, so the whole thing is a[br]colinear vector to T prime.

1:13:03.030,1:13:05.025
It's just T prime[br]times the scalar.

1:13:05.025,1:13:08.320
So he must be[br]perpendicular to T.

1:13:08.320,1:13:10.720
So T and N are perpendicular.

1:13:10.720,1:13:14.680
So I do have the[br]direction of motion.

1:13:14.680,1:13:19.482
I know that I must[br]have some scalar here.

1:13:19.482,1:13:22.796


1:13:22.796,1:13:26.630
How do I prove that this[br]scalar is the curvature?

1:13:26.630,1:13:30.510


1:13:30.510,1:13:35.995
So if I have-- if they[br]are colinear-- why are

1:13:35.995,1:13:36.740
they colinear?

1:13:36.740,1:13:42.200
T perpendicular to T[br]prime implies that T prime

1:13:42.200,1:13:46.020
is colinear to N. Say it again.

1:13:46.020,1:13:49.860
If T and T prime are[br]perpendicular to one another,

1:13:49.860,1:13:53.260
that means T prime is[br]calling it to the normal.

1:13:53.260,1:13:58.471
So here I may have[br]alph-- no alpha.

1:13:58.471,1:14:00.260
I don't know!

1:14:00.260,1:14:03.700
Alpha over [INAUDIBLE][br]sounds like a curve.

1:14:03.700,1:14:04.616
Give me some function.

1:14:04.616,1:14:08.864


1:14:08.864,1:14:09.530
STUDENT: u of s?

1:14:09.530,1:14:11.050
PROFESSOR TODA: Gamma of s.

1:14:11.050,1:14:15.360
u of s, I don't know.

1:14:15.360,1:14:17.410
So how did I conclude that?

1:14:17.410,1:14:19.500
From T perpendicular to T prime.

1:14:19.500,1:14:22.210
Now from here on, you[br]have to tell me why

1:14:22.210,1:14:29.430
gamma must be exactly kappa.

1:14:29.430,1:14:33.712
Well, let's take[br]T prime from here.

1:14:33.712,1:14:38.090
T prime from here[br]will give me what?

1:14:38.090,1:14:40.730
T prime is our prime prime.

1:14:40.730,1:14:42.300
Say what?

1:14:42.300,1:14:43.200
Our prime prime.

1:14:43.200,1:14:44.657
What is our prime prime?

1:14:44.657,1:14:46.565
Our [? problem ?] prime of s.

1:14:46.565,1:14:48.582
STUDENT: You have one[br]too many primes inside.

1:14:48.582,1:14:49.665
PROFESSOR TODA: Oh my god.

1:14:49.665,1:14:50.165
Yeah.

1:14:50.165,1:14:52.770


1:14:52.770,1:14:54.120
So R prime prime.

1:14:54.120,1:14:58.430
So T prime in[br]absolute value will

1:14:58.430,1:15:02.650
be exactly R double prime of s.

1:15:02.650,1:15:04.990
Oh, OK.

1:15:04.990,1:15:10.130
Note that from here also T[br]prime of s in absolute value,

1:15:10.130,1:15:13.840
in magnitude, I'm sorry,[br]has to be gamma of s.

1:15:13.840,1:15:14.820
Why is that?

1:15:14.820,1:15:17.470
Because the magnitude of N is 1.

1:15:17.470,1:15:20.930
N is unique vector[br]by definition.

1:15:20.930,1:15:24.870
So these two guys[br]have to coincide.

1:15:24.870,1:15:27.132
So R double prime,[br]the best thing

1:15:27.132,1:15:28.590
that I need to do,[br]it must coincide

1:15:28.590,1:15:30.500
with the scalar gamma of s.

1:15:30.500,1:15:32.840
So who is the[br]mysterious gamma of s?

1:15:32.840,1:15:36.340
He has no chance[br]but being this guy.

1:15:36.340,1:15:38.660
But this guy has a name.

1:15:38.660,1:15:41.920
This guy, he's the curvature[br][? cap ?] of s by definition.

1:15:41.920,1:15:45.738


1:15:45.738,1:15:49.126
Remember, Ryan, this[br]is the definition.

1:15:49.126,1:15:51.546
So by definition the[br]curvature was the magnitude

1:15:51.546,1:15:55.010
of the acceleration[br]in arclength.

1:15:55.010,1:15:55.910
OK.

1:15:55.910,1:15:58.460
Both of these guys are[br]T prime in magnitude.

1:15:58.460,1:16:01.770
So they must be equal[br]from here and here.

1:16:01.770,1:16:04.696
It implies that my[br]gamma must be kappa.

1:16:04.696,1:16:07.780
And I prove the formula.

1:16:07.780,1:16:09.130
OK.

1:16:09.130,1:16:10.911
How do you say[br]something is proved?

1:16:10.911,1:16:12.294
Because this is what we wanted.

1:16:12.294,1:16:16.235
We wanted to replace this[br]generic scalar function

1:16:16.235,1:16:20.080
to prove that this is[br]just the curvature.

1:16:20.080,1:16:20.580
QED.

1:16:20.580,1:16:24.420


1:16:24.420,1:16:26.960
That's exactly what[br]we wanted to prove.

1:16:26.960,1:16:29.046
Now, whatever scalar[br]function you have here,

1:16:29.046,1:16:30.170
that must be the curvature.

1:16:30.170,1:16:34.460


1:16:34.460,1:16:36.415
Very smart guy, this Mr. Frenet.

1:16:36.415,1:16:39.720


1:16:39.720,1:16:40.990
I'm now going to take a break.

1:16:40.990,1:16:43.830
If you want to go use the[br]bathroom really quickly,

1:16:43.830,1:16:45.025
feel free to do it.

1:16:45.025,1:16:47.590


1:16:47.590,1:16:49.470
I'm just going to[br]clean the board,

1:16:49.470,1:16:51.694
and I'll keep going[br]in a few minutes.

1:16:51.694,1:17:50.501


1:17:50.501,1:17:51.334
STUDENT: [INAUDIBLE]

1:17:51.334,1:17:55.799


1:17:55.799,1:17:57.298
PROFESSOR TODA: I[br]will do it-- well,

1:17:57.298,1:18:01.274
actually I want to do a[br]different example, simple one,

1:18:01.274,1:18:06.244
which is a plain curve, and show[br]that the curvature has a very

1:18:06.244,1:18:11.036
pretty formula that you[br]could [INAUDIBLE] memorize,

1:18:11.036,1:18:13.516
that in essence is the same.

1:18:13.516,1:18:17.484
But it depends on[br]y equals f of x.

1:18:17.484,1:18:19.468
[INAUDIBLE] So if[br]somebody gives you

1:18:19.468,1:18:22.444
a plane called y[br]equals f of x, can you

1:18:22.444,1:18:25.420
write that curvature[br][INAUDIBLE] function of f?

1:18:25.420,1:18:26.908
And you can.

1:18:26.908,1:18:30.545
And again, I was deep in[br]that when I was 18 or 19

1:18:30.545,1:18:31.868
as a freshman.

1:18:31.868,1:18:35.836
But unfortunately for me I[br]didn't learn it at that time.

1:18:35.836,1:18:41.340
And several years later when[br]I started teaching engineers,

1:18:41.340,1:18:43.776
well, they are[br]mostly mechanical.

1:18:43.776,1:18:46.770
And mechanical[br]engineering [INAUDIBLE].

1:18:46.770,1:18:50.263
They knew those, and they needed[br]those in every research paper.

1:18:50.263,1:18:54.255
So I had to learn it[br]together with them.

1:18:54.255,1:18:57.748
I'll worry about [INAUDIBLE].

1:18:57.748,1:19:01.241
STUDENT: Can you do a really[br]ugly one, like [INAUDIBLE]?

1:19:01.241,1:19:04.734
PROFESSOR TODA: I can[br]do some ugly ones.

1:19:04.734,1:20:37.240


1:20:37.240,1:20:47.670
And once you know the[br]general parametrization,

1:20:47.670,1:20:51.908
it will give you a curvature.

1:20:51.908,1:20:53.327
Now I'm testing your memory.

1:20:53.327,1:20:55.230
Let's see what you remember.

1:20:55.230,1:20:59.854
Um-- don't look at the notes.

1:20:59.854,1:21:03.270
A positive function,[br]absolute-- actually,

1:21:03.270,1:21:06.680
magnitude of what vector?

1:21:06.680,1:21:07.580
STUDENT: R prime.

1:21:07.580,1:21:17.060
PROFESSOR TODA: R prime velocity[br]plus acceleration speed cubed.

1:21:17.060,1:21:18.540
Right?

1:21:18.540,1:21:19.040
OK.

1:21:19.040,1:21:24.070
Now, can we take advantage[br]of what we just learned

1:21:24.070,1:21:30.055
and find-- you find[br]with me, of course, not

1:21:30.055,1:21:33.654
as professor and student,[br]but like a group of students

1:21:33.654,1:21:35.070
together.

1:21:35.070,1:21:46.286
Let's find a simple[br]formula corresponding

1:21:46.286,1:21:52.202
to the curvature[br]of a plane curve.

1:21:52.202,1:21:59.597


1:21:59.597,1:22:05.280
And the plane curve[br]could be [INAUDIBLE]

1:22:05.280,1:22:09.020
in two different ways,[br]just because I want

1:22:09.020,1:22:14.510
you to practice more on that.

1:22:14.510,1:22:18.360
Either given as a general[br]parametrization-- guys,

1:22:18.360,1:22:20.090
what is the general[br]parametrization

1:22:20.090,1:22:24.510
I'm talking about[br]for a plane curve?

1:22:24.510,1:22:26.400
x of t, y of t, right?

1:22:26.400,1:22:28.660
x equals x of t.

1:22:28.660,1:22:29.870
y equals y of t.

1:22:29.870,1:22:34.400
So one should not have[br]to do that all the time,

1:22:34.400,1:22:37.310
not have to do that for a[br]simplification like a playing

1:22:37.310,1:22:38.190
card.

1:22:38.190,1:22:41.710
We have to find another[br]formula that's pretty, right?

1:22:41.710,1:22:43.210
Well, maybe it's not as pretty.

1:22:43.210,1:22:45.250
But when is it really pretty?

1:22:45.250,1:22:49.090
I bet it's going to be really[br]pretty if you have a plane

1:22:49.090,1:22:54.610
curve even as you're used[br]to in an explicit form--

1:22:54.610,1:22:56.600
I keep going.

1:22:56.600,1:22:59.910
No stop. [INAUDIBLE].

1:22:59.910,1:23:01.110
I think it's better.

1:23:01.110,1:23:03.450
We make better use[br]of time this way.

1:23:03.450,1:23:06.890
Or y equals f of x.

1:23:06.890,1:23:12.600


1:23:12.600,1:23:17.953
This is an explicit way to[br]write the equation of a curve.

1:23:17.953,1:23:20.660


1:23:20.660,1:23:23.330
OK, so what do we need to do?

1:23:23.330,1:23:26.430
That should be really easy.

1:23:26.430,1:23:32.958
R of t being the first case of[br]our general parametrization,

1:23:32.958,1:23:40.720
x equals x of t, y equals y of[br]t will be-- who tells me, guys,

1:23:40.720,1:23:43.470
that-- this is in your hands.

1:23:43.470,1:23:47.700
Now you convinced me[br]that, for whatever reason,

1:23:47.700,1:23:49.880
you [INAUDIBLE].

1:23:49.880,1:23:51.800
You became friends[br]with these curves.

1:23:51.800,1:23:52.760
I don't know when.

1:23:52.760,1:23:54.680
I guess in the process[br]of doing homework.

1:23:54.680,1:23:55.650
Am I right?

1:23:55.650,1:24:00.480
I think you did not quite like[br]them before or the last week.

1:24:00.480,1:24:02.470
But I think you're[br]friends with them now.

1:24:02.470,1:24:06.556
x of t, y of t.

1:24:06.556,1:24:07.990
Let people talk.

1:24:07.990,1:24:12.770


1:24:12.770,1:24:13.740
STUDENT: 0.

1:24:13.740,1:24:15.670
PROFESSOR TODA: So.

1:24:15.670,1:24:16.170
Great.

1:24:16.170,1:24:21.110
And then R prime of t will be[br]x prime of t, y prime of t,

1:24:21.110,1:24:21.760
and 0.

1:24:21.760,1:24:24.487
I assume this to[br]be always non-zero.

1:24:24.487,1:24:26.235
I have a regular curve.

1:24:26.235,1:24:30.680
R double prime will be--[br]x double prime where

1:24:30.680,1:24:34.146
double prime-- we[br]did the review today

1:24:34.146,1:24:36.450
of the lasting acceleration.

1:24:36.450,1:24:39.630
Now, your friends over[br]here, are they nice or mean?

1:24:39.630,1:24:42.600
I hope they are not so mean.

1:24:42.600,1:24:45.810
The cross product is[br]a friendly fellow.

1:24:45.810,1:24:48.970
You have i, j, k, and[br]then the second row

1:24:48.970,1:24:50.640
would be x prime, y prime, 0.

1:24:50.640,1:24:54.846
The last row would be x double[br]prime, y double prime, 0.

1:24:54.846,1:24:58.550
And it's a piece of cake.

1:24:58.550,1:25:02.146


1:25:02.146,1:25:03.520
OK, piece of cake,[br]piece of cake.

1:25:03.520,1:25:08.960
But I want to know[br]what the answer is.

1:25:08.960,1:25:15.630
So you have exactly 15 seconds[br]to answer this question.

1:25:15.630,1:25:23.298
Who is R prime plus R double[br]prime as a [? coordinate. ?]

1:25:23.298,1:25:25.266
[INTERPOSING VOICES]

1:25:25.266,1:25:28.710


1:25:28.710,1:25:29.700
PROFESSOR TODA: Good.

1:25:29.700,1:25:35.372
x prime, y double prime minus x[br]double prime, y prime times k.

1:25:35.372,1:25:37.720
And it doesn't matter[br]when I take the magnitude,

1:25:37.720,1:25:40.600
because magnitude of k is 1.

1:25:40.600,1:25:42.380
So I discovered some.

1:25:42.380,1:25:46.700
This is how mathematicians like[br]to discover new formulas based

1:25:46.700,1:25:48.730
on the formulas they[br][? knew. ?] They

1:25:48.730,1:25:50.090
have a lot of satisfaction.

1:25:50.090,1:25:51.020
Look what I got.

1:25:51.020,1:25:56.630
Of course, they in general have[br]more complicated things to do,

1:25:56.630,1:25:58.952
and they have to[br]check and recheck.

1:25:58.952,1:26:06.305
But every piece of a[br]computation is a challenge.

1:26:06.305,1:26:10.310
And that gives[br]people satisfaction.

1:26:10.310,1:26:14.900
And when they make a mistake, it[br]brings a lot of tears as well.

1:26:14.900,1:26:21.190
So what-- could be written[br]on the bottom, what's

1:26:21.190,1:26:24.680
the speed cubed?

1:26:24.680,1:26:26.740
Speed is coming from this guy.

1:26:26.740,1:26:32.055
So the speed of the velocity,[br]the magnitude of the velocity

1:26:32.055,1:26:32.990
is the speed.

1:26:32.990,1:26:35.100
And that-- going[br]to give you square.

1:26:35.100,1:26:37.010
I'm not going to write[br]down [INAUDIBLE].

1:26:37.010,1:26:39.144
Square root of x squared,[br]x prime squared times

1:26:39.144,1:26:42.740
y prime squared,[br]and I cube that.

1:26:42.740,1:26:46.370
Many people, and I saw[br]that in engineering, they

1:26:46.370,1:26:49.760
don't like to put that[br]square root anymore.

1:26:49.760,1:26:53.950
And they just write x prime[br]squared plus y prime squared

1:26:53.950,1:26:55.110
to the what power?

1:26:55.110,1:26:55.690
STUDENT: 3/2.

1:26:55.690,1:26:56.540
PROFESSOR TODA: 3/2.

1:26:56.540,1:27:01.650
So this is very useful[br]for engineering styles,

1:27:01.650,1:27:05.110
when you have to deal[br]with plane curves, motions

1:27:05.110,1:27:08.560
in plane curves.

1:27:08.560,1:27:13.770
But now what do you[br]have in the case,

1:27:13.770,1:27:19.220
in the happy case, when[br]you have y equals f of x?

1:27:19.220,1:27:21.335
I'm going to do[br]that in a second.

1:27:21.335,1:27:26.460


1:27:26.460,1:27:29.144
I want to keep this[br]formula on the board.

1:27:29.144,1:27:38.108


1:27:38.108,1:27:40.598
What's the simplest[br]parametrization?

1:27:40.598,1:27:43.088
Because that's why we[br]need it, to look over

1:27:43.088,1:27:46.580
parametrizations[br]again and again.

1:27:46.580,1:27:52.190
R of t for this plane[br]curve will be-- what is t?

1:27:52.190,1:27:53.750
x is t, right?

1:27:53.750,1:27:56.000
x is t, y is f of t.

1:27:56.000,1:27:57.050
Piece of cake.

1:27:57.050,1:28:00.310
So you have t and f of t.

1:28:00.310,1:28:03.355
And how many of you watched[br]the videos that I sent you?

1:28:03.355,1:28:06.394


1:28:06.394,1:28:08.890
Do you prefer Khan[br]Academy, or do you

1:28:08.890,1:28:12.880
prefer the guys, [INAUDIBLE][br]guys who are lecturing?

1:28:12.880,1:28:15.585
The professors who are lecturing[br]in front of a board or in front

1:28:15.585,1:28:17.955
of a-- what is that?

1:28:17.955,1:28:21.350
A projector screen?

1:28:21.350,1:28:22.620
I like all of them.

1:28:22.620,1:28:25.140
I think they're very good.

1:28:25.140,1:28:27.572
I think you can learn[br]a lot from three

1:28:27.572,1:28:29.530
or four different[br]instructors at the same time.

1:28:29.530,1:28:32.010
That's ideal.

1:28:32.010,1:28:35.590
I guess that you have[br]this chance only now

1:28:35.590,1:28:36.980
in the past few years.

1:28:36.980,1:28:41.300
Because 20 years ago, if you're[br]didn't like your instructor

1:28:41.300,1:28:45.790
or just you couldn't stand[br]them, you had no other chance.

1:28:45.790,1:28:48.030
There was no[br]YouTube, no internet,

1:28:48.030,1:28:50.972
no way to learn from others.

1:28:50.972,1:29:00.220
R prime of t would[br]be 1 f prime of t.

1:29:00.220,1:29:02.840
But instead of t I'll[br]out x, because x is t.

1:29:02.840,1:29:03.930
I don't care.

1:29:03.930,1:29:07.470
R double prime of t would[br]be 0, f double prime of x.

1:29:07.470,1:29:12.430
So I feel that, hey, I know[br]what's going to come up.

1:29:12.430,1:29:15.390
And I'm ready.

1:29:15.390,1:29:17.680
Well, we are ready[br]to write it down.

1:29:17.680,1:29:20.190
This is going to be Mr. x prime.

1:29:20.190,1:29:22.790
This is going to be[br]replacing Mr. y prime.

1:29:22.790,1:29:25.780
This is going to replace[br]Mr. a double prime.

1:29:25.780,1:29:29.350
This is going to be replacing[br]Mr. y double prime of x.

1:29:29.350,1:29:31.120
Oh, OK, all right.

1:29:31.120,1:29:38.840
So k, our old friend from[br]here will become what?

1:29:38.840,1:29:42.490
And I'd better shut up,[br]because I'm talking too much.

1:29:42.490,1:29:45.020
STUDENT: [INAUDIBLE][br]double prime [INAUDIBLE].

1:29:45.020,1:29:48.092
PROFESSOR TODA: That is[br]the absolute value, mm-hmm.

1:29:48.092,1:29:54.041
[? n ?] double prime[br]of x, and nothing else.

1:29:54.041,1:29:54.540
Right, guys?

1:29:54.540,1:29:55.628
Are you with me?

1:29:55.628,1:29:56.960
Divided by--

1:29:56.960,1:29:57.850
STUDENT: [INAUDIBLE]

1:29:57.850,1:29:59.558
PROFESSOR TODA: Should[br]I add square root?

1:29:59.558,1:30:00.890
I love square roots.

1:30:00.890,1:30:01.880
I'm crazy about them.

1:30:01.880,1:30:11.950
So you go 1 plus f[br]prime squared cubed.

1:30:11.950,1:30:16.410
So that's going to[br]be-- any questions?

1:30:16.410,1:30:18.090
Are you guys with me?

1:30:18.090,1:30:21.296
That's going to be the[br]formula that I'm going

1:30:21.296,1:30:22.420
to use in the next example.

1:30:22.420,1:30:25.540


1:30:25.540,1:30:29.820
In case somebody[br]wants to know-- I got

1:30:29.820,1:30:32.010
this question from one of you.

1:30:32.010,1:30:35.060
Suppose we get a[br]parametrization of a circle

1:30:35.060,1:30:37.410
in the midterm or the final.

1:30:37.410,1:30:44.335
Somebody says, I have x[br]of t, just like we did it

1:30:44.335,1:30:47.555
today, a cosine t plus 0.

1:30:47.555,1:30:52.380
And y of t equals[br]a sine t plus y 0.

1:30:52.380,1:30:53.990
What is this, guys?

1:30:53.990,1:31:04.495
This is a circle, a center[br]at 0, y, 0, and radius a.

1:31:04.495,1:31:08.630


1:31:08.630,1:31:14.916
Can use a better formula-- that[br]anticipated my action today--

1:31:14.916,1:31:18.535
to actually prove that k[br]is going to be [? 1/a? ?]

1:31:18.535,1:31:19.260
Precisely.

1:31:19.260,1:31:20.550
Can we do that in the exam?

1:31:20.550,1:31:23.120
Yes.

1:31:23.120,1:31:25.490
So while I told[br]you a long time ago

1:31:25.490,1:31:28.590
that engineers and[br]mathematicians observed

1:31:28.590,1:31:31.000
hundreds of years[br]ago-- actually,

1:31:31.000,1:31:32.770
somebody said, no,[br]you're not right.

1:31:32.770,1:31:35.010
The Egyptians already saw that.

1:31:35.010,1:31:38.243
They had the notion of[br]inverse proportionality

1:31:38.243,1:31:42.390
in Egypt, which makes sense[br]if you look at the pyramids.

1:31:42.390,1:31:47.750
So one look at the radius,[br]it says if the radius is 2,

1:31:47.750,1:31:50.640
then the curvature[br]is not very bent.

1:31:50.640,1:31:52.760
So the curvature's inverse[br]proportion [INAUDIBLE]

1:31:52.760,1:31:53.480
the radius.

1:31:53.480,1:31:57.480
So if this is 2, we said[br]the curvature's 1/2.

1:31:57.480,1:32:01.680
If you take a big[br]circle, the bigger

1:32:01.680,1:32:04.060
the radius, the[br]smaller the bending

1:32:04.060,1:32:07.640
of the arc of the circle,[br]the smaller of the curvature.

1:32:07.640,1:32:10.670
Apparently the ancient[br]world knew that already.

1:32:10.670,1:32:12.195
They Egyptians knew that.

1:32:12.195,1:32:13.140
The Greeks knew that.

1:32:13.140,1:32:15.432
But I think they[br]never formalized it--

1:32:15.432,1:32:16.624
not that I know.

1:32:16.624,1:32:19.290


1:32:19.290,1:32:24.530
So if you are asked to[br]do this in any exam,

1:32:24.530,1:32:26.942
do you think that[br]would be a problem?

1:32:26.942,1:32:28.150
Of course we would do review.

1:32:28.150,1:32:31.730
Because people are going to[br]forget this formula, or even

1:32:31.730,1:32:33.370
the definition.

1:32:33.370,1:32:36.010
You can compute k[br]for this formula.

1:32:36.010,1:32:39.330
And we are going to[br]get k to the 1/a.

1:32:39.330,1:32:41.340
This is a piece[br]of cake, actually.

1:32:41.340,1:32:44.110
You may not believe me, but[br]once you plug in the equations

1:32:44.110,1:32:46.290
it's very easy.

1:32:46.290,1:32:48.425
Or you can do it[br]from the definition

1:32:48.425,1:32:50.640
that gives you k of s.

1:32:50.640,1:32:52.560
You'll reparametrize[br]this in arclength.

1:32:52.560,1:32:54.710
You can do that as well.

1:32:54.710,1:32:57.600
And you still get 1/a.

1:32:57.600,1:32:59.880
The question that[br]I got by email,

1:32:59.880,1:33:01.440
and I get a lot of email.

1:33:01.440,1:33:04.660
I told you, that[br]keeps me busy a lot,

1:33:04.660,1:33:08.490
about 200 emails every day.

1:33:08.490,1:33:10.680
I really like the emails[br]I get from students,

1:33:10.680,1:33:13.620
because I get emails from[br]all sorts of sources--

1:33:13.620,1:33:15.340
Got some spam also.

1:33:15.340,1:33:21.780
Anyway, what I'm trying to say,[br]I got this question last time

1:33:21.780,1:33:24.680
saying, if on the midterm[br]we get such a question,

1:33:24.680,1:33:28.073
can we say simply, curvature's[br]1/a, a is the radius.

1:33:28.073,1:33:30.820
Is that enough?

1:33:30.820,1:33:34.250
Depends on how the[br]problem was formulated.

1:33:34.250,1:33:39.360
Most likely I'm going to make[br]it through that or show that.

1:33:39.360,1:33:43.450
Even if you state something,[br]like, yes, it's 1/a,

1:33:43.450,1:33:46.270
with a little argument,[br]it's inverse proportional

1:33:46.270,1:33:50.380
to the radius, I will[br]still give partial credit.

1:33:50.380,1:33:53.650
For any argument that[br]is valid, especially

1:33:53.650,1:33:56.390
if it's based on[br]empirical observation,

1:33:56.390,1:33:58.920
I do give some extra[br]credit, even if you didn't

1:33:58.920,1:34:02.600
use the specific formula.

1:34:02.600,1:34:04.910
Let's see one example.

1:34:04.910,1:34:07.615
Let's take y equals e to the x.

1:34:07.615,1:34:11.870


1:34:11.870,1:34:15.726
No, let's take e[br]to the negative x.

1:34:15.726,1:34:16.690
Doesn't matter.

1:34:16.690,1:34:20.560


1:34:20.560,1:34:26.375
y equals e to the negative x.

1:34:26.375,1:34:30.830
And let's make x[br]between 0 and 1.

1:34:30.830,1:34:35.290


1:34:35.290,1:34:36.970
I'll say, write the curvature.

1:34:36.970,1:34:40.610


1:34:40.610,1:34:45.246
Write the equation or the[br]formula of the curvature.

1:34:45.246,1:34:50.230


1:34:50.230,1:34:54.700
And I know it's 2 o'clock[br]and I am answering questions.

1:34:54.700,1:34:58.070
This was a question that one of[br]you had during the short break

1:34:58.070,1:34:59.300
we took.

1:34:59.300,1:35:00.480
Can we do such a problem?

1:35:00.480,1:35:01.544
Like she said.

1:35:01.544,1:35:02.960
Yes, I [INAUDIBLE][br]to the negative

1:35:02.960,1:35:05.520
x because I want[br]to catch somebody

1:35:05.520,1:35:06.790
not knowing the derivative.

1:35:06.790,1:35:08.990
I don't know why I'm doing this.

1:35:08.990,1:35:10.510
Right?

1:35:10.510,1:35:15.480
So if I were to draw that, OK,[br]try and draw that, but not now.

1:35:15.480,1:35:18.620
Now, what formula[br]are you going to use?

1:35:18.620,1:35:21.860
Of course, you could[br]do this in many ways.

1:35:21.860,1:35:24.510
All those formulas are[br]equivalent for the curvature.

1:35:24.510,1:35:27.360
What's the simplest[br]way to do it?

1:35:27.360,1:35:30.020
Do y prime.

1:35:30.020,1:35:33.770
Minus it to the minus x.

1:35:33.770,1:35:36.790
Note here in this problem that[br]even if you mess up and forget

1:35:36.790,1:35:39.950
the minus sign, you still[br]get the final answer correct.

1:35:39.950,1:35:46.050
But I may subtract a few points[br]if I see something nonsensical.

1:35:46.050,1:35:47.300
y double prime equals--

1:35:47.300,1:35:48.870
[INTERPOSING VOICES]

1:35:48.870,1:35:51.660
--plus e to the minus x.

1:35:51.660,1:35:56.738
And what is the[br]curvature k of t?

1:35:56.738,1:35:59.200
STUDENT: y prime over--

1:35:59.200,1:36:01.340
PROFESSOR TODA: Oh, I[br]didn't say one more thing.

1:36:01.340,1:36:04.620
I want the curvature, but[br]I also want the curvature

1:36:04.620,1:36:08.486
in three separate moments,[br]in the beginning, in the end,

1:36:08.486,1:36:09.360
and in the middle.

1:36:09.360,1:36:11.430
STUDENT: Don't we[br]need to parametrize it

1:36:11.430,1:36:15.250
so we can [INAUDIBLE][br]x prime [INAUDIBLE]?

1:36:15.250,1:36:16.740
PROFESSOR TODA: No.

1:36:16.740,1:36:18.340
Did I erase it?

1:36:18.340,1:36:19.330
STUDENT: Yeah, you did.

1:36:19.330,1:36:20.710
PROFESSOR TODA: [INAUDIBLE].

1:36:20.710,1:36:24.356
And one of my colleagues[br]said, Magda, you are smart,

1:36:24.356,1:36:28.000
but you are like one[br]of those people who,

1:36:28.000,1:36:29.729
in the anecdotes[br]about math professors,

1:36:29.729,1:36:31.520
gets out of their office[br]and starts walking

1:36:31.520,1:36:32.930
and stops a student.

1:36:32.930,1:36:34.845
Was I going this[br]way or that way?

1:36:34.845,1:36:36.070
And that's me.

1:36:36.070,1:36:37.650
And I'm sorry about that.

1:36:37.650,1:36:41.933
I should not have erased that.

1:36:41.933,1:36:44.000
I'm going to go[br]ahead and rewrite it,

1:36:44.000,1:36:48.101
because I'm a goofball.

1:36:48.101,1:36:55.496
So the one that I wanted to use[br]k of x will be f double prime.

1:36:55.496,1:36:56.975
STUDENT: And cubed.

1:36:56.975,1:36:58.454
PROFESSOR TODA: Cubed!

1:36:58.454,1:36:59.440
Thank you.

1:36:59.440,1:37:02.920


1:37:02.920,1:37:09.636
So that 3/2, remember it,[br][INAUDIBLE] 3/2 [INAUDIBLE]

1:37:09.636,1:37:10.990
square root cubed.

1:37:10.990,1:37:13.750
Now, for this one, is it hard?

1:37:13.750,1:37:15.080
No.

1:37:15.080,1:37:16.190
That's a piece of cake.

1:37:16.190,1:37:18.610
I said I like it in[br]general, but I also

1:37:18.610,1:37:22.910
like it-- find the curvature[br]of this curve in the beginning.

1:37:22.910,1:37:24.100
You travel on me.

1:37:24.100,1:37:27.590
From time 0 to 1[br]o'clock, whatever.

1:37:27.590,1:37:28.430
One second.

1:37:28.430,1:37:32.770
That's saying this is in seconds[br]to make it more physical.

1:37:32.770,1:37:39.950
I want the k at 0, I want k[br]at 1/2, and I want k at 1.

1:37:39.950,1:37:42.430
And I'd like you to[br]compare those values.

1:37:42.430,1:37:46.270


1:37:46.270,1:37:49.120
And I'll give you one[br]more task after that.

1:37:49.120,1:37:50.580
But let me start working.

1:37:50.580,1:37:53.090
So you say you help me on that.

1:37:53.090,1:37:55.100
[INAUDIBLE]

1:37:55.100,1:38:02.516
Minus x over square[br]root of 1 plus--

1:38:02.516,1:38:04.380
STUDENT: [INAUDIBLE]

1:38:04.380,1:38:05.312
PROFESSOR TODA: Right.

1:38:05.312,1:38:08.190
So can I write this differently,[br]a little bit differently?

1:38:08.190,1:38:09.940
Like what?

1:38:09.940,1:38:12.020
I don't want to square[br]each of the minus 2x.

1:38:12.020,1:38:14.350
Can I do that?

1:38:14.350,1:38:19.540
And then the whole thing[br]I can say to the 3/2

1:38:19.540,1:38:24.530
or I can use the square root,[br]whichever is your favorite.

1:38:24.530,1:38:28.990
Now, what is k of 0?

1:38:28.990,1:38:29.490
STUDENT: 0.

1:38:29.490,1:38:32.990
Or 1.

1:38:32.990,1:38:34.462
PROFESSOR TODA: Really?

1:38:34.462,1:38:36.226
STUDENT: 1/2.

1:38:36.226,1:38:36.961
3/2.

1:38:36.961,1:38:38.710
PROFESSOR TODA: So[br]let's take this slowly.

1:38:38.710,1:38:43.584
Because we can all make[br]mistakes, goofy mistakes.

1:38:43.584,1:38:45.000
That doesn't mean[br]we're not smart.

1:38:45.000,1:38:46.770
We're very smart, right?

1:38:46.770,1:38:51.210
But it's just a matter of[br]book-keeping and paying

1:38:51.210,1:38:53.010
attention, being attentive.

1:38:53.010,1:38:55.420
OK.

1:38:55.420,1:39:00.240
When I take 0 and replace--[br]this is drying fast.

1:39:00.240,1:39:02.858
I'm trying to draw it.

1:39:02.858,1:39:10.000
I have 1 over 1[br]plus 1 to the 3/2.

1:39:10.000,1:39:15.160
I have a student in one exam[br]who was just-- I don't know.

1:39:15.160,1:39:16.580
He was rushing.

1:39:16.580,1:39:20.950
He didn't realize that[br]he had to take it slowly.

1:39:20.950,1:39:22.790
He was extremely smart, though.

1:39:22.790,1:39:29.760
1 over-- you have[br]that 1 plus 1 is 2.

1:39:29.760,1:39:33.710
2 to the 1/2 would be[br]square root of 2 cubed.

1:39:33.710,1:39:35.890
It would be exactly[br]2 square root of 2.

1:39:35.890,1:39:39.750
And more you can write[br]this as rationalized.

1:39:39.750,1:39:42.050
Now, I have a question for you.

1:39:42.050,1:39:43.010
[INAUDIBLE]

1:39:43.010,1:39:47.099
I'm When we were kids, if you[br]remember-- you are too young.

1:39:47.099,1:39:48.140
Maybe you don't remember.

1:39:48.140,1:39:52.710
But I remember when I was a kid,[br]my teacher would always ask me,

1:39:52.710,1:39:53.980
rationalize your answer.

1:39:53.980,1:39:56.920
Rationalize your answer.

1:39:56.920,1:40:00.362
Put the rational number[br]in the denominator.

1:40:00.362,1:40:02.740
Why do you think that was?

1:40:02.740,1:40:05.156
For hundreds of years[br]people did that.

1:40:05.156,1:40:07.430
STUDENT: [INAUDIBLE]

1:40:07.430,1:40:11.900
PROFESSOR TODA: Because they[br]didn't have a calculator.

1:40:11.900,1:40:16.303
So we used to, even I used to be[br]able to get the square root out

1:40:16.303,1:40:17.518
by hand.

1:40:17.518,1:40:20.920
Has anybody taught you how to[br]compute square root by hand?

1:40:20.920,1:40:21.649
You know that.

1:40:21.649,1:40:22.378
Who taught you?

1:40:22.378,1:40:23.350
STUDENT: I don't remember it.

1:40:23.350,1:40:24.940
My seventh grade[br]teacher taught us.

1:40:24.940,1:40:26.440
PROFESSOR TODA:[br]There is a technique

1:40:26.440,1:40:29.450
of taking groups of twos[br]and then fitting the--

1:40:29.450,1:40:31.020
and they still teach that.

1:40:31.020,1:40:33.250
I was amazed, they[br]still teach that

1:40:33.250,1:40:35.460
in half of the Asian countries.

1:40:35.460,1:40:39.360
And it's hard, but kids[br]in fifth and sixth grade

1:40:39.360,1:40:45.190
have that practice, which some[br]of us learned and forgot about.

1:40:45.190,1:40:50.029
So imagine that how people would[br]have done this, and of course,

1:40:50.029,1:40:51.070
square root of 2 is easy.

1:40:51.070,1:40:53.550
1.4142, blah blah blah.

1:40:53.550,1:40:54.250
Divide by 2.

1:40:54.250,1:40:56.350
You can do it by hand.

1:40:56.350,1:40:57.860
At least a good approximation.

1:40:57.860,1:41:01.970
But imagine having a nasty[br]square root there to compute,

1:41:01.970,1:41:05.850
and then you would divide[br]by that natural number.

1:41:05.850,1:41:09.140
You have to rely on your[br]own computation to do it.

1:41:09.140,1:41:11.190
There were no calculators.

1:41:11.190,1:41:14.098
How about k of 1?

1:41:14.098,1:41:14.598
How is that?

1:41:14.598,1:41:15.562
What is that?

1:41:15.562,1:41:20.382


1:41:20.382,1:41:23.756
e to the minus 1.

1:41:23.756,1:41:26.540
That's a little bit[br]harder to compute, right?

1:41:26.540,1:41:28.611
1 plus [INAUDIBLE].

1:41:28.611,1:41:31.760
What is that going to be?

1:41:31.760,1:41:34.040
Minus 2.

1:41:34.040,1:41:37.143
Replace it by 1 to the 3/2.

1:41:37.143,1:41:41.930
I would like you to go[br]home and do the following.

1:41:41.930,1:41:45.810
[INAUDIBLE]-- Not now, not now.

1:41:45.810,1:41:48.310
We stay a little[br]bit longer together.

1:41:48.310,1:41:51.900
k of 0, k of 1/2, and k of 1.

1:41:51.900,1:41:53.034
Which one is bigger?

1:41:53.034,1:41:59.930


1:41:59.930,1:42:03.395
And one last question about[br]that, how much extra credit

1:42:03.395,1:42:04.220
should I give you?

1:42:04.220,1:42:05.870
One point?

1:42:05.870,1:42:08.420
One point if you turn this in.

1:42:08.420,1:42:11.200
Um, yeah.

1:42:11.200,1:42:13.380
Four, [? maybe ?] two points.

1:42:13.380,1:42:19.430
Compare all these[br]three values, and find

1:42:19.430,1:42:32.792
the maximum and the[br]minimum of kappa of t,

1:42:32.792,1:42:37.530
kappa of x, for[br]the interval where

1:42:37.530,1:42:46.690
x is in the interval 0, 1.

1:42:46.690,1:42:48.870
0, closed 1.

1:42:48.870,1:42:49.940
Close it.

1:42:49.940,1:42:53.930
Now, don't ask me,[br]because it's extra credit.

1:42:53.930,1:42:58.740
One question was, by email,[br]can I ask my tutor to help me?

1:42:58.740,1:43:02.170
As long as your tutor doesn't[br]write down your solution,

1:43:02.170,1:43:03.720
you are in good shape.

1:43:03.720,1:43:07.020
Your tutor should help you[br]understand some constants,

1:43:07.020,1:43:08.290
spend time with you.

1:43:08.290,1:43:12.773
But they should not write[br]your assignment themselves.

1:43:12.773,1:43:13.272
OK?

1:43:13.272,1:43:16.540
So it's not a big deal.

1:43:16.540,1:43:22.030
Not I want to tell you one[br]secret that I normally don't

1:43:22.030,1:43:26.730
tell my Calculus 3 students.

1:43:26.730,1:43:29.370
But the more I get[br]to know you, the more

1:43:29.370,1:43:34.020
I realize that you are worth[br]me telling you about that.

1:43:34.020,1:43:35.420
STUDENT: [INAUDIBLE]

1:43:35.420,1:43:38.080
PROFESSOR TODA: No.

1:43:38.080,1:43:41.900
There is a beautiful[br]theory that engineers

1:43:41.900,1:43:48.142
use when they start the motions[br]of curves and parametrizations

1:43:48.142,1:43:51.058
in space.

1:43:51.058,1:43:53.420
And that includes[br]the Frenet formulas.

1:43:53.420,1:43:56.054


1:43:56.054,1:43:58.510
And you already[br]know the first one.

1:43:58.510,1:44:04.520
And I was debating, I was just[br]reviewing what I taught you,

1:44:04.520,1:44:07.321
and I was happy with[br]what I taught you.

1:44:07.321,1:44:10.580
And I said, they know[br]about position vector.

1:44:10.580,1:44:13.100
They know about[br]velocity, acceleration.

1:44:13.100,1:44:15.927
They know how to get back[br]and forth from one another.

1:44:15.927,1:44:16.760
They know our claim.

1:44:16.760,1:44:18.468
They know how to[br][? reparameterize our ?]

1:44:18.468,1:44:19.940
claims.

1:44:19.940,1:44:25.040
They know the [INAUDIBLE][br]and B. They know already

1:44:25.040,1:44:27.010
the first Frenet formula.

1:44:27.010,1:44:28.010
They know the curvature.

1:44:28.010,1:44:29.992
What else can I teach them?

1:44:29.992,1:44:34.070
I want to show you--[br]one of you asked me,

1:44:34.070,1:44:38.280
is this all that we should know?

1:44:38.280,1:44:41.875
This is all that a regular[br]student should know in Calculus

1:44:41.875,1:44:43.746
3, but there is more.

1:44:43.746,1:44:44.870
And you are honor students.

1:44:44.870,1:44:49.930
And I want to show you some[br]beautiful equations here.

1:44:49.930,1:44:54.930
So do you remember that[br]if I introduce r of s

1:44:54.930,1:45:03.900
as a curving arclength,[br]that is a regular curve.

1:45:03.900,1:45:11.050
I said there is a certain famous[br]formula that is T prime of s

1:45:11.050,1:45:13.900
called-- leave space.

1:45:13.900,1:45:15.320
Leave a little bit of space.

1:45:15.320,1:45:15.945
You'll see why.

1:45:15.945,1:45:17.850
It's a surprise.

1:45:17.850,1:45:22.950
k times-- why[br]don't I say k of s?

1:45:22.950,1:45:26.220
Because I want to point[br]out that k is an invariant.

1:45:26.220,1:45:28.580
Even if you have[br]another parameter,

1:45:28.580,1:45:29.905
would be the same function.

1:45:29.905,1:45:38.565
But yes, as a function of s,[br]would be k times N bar, bar.

1:45:38.565,1:45:40.906
More bars because[br]they are free vectors.

1:45:40.906,1:45:42.920
They are not bound[br]to a certain point.

1:45:42.920,1:45:44.550
They're not married[br]to a certain point.

1:45:44.550,1:45:49.180
They are free to shift[br]by parallelism in space.

1:45:49.180,1:45:54.250
However, I'm going to review[br]them as bound at the point

1:45:54.250,1:45:55.200
where they are.

1:45:55.200,1:45:58.110
So they-- no way they[br]are married to the point

1:45:58.110,1:46:03.640
that they belong to.

1:46:03.640,1:46:07.200
Maybe the [? bend ?][br]will change.

1:46:07.200,1:46:09.394
I don't know how it's[br]going to change like crazy.

1:46:09.394,1:46:18.170


1:46:18.170,1:46:19.250
Something like that.

1:46:19.250,1:46:26.819
At every point you have a T, an[br]N, and it's a 90 degree angle.

1:46:26.819,1:46:30.900
Then you have the binormal,[br]which makes a 90 degree

1:46:30.900,1:46:33.280
angle-- [INAUDIBLE].

1:46:33.280,1:46:36.860
So the way you should[br]imagine these corners

1:46:36.860,1:46:39.360
would be something[br]like that, right?

1:46:39.360,1:46:40.860
90-90-90.

1:46:40.860,1:46:43.360
It's just hard to draw them.

1:46:43.360,1:46:51.730
Between the vectors you have--[br]If you draw T and N, am I

1:46:51.730,1:46:53.430
right, that is coming out?

1:46:53.430,1:46:54.110
No.

1:46:54.110,1:46:56.050
I have to switch them.

1:46:56.050,1:46:57.650
T and N. Now, am I right?

1:46:57.650,1:46:59.340
Now I'm thinking of[br]the [? faucet. ?]

1:46:59.340,1:47:02.440
If I move T-- yeah,[br]now it's coming out.

1:47:02.440,1:47:08.120
So this is not getting[br]into the formula.

1:47:08.120,1:47:09.490
So this is the first formula.

1:47:09.490,1:47:10.350
You say, so what?

1:47:10.350,1:47:11.185
You've taught that.

1:47:11.185,1:47:12.450
We proved it together.

1:47:12.450,1:47:14.070
What do you want from us?

1:47:14.070,1:47:17.560
I want to teach you[br]two more formulas.

1:47:17.560,1:47:18.540
N prime.

1:47:18.540,1:47:21.970


1:47:21.970,1:47:24.420
And I'd like you to[br]leave more space here.

1:47:24.420,1:47:27.360


1:47:27.360,1:47:30.746
So you have like an empty field[br]here and an empty field here

1:47:30.746,1:47:32.024
[INAUDIBLE].

1:47:32.024,1:47:35.815
If you were to compute[br]T prime, the magic thing

1:47:35.815,1:47:40.452
is that T prime is a vector.

1:47:40.452,1:47:41.378
N prime is a vector.

1:47:41.378,1:47:42.770
B prime is a vector.

1:47:42.770,1:47:44.430
They're all vectors.

1:47:44.430,1:47:48.630
They are the derivatives[br]of the vectors T and NB.

1:47:48.630,1:47:50.970
And you say, why would I[br]care about the derivatives

1:47:50.970,1:47:52.210
of the vectors T and NB?

1:47:52.210,1:47:54.310
I'll tell you in a second.

1:47:54.310,1:47:58.070
So if you were to[br]compute in prime,

1:47:58.070,1:48:00.050
you're going to get here.

1:48:00.050,1:48:04.357
Minus k of s times T of s.

1:48:04.357,1:48:07.219
Leave room.

1:48:07.219,1:48:09.570
Leave room, because there[br]is no component that

1:48:09.570,1:48:13.760
depends on N. No such component[br]that that depends on N.

1:48:13.760,1:48:14.680
This is [INAUDIBLE].

1:48:14.680,1:48:17.260
There is nothing in[br]N. And then in the end

1:48:17.260,1:48:28.580
you'll say, plus tau of s[br]times B. There is missing--

1:48:28.580,1:48:30.350
something is.

1:48:30.350,1:48:32.990
And finally, if[br]you take B prime,

1:48:32.990,1:48:34.950
there is nothing[br]here, nothing here.

1:48:34.950,1:48:42.865
In the middle you have[br]minus tau of s times N of s.

1:48:42.865,1:48:45.700


1:48:45.700,1:48:49.730
And now you know that nobody[br]else but you knows that.

1:48:49.730,1:48:54.060
The other regular sections[br]don't know these formulas.

1:48:54.060,1:48:57.420


1:48:57.420,1:49:02.220
What do you observe about[br]this bunch of equations?

1:49:02.220,1:49:04.160
Say, oh, wait a minute.

1:49:04.160,1:49:06.520
First of all, why did[br]you put it like that?

1:49:06.520,1:49:07.840
Looks like a cross.

1:49:07.840,1:49:09.160
It is a cross.

1:49:09.160,1:49:12.830
It is like one is shaped in the[br]name of the Father, of the Son,

1:49:12.830,1:49:13.730
and so on.

1:49:13.730,1:49:17.040
So does it have anything[br]to do with religion?

1:49:17.040,1:49:17.540
No.

1:49:17.540,1:49:23.260
But it's going to help you[br]memorize better the equations.

1:49:23.260,1:49:27.004
These are the famous[br]Frenet equations.

1:49:27.004,1:49:30.310


1:49:30.310,1:49:33.980
You only saw the first one.

1:49:33.980,1:49:35.062
What do they represent?

1:49:35.062,1:49:38.230


1:49:38.230,1:49:40.090
If somebody asks you, what is k?

1:49:40.090,1:49:42.910
What it is k of s?

1:49:42.910,1:49:44.130
What's the curvature?

1:49:44.130,1:49:44.880
You go to a party.

1:49:44.880,1:49:46.820
There are only nerds.

1:49:46.820,1:49:47.440
It's you.

1:49:47.440,1:49:50.370
Some people taking advanced[br]calculus or some people

1:49:50.370,1:49:54.790
from Physics, and they say, OK,[br]have you heard of the Frenet

1:49:54.790,1:49:56.920
motion, Frenet[br]formulas, and you say,

1:49:56.920,1:49:58.760
I know everything about it.

1:49:58.760,1:50:02.310
What if they ask you, what[br]is the curvature of k?

1:50:02.310,1:50:07.640
You say, curvature measures[br]how a curve is bent.

1:50:07.640,1:50:11.820
And they say, yeah, but the[br]Frenet formula tells you

1:50:11.820,1:50:13.610
more about that.

1:50:13.610,1:50:17.720
Not only k shows you[br]how bent the curve is.

1:50:17.720,1:50:27.080
But k is a measure of[br]how fast T changes.

1:50:27.080,1:50:28.240
And he sees why.

1:50:28.240,1:50:31.030
Practically, if you take[br]the [INAUDIBLE] to the bat,

1:50:31.030,1:50:37.310
this is the speed of T. So how[br]fast the teaching will change.

1:50:37.310,1:50:39.890
That will be magnitude,[br]will be just k.

1:50:39.890,1:50:42.440
Because magnitude of N is 1.

1:50:42.440,1:50:48.820
So note that k of s is[br]the length of T prime.

1:50:48.820,1:51:04.387
This measures the change[br]in T. So how fast T varies.

1:51:04.387,1:51:08.610


1:51:08.610,1:51:11.320
What does the torsion represent?

1:51:11.320,1:51:16.690
Well, how fast the[br]binormal varies.

1:51:16.690,1:51:20.596
But if you want to[br]think of a helix,

1:51:20.596,1:51:25.640
and it's a little[br]bit hard to imagine,

1:51:25.640,1:51:30.160
the curvature measures how[br]bent a certain curve is.

1:51:30.160,1:51:33.800
And it measures how[br]bent a plane curve is.

1:51:33.800,1:51:38.720
For example, for the circle you[br]have radius a, 1/a, and so on.

1:51:38.720,1:51:40.870
But there must be[br]also a function that

1:51:40.870,1:51:46.090
shows you how a curve twists.

1:51:46.090,1:51:50.060
Because you have not[br]just a plane curve where

1:51:50.060,1:51:52.370
you care about curvature only.

1:51:52.370,1:51:58.570
But in the space curve you[br]care how the curves twist.

1:51:58.570,1:52:03.190
How fast do they move[br]away from a certain plane?

1:52:03.190,1:52:10.956
Now, if I were to draw-- is[br]it hard to memorize these?

1:52:10.956,1:52:11.456
No.

1:52:11.456,1:52:14.060
I memorized them easily[br]based on the fact

1:52:14.060,1:52:19.850
that everything looks[br]like a decomposition

1:52:19.850,1:52:23.920
of a vector in terms of[br]T, N, and B. So in my mind

1:52:23.920,1:52:28.470
it was like, I take any vector[br]I want, B. And this is T,

1:52:28.470,1:52:32.700
this is N, and this is B.[br]Just the weight was IJK.

1:52:32.700,1:52:36.680
Instead if I, I have T. Instead[br]of J, I have N. Instead of K,

1:52:36.680,1:52:40.040
I have B. They are[br]still unit vectors.

1:52:40.040,1:52:42.610
So locally at the[br]point I have this frame

1:52:42.610,1:52:44.230
and I have any vector.

1:52:44.230,1:52:46.950
This vector-- I'm a physicist.

1:52:46.950,1:52:50.940
So let's say I'm going to[br]represent that as v1 times

1:52:50.940,1:52:54.050
the T plus v2[br]times-- instead of J,

1:52:54.050,1:52:57.790
we'll use that N plus[br]B3 times-- that's

1:52:57.790,1:52:59.980
the last element of the bases.

1:52:59.980,1:53:03.605
Instead of k I have v.[br]So it's the same here.

1:53:03.605,1:53:06.360
You try to pick a[br]vector and decompose

1:53:06.360,1:53:09.950
that in terms of T, N, and B.[br]Will I put that on the final?

1:53:09.950,1:53:10.720
No.

1:53:10.720,1:53:12.925
But I would like you to[br]remember it, especially

1:53:12.925,1:53:17.070
if you are an engineering[br]major or physics major,

1:53:17.070,1:53:19.692
that there is this[br]kind of Frenet frame.

1:53:19.692,1:53:26.010
For those of you who are taking[br]a-- for differential equations,

1:53:26.010,1:53:28.760
you already do some matrices[br]and built-in systems

1:53:28.760,1:53:31.590
of equations, systems of[br]differential equations.

1:53:31.590,1:53:33.215
I'm not going to get there.

1:53:33.215,1:53:38.072
But suppose you don't know[br]differential equations,

1:53:38.072,1:53:41.730
but you know a little[br]bit of linear algebra.

1:53:41.730,1:53:44.950
And I know you know how[br]to multiply matrices.

1:53:44.950,1:53:47.120
You know how I know[br]you multiply matrices,

1:53:47.120,1:53:49.540
no matter how much[br]mathematics you learn.

1:53:49.540,1:53:52.670
And most of you, you are not in[br]general algebra this semester.

1:53:52.670,1:53:55.070
Only two of you are[br]in general algebra.

1:53:55.070,1:54:03.250
When I took a C++ course,[br]the first homework I got was

1:54:03.250,1:54:06.530
to program a matrix[br]multiplication.

1:54:06.530,1:54:07.730
I have to give in matrices.

1:54:07.730,1:54:10.900
I have to program that in C++.

1:54:10.900,1:54:14.600
And freshmen knew that.

1:54:14.600,1:54:20.440
So that means you know how[br]to write this as a matrix

1:54:20.440,1:54:21.410
multiplication.

1:54:21.410,1:54:23.050
Can anybody help me?

1:54:23.050,1:54:25.880
So T, N, B is the magic triple.

1:54:25.880,1:54:28.980
T, N, B's the magic corner.

1:54:28.980,1:54:32.000
T, N, and B are the Three[br]Musketeers who are all

1:54:32.000,1:54:34.326
orthogonal to one another.

1:54:34.326,1:54:37.985
And then I do derivative[br]with respect to s.

1:54:37.985,1:54:42.290
If I want to be[br]elegant, I'll put d/ds.

1:54:42.290,1:54:44.280
OK.

1:54:44.280,1:54:47.330
How am I going to[br]fill in this matrix?

1:54:47.330,1:54:50.650
So somebody who wants to know[br]about differential equations,

1:54:50.650,1:54:51.610
this would be a--

1:54:51.610,1:54:52.790
STUDENT: 0, k, 0.

1:54:52.790,1:54:53.873
PROFESSOR TODA: Very good.

1:54:53.873,1:55:04.770
0, k, 0, minus k 0[br]tau, 0 minus tau 0.

1:55:04.770,1:55:07.362
This is called the[br]skew symmetric matrix.

1:55:07.362,1:55:11.810


1:55:11.810,1:55:14.740
Such matrices are very[br]important in robotics.

1:55:14.740,1:55:17.430
If you've ever been[br]to a robotics team,

1:55:17.430,1:55:20.040
like one of those[br]projects, you should

1:55:20.040,1:55:22.990
know that when we study[br]motions of-- let's say

1:55:22.990,1:55:26.620
that my arm performs[br]two rotations in a row.

1:55:26.620,1:55:30.500
All these motions[br]are described based

1:55:30.500,1:55:35.320
on some groups of rotations.

1:55:35.320,1:55:39.950
And if I go into details,[br]it's going to be really hard.

1:55:39.950,1:55:45.580
But practically[br]in such a setting

1:55:45.580,1:55:49.800
we have to deal with matrices[br]that either have determined

1:55:49.800,1:55:53.520
one, like all rotations[br]actually have,

1:55:53.520,1:55:58.300
or have some other[br]properties, like this guy.

1:55:58.300,1:56:00.410
What's the determinant[br]of this guy?

1:56:00.410,1:56:02.010
What do you guys think?

1:56:02.010,1:56:02.750
Just look at it.

1:56:02.750,1:56:03.250
STUDENT: 0?

1:56:03.250,1:56:04.000
PROFESSOR TODA: 0.

1:56:04.000,1:56:05.660
It has determinant 0.

1:56:05.660,1:56:08.470
And moreover, it[br]looks in the mirror.

1:56:08.470,1:56:11.195
So this comes from[br]a group of motion,

1:56:11.195,1:56:14.690
which is little s over 3,[br]the linear algebra, actually.

1:56:14.690,1:56:17.190
So when k is looking[br]in the mirror,

1:56:17.190,1:56:20.820
it becomes minus k tau,[br]is becoming minus tau.

1:56:20.820,1:56:24.190
It is antisymmetric[br]or skew symmetric.

1:56:24.190,1:56:27.010
Skew symmetric or[br]antisymmetric is the same.

1:56:27.010,1:56:29.960
STUDENT: Antisymmetric,[br]skew symmetric matrix.

1:56:29.960,1:56:31.935
PROFESSOR TODA: Skew[br]symmetric or antisymmetric

1:56:31.935,1:56:33.370
is exactly the same thing.

1:56:33.370,1:56:34.292
They are synonyms.

1:56:34.292,1:56:37.060


1:56:37.060,1:56:40.130
So it looks in the mirror[br]and picks up the minus sign,

1:56:40.130,1:56:41.820
has 0 in the bag.

1:56:41.820,1:56:43.135
What am I going to put here?

1:56:43.135,1:56:44.460
You already got the idea.

1:56:44.460,1:56:47.060
So when Ryan gave[br]me this, he meant

1:56:47.060,1:56:50.460
that he knew what I'm going[br]to put here, as a vector,

1:56:50.460,1:56:54.024
as a column vector.

1:56:54.024,1:56:54.936
STUDENT: [INAUDIBLE]

1:56:54.936,1:56:56.019
PROFESSOR TODA: No, no no.

1:56:56.019,1:56:57.040
How do I multiply?

1:56:57.040,1:56:58.510
TNB, right?

1:56:58.510,1:57:01.300
So guys, how do you[br]multiply matrices?

1:57:01.300,1:57:05.420
You go first row[br]and first column.

1:57:05.420,1:57:06.400
So you go like this.

1:57:06.400,1:57:13.629
0 times T plus k times 10[br]plus 0 times B. Here it is.

1:57:13.629,1:57:15.170
So I'm teaching you[br]a little bit more

1:57:15.170,1:57:18.620
than-- if you are going to[br]stick with linear algebra

1:57:18.620,1:57:21.250
and stick with[br]differential equations,

1:57:21.250,1:57:25.440
this is a good introduction[br]to more of those mathematics.

1:57:25.440,1:57:26.111
Yes, sir?

1:57:26.111,1:57:28.059
STUDENT: Why don't[br]you use Cramer's rule?

1:57:28.059,1:57:28.850
PROFESSOR TODA: Uh?

1:57:28.850,1:57:31.270
STUDENT: Why don't you[br]use the Cramer's rule?

1:57:31.270,1:57:32.722
PROFESSOR TODA:[br]The Cramer's rule?

1:57:32.722,1:57:34.835
STUDENT: Yeah. [INAUDIBLE].

1:57:34.835,1:57:35.626
PROFESSOR TODA: No.

1:57:35.626,1:57:44.070
First of all, Crarmer's rule is[br]to solve systems of equations

1:57:44.070,1:57:47.810
that don't involve derivatives,[br]like a linear system

1:57:47.810,1:57:51.960
like Ax equals B.[br]I'm going to have,

1:57:51.960,1:57:56.760
for example, 3x1[br]plus 2x3 equals 1.

1:57:56.760,1:58:01.000
5x1 plus x2 plus x3[br]equals something else.

1:58:01.000,1:58:03.400
So for that I can[br]use Cramer's rule.

1:58:03.400,1:58:04.690
But look at that!

1:58:04.690,1:58:06.060
This is really complicated.

1:58:06.060,1:58:07.840
It's a dynamical system.

1:58:07.840,1:58:11.580
At every moment of time[br]the vectors are changing.

1:58:11.580,1:58:13.420
So it's a crazy [INAUDIBLE].

1:58:13.420,1:58:19.095
Like A of t times[br]something, so some vector

1:58:19.095,1:58:22.858
that is also depending on[br]time equals the derivative

1:58:22.858,1:58:25.000
of that vector that [INAUDIBLE].

1:58:25.000,1:58:31.560
So that's a OD system that[br]you should learn in 3351.

1:58:31.560,1:58:33.360
So I don't know what[br]your degree plan is,

1:58:33.360,1:58:35.162
but most of you in[br]engineering will

1:58:35.162,1:58:43.802
take my class, 2316 in algebra,[br]OD1 3350 where they teach you

1:58:43.802,1:58:45.010
about differential equations.

1:58:45.010,1:58:48.070
These are all differential[br]equations, all three of them.

1:58:48.070,1:58:51.326
In 3351 you learn[br]about this system

1:58:51.326,1:58:54.460
which is a system of[br]differential equation.

1:58:54.460,1:58:57.210
And then you[br]practically say, now I

1:58:57.210,1:58:59.840
know everything I need to[br]know in math, and you say,

1:58:59.840,1:59:01.100
goodbye math.

1:59:01.100,1:59:02.740
If you guys wanted[br]to learn more,

1:59:02.740,1:59:06.223
of course I would be very[br]happy to learn that, hey, I

1:59:06.223,1:59:08.810
like math, I'd like[br]to be a double major.

1:59:08.810,1:59:12.320
I'd like to be not just an[br]engineering, but also math

1:59:12.320,1:59:14.550
major if you really like it.

1:59:14.550,1:59:18.170
Many people already[br]have a minor.

1:59:18.170,1:59:20.240
Many of you have a[br]minor in your plan.

1:59:20.240,1:59:22.835
Like for that minor[br]you only need--

1:59:22.835,1:59:24.170
STUDENT: One extra math course.

1:59:24.170,1:59:25.753
PROFESSOR TODA: One[br]extra math course.

1:59:25.753,1:59:30.720
For example, with 3350 you[br]don't need 3351 for a minor.

1:59:30.720,1:59:31.380
Why?

1:59:31.380,1:59:34.380
Because you are taking the[br]probability in stats anyway.

1:59:34.380,1:59:35.260
You have to.

1:59:35.260,1:59:38.600
They force you to do that, 3342.

1:59:38.600,1:59:44.970
So if you take 3351 it's on top[br]of the minor that we give you.

1:59:44.970,1:59:46.400
I know because that's what I do.

1:59:46.400,1:59:47.980
I look at the degree plans.

1:59:47.980,1:59:51.960
And I work closely to the[br]math adviser, with Patty.

1:59:51.960,1:59:54.265
She has all the [INAUDIBLE].

1:59:54.265,1:59:55.390
STUDENT: So is [INAUDIBLE]?

1:59:55.390,1:59:59.527


1:59:59.527,2:00:00.860
PROFESSOR TODA: You mean double?

2:00:00.860,2:00:01.840
Double degree?

2:00:01.840,2:00:04.214
We have this already in place.

2:00:04.214,2:00:05.380
We've had it for many years.

2:00:05.380,2:00:07.400
It's an excellent plan.

2:00:07.400,2:00:09.870
162 hours it is now.

2:00:09.870,2:00:12.870
It used to be 159.

2:00:12.870,2:00:17.700
Double major, computer[br]science and mathematics.

2:00:17.700,2:00:22.840
And I could say they were[br]some of the most successful

2:00:22.840,2:00:26.520
in terms of finding jobs.

2:00:26.520,2:00:28.660
What would you take[br]on top of that?

2:00:28.660,2:00:30.945
Well, as a math major you[br]have a few more courses

2:00:30.945,2:00:32.860
to take one top of that.

2:00:32.860,2:00:36.720
You can link your computer[br]science with the mathematics,

2:00:36.720,2:00:39.630
for example, by taking[br]numerical analysis.

2:00:39.630,2:00:42.300
If you love computers[br]and you like calculus

2:00:42.300,2:00:46.700
and you want to put[br]together all the information

2:00:46.700,2:00:49.140
you have in both, then[br]numerical analysis

2:00:49.140,2:00:50.430
would be your best bet.

2:00:50.430,2:00:55.215
And they require that in[br]both computer science degree

2:00:55.215,2:00:58.230
if you are a double major,[br]and your math degree.

2:00:58.230,2:01:03.050
So the good thing is that some[br]things count for both degrees.

2:01:03.050,2:01:06.746
And so with those 160[br]hours you are very happy.

2:01:06.746,2:01:10.060
Oh, I'm done, I got[br]a few more hours.

2:01:10.060,2:01:12.420
Many math majors[br]already have around 130.

2:01:12.420,2:01:13.830
They're not supposed to.

2:01:13.830,2:01:16.080
They are supposed[br]to stop at 120.

2:01:16.080,2:01:19.690
So why not go the extra 20 hours[br]and get two degrees in one?

2:01:19.690,2:01:21.133
STUDENT: It's a semester.

2:01:21.133,2:01:22.095
PROFESSOR TODA: Yeah.

2:01:22.095,2:01:23.428
Of course, it's a lot more work.

2:01:23.428,2:01:26.430
But we have people[br]who like-- really they

2:01:26.430,2:01:30.170
are nerdy people who loved[br]computer science from when

2:01:30.170,2:01:31.990
they were three or four.

2:01:31.990,2:01:33.410
And they also like math.

2:01:33.410,2:01:37.160
And they say, OK,[br]I want to do both.

2:01:37.160,2:01:41.640
OK, a little bit more[br]and I'll let you go.

2:01:41.640,2:01:44.822
Now I want you to ask[br]me other questions

2:01:44.822,2:01:48.486
you may have had about the[br]homework, anything that

2:01:48.486,2:01:59.224
gave you headache, anything that[br]you feel you need a little bit

2:01:59.224,2:02:00.712
more of an explanation about.

2:02:00.712,2:02:12.471


2:02:12.471,2:02:12.970
Yes?

2:02:12.970,2:02:14.011
STUDENT: I just have one.

2:02:14.011,2:02:15.795
In WeBWork, what[br]is the easiest way

2:02:15.795,2:02:17.650
to take the square[br]root of something?

2:02:17.650,2:02:18.610
STUDENT: sqrt.

2:02:18.610,2:02:21.966
PROFESSOR TODA: sqrt[br]is what you type.

2:02:21.966,2:02:24.870
But of course you can[br]also go to the caret 1/2.

2:02:24.870,2:02:27.930


2:02:27.930,2:02:29.490
Something non-technical?

2:02:29.490,2:02:34.354
Any question, yes sir,[br]from the homework?

2:02:34.354,2:02:38.690
Or in relation to [INAUDIBLE]?

2:02:38.690,2:02:41.506
STUDENT: I don't understand[br]why is the tangent unit vector,

2:02:41.506,2:02:44.162
it's just the slope off[br]of that line, right?

2:02:44.162,2:02:45.144
The drunk bug?

2:02:45.144,2:02:47.100
Whatever line the[br]drunk bug is on?

2:02:47.100,2:02:49.120
PROFESSOR TODA: So it[br]would be the tangent

2:02:49.120,2:02:52.040
to the directional[br]motion, which is a curve.

2:02:52.040,2:02:54.620


2:02:54.620,2:02:58.140
And normalized to[br]have length one.

2:02:58.140,2:03:01.700
Because otherwise our[br]prime is-- you may say,

2:03:01.700,2:03:04.210
why do you need T to be unitary?

2:03:04.210,2:03:07.150


2:03:07.150,2:03:11.355
OK, computations become[br]horrible unless your speed

2:03:11.355,2:03:13.815
is 1 or 5 or 9.

2:03:13.815,2:03:18.140
If the speed is a constant,[br]everything else becomes easier.

2:03:18.140,2:03:20.182
So that's one reason.

2:03:20.182,2:03:22.086
STUDENT: And why[br]is the derivative

2:03:22.086,2:03:24.466
of T then perpendicular?

2:03:24.466,2:03:26.502
Why does it always turn into--

2:03:26.502,2:03:27.960
PROFESSOR TODA:[br]Perpendicular to T?

2:03:27.960,2:03:30.940
We've done that last time,[br]but I'm glad to do it again.

2:03:30.940,2:03:34.430
And I forgot what we[br]wrote in the book,

2:03:34.430,2:03:36.720
and I also saw in[br]the book this thing

2:03:36.720,2:03:42.800
that if you have R, in[br]absolute value, constant--

2:03:42.800,2:03:44.810
and I've done that[br]with you guys--

2:03:44.810,2:03:52.020
prove that R and R prime had[br]every point perpendicular.

2:03:52.020,2:03:54.850
So if you have-- we've[br]done that before.

2:03:54.850,2:03:57.090
Now, what do you do then?

2:03:57.090,2:04:00.564
T [INAUDIBLE] T is 1.

2:04:00.564,2:04:04.500
The scalar [INAUDIBLE][br]the product.

2:04:04.500,2:04:09.540
T prime times T plus[br]T prime T prime.

2:04:09.540,2:04:12.090
So 0.

2:04:12.090,2:04:16.540
And T is perpendicular[br]to T prime,

2:04:16.540,2:04:20.610
because that means T[br]or T prime equals 0.

2:04:20.610,2:04:27.860


2:04:27.860,2:04:30.360
When you run in a[br]circle, you say--

2:04:30.360,2:04:33.790
OK, let's run in a circle.

2:04:33.790,2:04:40.650
I say, this is my T. I can feel[br]that there is something that's

2:04:40.650,2:04:42.530
trying to bend me this way.

2:04:42.530,2:04:44.366
That is my acceleration.

2:04:44.366,2:04:49.040
And I have to-- but I don't[br]know-- how familiar are you

2:04:49.040,2:04:51.385
with the winter sports?

2:04:51.385,2:04:54.150


2:04:54.150,2:04:58.070
In many winter sports, the[br]Frenet Trihedron is crucial.

2:04:58.070,2:05:01.090
Imagine that you have[br]one of those slopes,

2:05:01.090,2:05:04.890
and all of the sudden the[br]torsion becomes too weak.

2:05:04.890,2:05:06.680
That means it becomes dangerous.

2:05:06.680,2:05:09.870
That means that the[br]vehicle you're in,

2:05:09.870,2:05:15.010
the snow vehicle or any kind[br]of-- your skis, [INAUDIBLE],

2:05:15.010,2:05:20.850
if the torsion of your body[br]moving can become too big,

2:05:20.850,2:05:21.880
that will be a problem.

2:05:21.880,2:05:24.670
So you have to redesign[br]that some more.

2:05:24.670,2:05:26.660
And this is what they do.

2:05:26.660,2:05:28.570
You know there have[br]been many accidents.

2:05:28.570,2:05:32.360
And many times they say,[br]even in Formula One,

2:05:32.360,2:05:38.170
the people who project[br]a certain racetrack,

2:05:38.170,2:05:41.620
like a track in[br]Indianapolis or Montecarlo

2:05:41.620,2:05:44.460
or whatever, they[br]have to have in mind

2:05:44.460,2:05:47.660
that Frenet frame every second.

2:05:47.660,2:05:50.690
So there are[br]simulators showing how

2:05:50.690,2:05:52.874
the Frenet frame is changing.

2:05:52.874,2:05:55.718
There are programs that[br]measure the curvature

2:05:55.718,2:05:59.950
in a torsion for those[br]simulators at every point.

2:05:59.950,2:06:02.690
Neither the curvature[br]nor the torsion

2:06:02.690,2:06:04.360
can exceed a certain value.

2:06:04.360,2:06:06.900
Otherwise it becomes dangerous.

2:06:06.900,2:06:09.545
You say, oh, I thought[br]only the speed is a danger.

2:06:09.545,2:06:10.910
Nope.

2:06:10.910,2:06:14.846
It's also the way that the[br]motion, if it's a skew curve,

2:06:14.846,2:06:16.710
it's really complicated.

2:06:16.710,2:06:20.030
Because you twist and turn[br]and bend in many ways.

2:06:20.030,2:06:22.239
And it can become[br]really dangerous.

2:06:22.239,2:06:23.280
Speed is not [INAUDIBLE].

2:06:23.280,2:06:26.262


2:06:26.262,2:06:30.252
STUDENT: So the torsion was[br]the twists in the track?

2:06:30.252,2:06:31.960
PROFESSOR TODA: The[br]torsion is the twist.

2:06:31.960,2:06:34.580
And by the way, keep your idea.

2:06:34.580,2:06:37.290
You wanted to ask[br]something more?

2:06:37.290,2:06:43.090
When you twist-- suppose you[br]have something like a race car.

2:06:43.090,2:06:47.190
And the race car is at[br]the walls of the track.

2:06:47.190,2:06:57.980
And here's-- when you have[br]a very abrupt curvature

2:06:57.980,2:07:03.760
and torsion, and you can have[br]that in Formula One as well,

2:07:03.760,2:07:09.922
why do they build one wall[br]a lot higher than the other?

2:07:09.922,2:07:13.980
Because the poor car-- I[br]don't know how passionate you

2:07:13.980,2:07:19.552
are about Formula[br]One or car races--

2:07:19.552,2:07:24.690
the poor car is going[br]to be close to the wall.

2:07:24.690,2:07:28.382
It's going to bend like that,[br]that wall would be round.

2:07:28.382,2:07:32.850
And as a builder, you have to[br]build the wall really high.

2:07:32.850,2:07:35.818
Because that kind of high[br]speed, high velocity,

2:07:35.818,2:07:39.350
high curvature, the poor[br]car's going szhhhhh-- then

2:07:39.350,2:07:42.050
again on a normal track.

2:07:42.050,2:07:45.140
Imagine what happens if the[br]wall is not high enough.

2:07:45.140,2:07:48.490
The wheels of the car[br]will go up and get over.

2:07:48.490,2:07:50.041
And it's going to be a disaster.

2:07:50.041,2:07:52.740


2:07:52.740,2:07:57.490
So that engineer ha to study[br]all the parametric equations

2:07:57.490,2:08:01.240
and the Frenet frame and[br]deep down make a simulator,

2:08:01.240,2:08:04.450
compute how tall the walls[br]should be in order for the car

2:08:04.450,2:08:10.220
not to get over on the other[br]side or get off the track.

2:08:10.220,2:08:12.350
It's really complicated stuff.

2:08:12.350,2:08:14.890
It's all mathematics[br]and physics,

2:08:14.890,2:08:18.680
but all the applications are[br]run by engineers and-- yes, sir?

2:08:18.680,2:08:22.033
STUDENT: What's the difference[br][INAUDIBLE] centrifugal force?

2:08:22.033,2:08:23.950
PROFESSOR TODA: The[br]centrifugal force

2:08:23.950,2:08:26.380
is related to our double prime.

2:08:26.380,2:08:32.130
Our double prime is related[br]to N and T at the same time.

2:08:32.130,2:08:36.160
So at some point, let me ask you[br]one last question and I'm done.

2:08:36.160,2:08:39.210


2:08:39.210,2:08:43.236
What's the relationship between[br]acceleration or double prime?

2:08:43.236,2:08:45.982
And are they the same thing?

2:08:45.982,2:08:50.297
And when are they[br]not the same thing?

2:08:50.297,2:08:52.380
Because you say, OK,[br]practically the centrifugal--

2:08:52.380,2:08:54.180
STUDENT: They're[br]the same on a curve.

2:08:54.180,2:08:55.740
PROFESSOR TODA:[br]They are the same--

2:08:55.740,2:08:56.823
STUDENT: Like on a circle.

2:08:56.823,2:08:58.520
PROFESSOR TODA: On a circle!

2:08:58.520,2:09:00.090
And you are getting so close.

2:09:00.090,2:09:01.370
It's hot, hot, hot.

2:09:01.370,2:09:08.100
On a circle and on a helix they[br]are the same up to a constant.

2:09:08.100,2:09:11.310
So what do you think the[br]magic answer will be?

2:09:11.310,2:09:12.480
N was what, guys?

2:09:12.480,2:09:15.220
N was-- remind me again.

2:09:15.220,2:09:18.475
That was T prime over[br]absolute value of T prime.

2:09:18.475,2:09:22.290
But that doesn't mean,[br]does not equal, in general,

2:09:22.290,2:09:26.350
does not equal to[br]R double prime.

2:09:26.350,2:09:28.096
When is it equal?

2:09:28.096,2:09:29.930
In general it's not equal.

2:09:29.930,2:09:31.070
When is it equal?

2:09:31.070,2:09:35.440
If you are in aclength, you[br]see the advantage of aclength.

2:09:35.440,2:09:36.965
It's wonderful.

2:09:36.965,2:09:40.067
In arclength, T is R prime of s.

2:09:40.067,2:09:45.940
And in arclength that means T[br]prime is R double prime of s.

2:09:45.940,2:09:48.590
And in arclength[br]I just told you,

2:09:48.590,2:09:50.330
T prime is the first[br]Frenet formula.

2:09:50.330,2:09:55.510
It'll be curvature times the N.

2:09:55.510,2:10:02.180
So the acceleration[br]practically and the N

2:10:02.180,2:10:06.560
will be the same in arclength,[br]up to a scalar multiplication.

2:10:06.560,2:10:11.678
But what if your speed[br]is not even constant?

2:10:11.678,2:10:12.530
Then God help you.

2:10:12.530,2:10:16.850
Because the acceleration[br]R double prime and N

2:10:16.850,2:10:19.780
are not colinear.

2:10:19.780,2:10:24.370
So if I were to draw-- and[br]that's my last picture--

2:10:24.370,2:10:26.950
let me give you a[br]wild motion here.

2:10:26.950,2:10:32.466
You start slow and then you go[br]crazy and fast and slow down.

2:10:32.466,2:10:35.657
Just like most of the[br]physical models from the bugs

2:10:35.657,2:10:38.900
and the flies and so on.

2:10:38.900,2:10:44.950
In that kind of crazy motion you[br]have a T and N at every point.

2:10:44.950,2:10:45.450
[INAUDIBLE]

2:10:45.450,2:10:48.430


2:10:48.430,2:10:50.860
[? v ?] will be down.

2:10:50.860,2:10:53.360
And T is here.

2:10:53.360,2:10:56.940
So can you draw arc[br]double prime for me?

2:10:56.940,2:10:59.433
It will still be[br]towards the inside.

2:10:59.433,2:11:04.200
But it's still going to[br]coincide with N. Maybe this one.

2:11:04.200,2:11:12.640
What's the magic thing is[br]that T, N, and R double prime

2:11:12.640,2:11:15.745
are in the same plane always.

2:11:15.745,2:11:18.170
That's another[br]secret other students

2:11:18.170,2:11:19.630
don't know in Calculus 3.

2:11:19.630,2:11:22.456
That same thing is[br]called osculating plane.

2:11:22.456,2:11:25.630


2:11:25.630,2:11:31.270
We have a few magic[br]names for these things.

2:11:31.270,2:11:36.509
So T and N, the plane that[br]is-- how shall I say that?

2:11:36.509,2:11:37.050
I don't know.

2:11:37.050,2:11:43.638
The plane given by T and N[br]is called osculating plane.

2:11:43.638,2:11:46.710


2:11:46.710,2:11:49.080
The acceleration is[br]always on that plane.

2:11:49.080,2:11:52.460
So imagine T and N are[br]in the same shaded plane.

2:11:52.460,2:11:55.500
R double prime is[br]in the same plane.

2:11:55.500,2:11:56.550
OK?

2:11:56.550,2:11:58.510
Now, can you guess[br]the other two names?

2:11:58.510,2:12:03.639
So this is T, this[br]is N. And B is up.

2:12:03.639,2:12:04.805
This is my body's direction.

2:12:04.805,2:12:06.200
T and N, look at me.

2:12:06.200,2:12:10.160
T, N, and B. I'm the[br]Frenet Trihedron.

2:12:10.160,2:12:13.360
Which one is the[br]osculating plane?

2:12:13.360,2:12:16.740
It's the horizontal xy plane.

2:12:16.740,2:12:20.940
OK, do you know-- maybe you're[br]a mechanical engineering major,

2:12:20.940,2:12:23.650
and after that I[br]will let you go.

2:12:23.650,2:12:25.920
No extra credit,[br]though for this task.

2:12:25.920,2:12:29.130
Maybe I'm going to start asking[br]questions and give you $1.

2:12:29.130,2:12:31.330
I used to do that a lot[br]in differential equations,

2:12:31.330,2:12:34.580
like ask a hard question,[br]whoever gets it first,

2:12:34.580,2:12:36.420
give her a dollar.

2:12:36.420,2:12:41.595
Until a point when they asked[br]me to teach Honors 3350 when

2:12:41.595,2:12:44.210
I started having three or four[br]people answering the question

2:12:44.210,2:12:44.925
at the same time.

2:12:44.925,2:12:49.020
And that was a[br]significant expense,

2:12:49.020,2:12:52.215
because I had to give $4[br]away at the same time.

2:12:52.215,2:12:53.840
STUDENT: I feel like[br]you should've just

2:12:53.840,2:12:54.590
split it between--

2:12:54.590,2:12:57.950
PROFESSOR TODA: So that's[br]normal and binormal.

2:12:57.950,2:13:00.800
This is me, the binormal,[br]and this is the normal.

2:13:00.800,2:13:03.080
Does anybody know the[br]name of this plane,

2:13:03.080,2:13:05.850
between normal and bionormal?

2:13:05.850,2:13:08.170
This would be this plane.

2:13:08.170,2:13:10.750
STUDENT: The skew [INAUDIBLE].

2:13:10.750,2:13:12.250
PROFESSOR TODA:[br]Normal and binormal.

2:13:12.250,2:13:13.884
They call that normal plane.

2:13:13.884,2:13:16.540


2:13:16.540,2:13:22.510
So it's tricky if you are not[br]a mechanical engineering major.

2:13:22.510,2:13:28.460
But some of you are maybe[br]and will learn that later.

2:13:28.460,2:13:29.940
Any other questions for me?

2:13:29.940,2:13:33.890
Now, in my office I'm[br]going to do review.

2:13:33.890,2:13:37.836
I was wondering[br]if you have time,

2:13:37.836,2:13:39.960
I don't know if you have[br]time to come to my office,

2:13:39.960,2:13:43.340
but should you have any kind[br]of homework related question,

2:13:43.340,2:13:46.290
I'll be very happy[br]to answer it now.

2:13:46.290,2:13:49.000
3:00 to 5:00.

2:13:49.000,2:13:51.020
Now, one time I[br]had a student who

2:13:51.020,2:13:53.175
only had seven questions left.

2:13:53.175,2:13:55.690
He came to my office and[br]he left with no homework.

2:13:55.690,2:13:57.593
We finished all of them.

2:13:57.593,2:13:58.380
And I felt guilty.

2:13:58.380,2:14:00.870
But at the same, he[br]said, well, no, it's

2:14:00.870,2:14:03.170
better I came to you instead[br]of going to my tutor.

2:14:03.170,2:14:05.180
It was fine.

2:14:05.180,2:14:08.670
So we can try some[br]problems together today

2:14:08.670,2:14:11.850
if you want between 3:00 and[br]5:00, if you have the time.

2:14:11.850,2:14:13.838
Some of you don't have the time.

2:14:13.838,2:14:14.832
All right?

2:14:14.832,2:14:16.323
If you don't have[br]the time today,

2:14:16.323,2:14:19.305
and you would like to[br]be helped [INAUDIBLE],

2:14:19.305,2:14:21.293
click Email Instructor.

2:14:21.293,2:14:24.275
I'm going to get the[br]questions [INAUDIBLE].

2:14:24.275,2:14:26.014
You're welcome to[br]ask me anything

2:14:26.014,2:14:27.257
at any time over there.

2:14:27.257,2:14:38.191


2:14:38.191,2:14:41.173
[CLASSROOM CHATTER]

2:14:41.173,2:15:12.348


2:15:12.348,2:15:14.472
PROFESSOR TODA: I have[br]somebody who's taking notes.

2:15:14.472,2:15:15.167
STUDENT: Yeah, I know.

2:15:15.167,2:15:15.963
And that's why I was like--

2:15:15.963,2:15:16.957
PROFESSOR TODA: He's[br]going to make a copy

2:15:16.957,2:15:18.448
and I'll give you a copy.

2:15:18.448,2:15:19.442
STUDENT: Yeah.

2:15:19.442,2:15:23.621
My Cal 1 teacher,[br]Dr. [INAUDIBLE].

2:15:23.621,2:15:24.412
STUDENT: Thank you.

2:15:24.412,2:15:25.495
PROFESSOR TODA: Yes, yeah.

2:15:25.495,2:15:26.897
Have a nice day.

2:15:26.897,2:15:29.030
STUDENT: --got really mad[br]when I don't take notes.

2:15:29.030,2:15:34.680
Because he felt like[br]I was not, I guess--

2:15:34.680,2:15:36.503