0:00:00.000,0:00:00.500


0:00:00.500,0:00:01.900
PROFESSOR: Plus 1.

0:00:01.900,0:00:04.730
And next would be[br]between-- this is where

0:00:04.730,0:00:06.900
most people have the problem.

0:00:06.900,0:00:09.000
They thought x is[br]any real number.

0:00:09.000,0:00:10.830
No-- no, no, no, no, no.

0:00:10.830,0:00:12.420
You wanted a segment.

0:00:12.420,0:00:15.750
x has the values[br]between this value,

0:00:15.750,0:00:18.190
whatever value's on this[br]axis and that value.

0:00:18.190,0:00:23.730
So x equals 1, x equals[br]2 are the end points.

0:00:23.730,0:00:29.810
How do you write a[br]parameterized equation?

0:00:29.810,0:00:32.479
And that should help you[br]very much on the web work

0:00:32.479,0:00:38.337
homework on that problem[br]for such a function.

0:00:38.337,0:00:39.545
Well, you say, wait a minute.

0:00:39.545,0:00:41.675
Magdalena, this is[br]a linear function.

0:00:41.675,0:00:42.610
It's a piece of cake.

0:00:42.610,0:00:45.230
I have just x plus 1.

0:00:45.230,0:00:47.690
I know how to deal with that.

0:00:47.690,0:00:49.750
Yes, but I'm asking[br]you something else.

0:00:49.750,0:00:53.770
Rather than writing[br]the explicit equation

0:00:53.770,0:00:58.870
in Cartesian coordinates x and[br]y, tell me what time it is.

0:00:58.870,0:01:01.110
And then I'm going[br]to travel in time.

0:01:01.110,0:01:06.590
I want to travel in time, in[br]space-time, on the segment,

0:01:06.590,0:01:08.190
right?

0:01:08.190,0:01:13.700
So why if x equals[br]x plus 1 has what

0:01:13.700,0:01:15.910
is that-- what[br]parameterization has infinitely

0:01:15.910,0:01:18.080
many parameterization?

0:01:18.080,0:01:22.620
Somebody will say, ha, you told[br]us that it has infinitely many.

0:01:22.620,0:01:24.200
Why do you insist on one?

0:01:24.200,0:01:27.765
Which one is the most natural[br]and the easiest to grasp?

0:01:27.765,0:01:30.425
STUDENT: Zero to one.

0:01:30.425,0:01:32.790
PROFESSOR: Zero to one is[br]not a parameterization.

0:01:32.790,0:01:34.560
STUDENT: Times zero one.

0:01:34.560,0:01:39.280
PROFESSOR: So, so, so what[br]is the parametric equation

0:01:39.280,0:01:41.190
of a curve in general?

0:01:41.190,0:01:46.530
If I have a curve, y equals--[br]oh, I'll start with x.

0:01:46.530,0:01:49.850
X equals x of t[br]and y equals y of t

0:01:49.850,0:01:56.700
represent the two[br]parametric questions that

0:01:56.700,0:02:00.080
give that curve's[br]equation in plane--

0:02:00.080,0:02:03.620
in plane where[br]the i of t belongs

0:02:03.620,0:02:05.270
to a certain interval i.

0:02:05.270,0:02:06.720
That's the mysterious interval.

0:02:06.720,0:02:10.100
I don't really care[br]about that in general.

0:02:10.100,0:02:15.280
In my case, which one is the[br]most natural parametrization,

0:02:15.280,0:02:17.940
guys?

0:02:17.940,0:02:19.690
Take x to be time.

0:02:19.690,0:02:20.790
Say again, Magdalena.

0:02:20.790,0:02:23.040
Take x to be time.

0:02:23.040,0:02:26.870
And that will make[br]your life easier.

0:02:26.870,0:02:29.550
I take x to be time.

0:02:29.550,0:02:33.460
And then y would be time plus 1.

0:02:33.460,0:02:34.570
And I'm happy.

0:02:34.570,0:02:39.380
So the way they asked you to[br]enter your answer in web work

0:02:39.380,0:02:45.050
was as r of t equals-- and[br]it's blinking, blinking,

0:02:45.050,0:02:46.885
interactive field for you.

0:02:46.885,0:02:48.820
You say, OK, t?

0:02:48.820,0:02:51.490
T what?

0:02:51.490,0:02:53.660
And I'm not going to[br]solve your problem.

0:02:53.660,0:02:55.530
But your problem is similar.

0:02:55.530,0:02:56.030
Why?

0:02:56.030,0:03:04.020
Because r of t, which is the[br]vector equation of your y

0:03:04.020,0:03:09.670
or curve would give you the[br]position vector, which is what?

0:03:09.670,0:03:10.265
Wait a second.

0:03:10.265,0:03:14.440
Let me finish. x of t[br]times i plus y of t times

0:03:14.440,0:03:17.320
j is the definition[br]I gave last time.

0:03:17.320,0:03:18.280
Go ahead.

0:03:18.280,0:03:20.900
STUDENT: Where'd you get[br]r of t and what is it?

0:03:20.900,0:03:23.550
PROFESSOR: I already[br]discussed it last time.

0:03:23.550,0:03:27.300
So since I'm[br]reviewing today, just

0:03:27.300,0:03:30.020
reviewing today[br]chapter 10, I really

0:03:30.020,0:03:32.340
don't mind going over with you.

0:03:32.340,0:03:34.780
But please keep in[br]mind this is the first

0:03:34.780,0:03:37.629
and the last time I'm[br]going to review things

0:03:37.629,0:03:38.420
with you last time.

0:03:38.420,0:03:44.150
So what did you say a position[br]vector is for a curve?

0:03:44.150,0:03:46.953
When we talked about[br]the drunken bug,

0:03:46.953,0:03:50.720
we say the drunken bug is[br]following a trajectory.

0:03:50.720,0:03:53.870
He or she is struggling in time.

0:03:53.870,0:04:00.620
I have a given frame xyz[br]system of coordinates-- system

0:04:00.620,0:04:03.650
of axes of coordinates[br]with a certain origin.

0:04:03.650,0:04:07.660
Thank God for this origin[br]because you cannot refer

0:04:07.660,0:04:11.070
to a position vector[br]unless you have a frame--

0:04:11.070,0:04:14.190
an original frame, a position[br]frame, initial frame.

0:04:14.190,0:04:21.856
So r of t represents the vector[br]that originates at the origin o

0:04:21.856,0:04:28.480
and ends exactly at the position[br]of your particle at time t.

0:04:28.480,0:04:30.925
If you want, if[br]you hate bugs, this

0:04:30.925,0:04:35.370
is just the particle from[br]physics that travels in time t.

0:04:35.370,0:04:35.990
So--

0:04:35.990,0:04:39.731
STUDENT: OK, so the r of t[br]is represented in the parent

0:04:39.731,0:04:40.230
equation

0:04:40.230,0:04:41.500
PROFESSOR: Yes, sir.

0:04:41.500,0:04:42.530
Exactly.

0:04:42.530,0:04:45.820
In a plane where z[br]is 0-- so you imagine

0:04:45.820,0:04:48.520
the z-axis coming at z0.

0:04:48.520,0:04:50.520
This is the xy plane.

0:04:50.520,0:04:52.960
And I'm very happy[br]I have on the floor.

0:04:52.960,0:04:54.570
This bug is on the floor.

0:04:54.570,0:04:56.469
He doesn't want to know[br]what's the dimension.

0:04:56.469,0:04:57.510
So what's he going to do?

0:04:57.510,0:05:02.480
He's going to say plus 0 times[br]k that I don't care about

0:05:02.480,0:05:05.919
because the position[br]vector will be given by--

0:05:05.919,0:05:06.460
STUDENT: So--

0:05:06.460,0:05:07.876
PROFESSOR: --or[br]for a plane curve.

0:05:07.876,0:05:09.470
STUDENT: So if this[br]was in 3D space

0:05:09.470,0:05:14.180
and we had three equations[br]so it was like z equals--

0:05:14.180,0:05:19.392
is equal to 2y plus x plus 1,[br]then it would be-- then how

0:05:19.392,0:05:20.605
would we do that?

0:05:20.605,0:05:23.400
PROFESSOR: Let me remind us in[br]general the way I pointed it

0:05:23.400,0:05:24.780
out last.

0:05:24.780,0:05:27.800
R of t in general as[br]a position vector,

0:05:27.800,0:05:29.540
we said many things about it.

0:05:29.540,0:05:33.590
We said it is a smooth function.

0:05:33.590,0:05:36.170
What does it mean[br]differential role

0:05:36.170,0:05:38.810
with derivative continuous?

0:05:38.810,0:05:41.072
What did-- actually, that's c1.

0:05:41.072,0:05:42.030
What else did they say?

0:05:42.030,0:05:43.520
He said it's a regular.

0:05:43.520,0:05:45.540
It's a regular vector function.

0:05:45.540,0:05:46.700
What does it mean?

0:05:46.700,0:05:49.040
It never stops, not[br]even for a second.

0:05:49.040,0:05:51.950
Well, the velocity[br]of that is zero.

0:05:51.950,0:05:53.850
When we introduced[br]it-- all right,

0:05:53.850,0:05:56.015
I cannot teach the whole[br]thing all over again,

0:05:56.015,0:05:59.990
but I'll be happy to[br]do review just today.

0:05:59.990,0:06:05.420
It's going to be x of ti[br]plus y of tj plus z over k.

0:06:05.420,0:06:07.220
That is a way to[br]write it like that.

0:06:07.220,0:06:13.220
Or the simpler way to write[br]it as x of t, y of t, z of t.

0:06:13.220,0:06:15.830
Now, if it involves[br]using different notation,

0:06:15.830,0:06:17.720
I want to warn you about that.

0:06:17.720,0:06:21.700
Some people like to put[br]braces like angular brackets.

0:06:21.700,0:06:25.460
Or some people like[br]because it's a vector.

0:06:25.460,0:06:29.486
And that's the way they define[br]vector Some people like just

0:06:29.486,0:06:30.236
round parentheses.

0:06:30.236,0:06:31.710
This is more practically.

0:06:31.710,0:06:34.450
These are the coordinates[br]of a position vector

0:06:34.450,0:06:37.240
with respect to the ijk frame.

0:06:37.240,0:06:40.420
So since we talked[br]about this already,

0:06:40.420,0:06:43.020
some simple examples[br]have been given.

0:06:43.020,0:06:45.160
One of them was[br]a circling plane,

0:06:45.160,0:06:48.070
another circling plane[br]of a different speed,

0:06:48.070,0:06:49.500
a segment of a line.

0:06:49.500,0:06:50.970
This is the segment of a line.

0:06:50.970,0:06:52.260
What else have we discussed?

0:06:52.260,0:06:54.360
We discuss about[br]something wilder,

0:06:54.360,0:06:57.690
which was the helix[br]at different speeds?

0:06:57.690,0:07:01.190
All right, so very good[br]question for him was-- so

0:07:01.190,0:07:02.830
is this x of tt?

0:07:02.830,0:07:03.380
Yes.

0:07:03.380,0:07:05.420
Is this y of tt plus 1?

0:07:05.420,0:07:05.920
Yes.

0:07:05.920,0:07:08.760
Is this z of t 0 in my case?

0:07:08.760,0:07:09.722
Precisely

0:07:09.722,0:07:13.166
STUDENT: So if you[br]gave value to z,

0:07:13.166,0:07:16.610
what would you chose to[br]make t parameterized?

0:07:16.610,0:07:20.314
PROFESSOR: OK, t in[br]general, if you are moving,

0:07:20.314,0:07:22.480
you have an infinite motion[br]that comes from nowhere,

0:07:22.480,0:07:24.220
goes nowhere, right?

0:07:24.220,0:07:28.770
OK, then you can say[br]t is between minus

0:07:28.770,0:07:29.920
infinity plus infinity.

0:07:29.920,0:07:31.050
And that's your i--

0:07:31.050,0:07:32.300
STUDENT: But what I'm saying--

0:07:32.300,0:07:36.510
PROFESSOR: But-- but in[br]your case-- in your case,

0:07:36.510,0:07:40.370
you think oh, I know[br]where I'm starting.

0:07:40.370,0:07:44.230
So to that equals[br]to 1, t must be 1.

0:07:44.230,0:07:47.060
So I start my[br]movement at 1 second

0:07:47.060,0:07:52.690
and I end my movement at 2[br]seconds where x will be 2,

0:07:52.690,0:07:54.580
and y will be 3.

0:07:54.580,0:07:57.431
STUDENT: Well, I mean--[br]so you said x equals t.

0:07:57.431,0:07:59.816
You took that from[br]the y equals x plus 1.

0:07:59.816,0:08:02.439
If you had the third[br]variable t, what would you--

0:08:02.439,0:08:03.980
PROFESSOR: It's not[br]a third variable.

0:08:03.980,0:08:05.860
It's the time parameter.

0:08:05.860,0:08:08.770
So I work in three[br]variables-- xyz in space.

0:08:08.770,0:08:10.810
Those are my space coordinates.

0:08:10.810,0:08:14.090
The space coordinates[br]are function of time.

0:08:14.090,0:08:17.130
So it's all about physics.

0:08:17.130,0:08:19.690
So mathematics sometimes[br]becomes physics.

0:08:19.690,0:08:22.945
Thank God we are sisters,[br]even step-sisters.

0:08:22.945,0:08:24.760
X is a function of t.

0:08:24.760,0:08:26.102
Y is a function of t.

0:08:26.102,0:08:28.430
Z is a function of t.

0:08:28.430,0:08:29.030
Right?

0:08:29.030,0:08:30.740
Am I answering your[br]question or maybe

0:08:30.740,0:08:33.010
I didn't quite understand the--

0:08:33.010,0:08:35.474
STUDENT: Well, I understand[br]how to parameterize

0:08:35.474,0:08:36.826
the idea of a plane.

0:08:36.826,0:08:39.179
How do you do it[br]in space though?

0:08:39.179,0:08:42.220
PROFESSOR: In space-- in[br]space, you're already here.

0:08:42.220,0:08:46.370
So if you want to ride this[br]not in plane but in space,

0:08:46.370,0:08:51.380
your parametric equation is[br]ti plus t plus 1j plus 0k,

0:08:51.380,0:08:54.430
for this example,[br]anywhere in r3.

0:08:54.430,0:08:56.570
We live in r3.

0:08:56.570,0:08:58.380
All righty?

0:08:58.380,0:09:00.850
We live in r3.

0:09:00.850,0:09:03.430
OK, let me give[br]you more examples.

0:09:03.430,0:09:05.920
Because I think I'm[br]running out of time.

0:09:05.920,0:09:09.170
But I still have to[br]cover the material,

0:09:09.170,0:09:11.120
eventually get somewhere.

0:09:11.120,0:09:15.740
However, I want you to see[br]more examples that will help

0:09:15.740,0:09:18.610
you grasp this notion better.

0:09:18.610,0:09:25.190
So guys, imagine that[br]we have space r3-- that

0:09:25.190,0:09:28.634
could be rn-- in[br]which I have an origin

0:09:28.634,0:09:31.586
and I have a [INAUDIBLE].

0:09:31.586,0:09:35.030
And somebody gives[br]me a position vector

0:09:35.030,0:09:38.500
for a motion that's[br]a regular curve.

0:09:38.500,0:09:44.760
And that's x of tri plus[br]y is tj plus z of tk.

0:09:44.760,0:09:49.180
And since his question[br]is a very valid one,

0:09:49.180,0:09:52.760
let's see what happens[br]in a later case.

0:09:52.760,0:09:56.410
So I'm going to deviate a[br]little from my lesson plan.

0:09:56.410,0:09:59.690
And I say let us be[br]flexible and compare

0:09:59.690,0:10:02.029
that with the inner curve.

0:10:02.029,0:10:04.025
Because in the[br]process of comparison,

0:10:04.025,0:10:06.520
you learn a lot more.

0:10:06.520,0:10:11.020
If I were to be right above[br]my [INAUDIBLE] like that.

0:10:11.020,0:10:17.771
So this is the spacial curve in[br]our three imaginary trajectory

0:10:17.771,0:10:20.206
run of a bug or a particle.

0:10:20.206,0:10:24.030
As we said, this is the[br]planar curve-- planar,

0:10:24.030,0:10:28.706
parametrized curve in r2.

0:10:28.706,0:10:29.550
What's different?

0:10:29.550,0:10:31.390
What do we know about them?

0:10:31.390,0:10:35.470
We clearly know section 10.2.

0:10:35.470,0:10:38.900
What I hate in general[br]about processors

0:10:38.900,0:10:43.230
is if they are way[br]too structured.

0:10:43.230,0:10:47.220
Mathematics cannot be talking[br]sections where you say, oh,

0:10:47.220,0:10:51.786
section 10.2 is only about[br]velocity and acceleration.

0:10:51.786,0:10:55.130
But section 10.4 is[br]about tangent unit vector

0:10:55.130,0:10:56.740
and principle normal.

0:10:56.740,0:10:58.840
Well, they are related.

0:10:58.840,0:11:03.740
So it's only natural when[br]we talk about section 10.2

0:11:03.740,0:11:11.470
acceleration and velocity[br]that from acceleration, you

0:11:11.470,0:11:22.290
have a induced line to tangent[br]unit vector-- tangent unit

0:11:22.290,0:11:23.400
vector.

0:11:23.400,0:11:28.450
And later on, you're going[br]to compare acceleration

0:11:28.450,0:11:30.610
with a normal principal vector.

0:11:30.610,0:11:32.490
Sometimes, they[br]are the same thing.

0:11:32.490,0:11:35.020
Sometimes, they are[br]not the same thing.

0:11:35.020,0:11:38.560
It's a good idea to see[br]when they are the same thing

0:11:38.560,0:11:40.600
and when they are not.

0:11:40.600,0:11:44.890
So in section 10.4, we[br]will focus practically

0:11:44.890,0:11:48.430
or t, n, and v, the Frenet[br]frame and its consequences

0:11:48.430,0:11:51.820
on curvature, we already[br]talked about that a little bit.

0:11:51.820,0:11:56.300
In 10.2, practically,[br]we didn't cover much.

0:11:56.300,0:11:59.380
I only told you about[br]velocity, acceleration.

0:11:59.380,0:12:03.150
However, I would like[br]to review that for you.

0:12:03.150,0:12:05.990
Because I don't want[br]to risk losing you.

0:12:05.990,0:12:07.900
I'm going to lose[br]some of you anyway.

0:12:07.900,0:12:10.120
Two people said this[br]course is too hard for me.

0:12:10.120,0:12:11.950
I'm going to drop.

0:12:11.950,0:12:14.270
You are free to drop and I[br]think it's better for you

0:12:14.270,0:12:16.360
to drop than struggle.

0:12:16.360,0:12:21.442
But as long as you can still[br]learn and you can follow,

0:12:21.442,0:12:22.660
you shouldn't drop.

0:12:22.660,0:12:26.870
So try to see exactly[br]how much you can handle.

0:12:26.870,0:12:30.400
If you can handle just the[br]regular section of calc three,

0:12:30.400,0:12:32.150
go to that regular section.

0:12:32.150,0:12:36.430
If you can handle more, if[br]you are good at mathematics,

0:12:36.430,0:12:39.089
if you have always[br]been considered bright

0:12:39.089,0:12:41.780
in mathematics in high[br]school, let us stay here.

0:12:41.780,0:12:43.030
Otherwise, go.

0:12:43.030,0:12:44.280
Don't stay.

0:12:44.280,0:12:48.790
All right, so the[br]velocities are prime of t.

0:12:48.790,0:12:51.680
The acceleration is[br]our double prime of t.

0:12:51.680,0:12:53.610
We have done that last time.

0:12:53.610,0:12:55.330
We were very happy.

0:12:55.330,0:12:58.430
What would happen in a[br]planar curve seen on 2?

0:12:58.430,0:13:02.460
The same thing, of course,[br]except the last component

0:13:02.460,0:13:03.800
is not there.

0:13:03.800,0:13:06.820
It's part of ti[br]plus y prime of tj.

0:13:06.820,0:13:10.780
And there is a 0k in both cases.

0:13:10.780,0:13:12.870
So all these are factors.

0:13:12.870,0:13:15.295
At times, I'm not going[br]to point that out anymore.

0:13:15.295,0:13:18.000


0:13:18.000,0:13:20.320
The derivation goes[br]component-wise.

0:13:20.320,0:13:24.890
So if you forgot how to derive[br]or you want to drink and derive

0:13:24.890,0:13:28.370
or something, then you[br]don't belong in this class.

0:13:28.370,0:13:32.250
So again, make sure you know[br]the derivations and integrations

0:13:32.250,0:13:33.779
really well.

0:13:33.779,0:13:35.445
I'm going to work[br]some examples out just

0:13:35.445,0:13:36.720
to refresh your memory.

0:13:36.720,0:13:40.450
But if you have struggled with[br]differentiation and integration

0:13:40.450,0:13:45.170
in Calc 1, then you do not[br]do belong in this class.

0:13:45.170,0:13:53.136
All right, let's[br]see about speed.

0:13:53.136,0:13:54.620
It's about speed.

0:13:54.620,0:13:56.030
It's about time.

0:13:56.030,0:14:00.090
It's about time to remember[br]what the speed was.

0:14:00.090,0:14:04.320
The speed was the absolute[br]value or the magnitude.

0:14:04.320,0:14:07.015
It's not an absolute[br]value, but it's a magnitude

0:14:07.015,0:14:08.590
of the velocity factor.

0:14:08.590,0:14:11.100
This is the speed.

0:14:11.100,0:14:13.552
And the same in this case.

0:14:13.552,0:14:17.600
If I want to write an explicit[br]formula because somebody

0:14:17.600,0:14:21.130
asked me by email, can I write[br]an explicit formula, of course.

0:14:21.130,0:14:24.246
That's a piece of cake and you[br]should know that from before.

0:14:24.246,0:14:29.780
X prime of t squared plus[br]y prime of t squared plus z

0:14:29.780,0:14:34.110
prime of t squared[br]under the square root.

0:14:34.110,0:14:37.265
I was not going to insist[br]on the planar curve.

0:14:37.265,0:14:41.240
Of course the planar curve will[br]have a speed that all of you

0:14:41.240,0:14:42.420
know about.

0:14:42.420,0:14:44.950
And that's going to be[br]square root of x prime of t

0:14:44.950,0:14:49.070
squared plus y root[br]prime of t squared.

0:14:49.070,0:14:53.340
You should do your own thinking[br]to see what the particular case

0:14:53.340,0:14:55.710
will become.

0:14:55.710,0:14:58.430
However, I want to[br]see if you understood

0:14:58.430,0:15:01.780
what derives from[br]that in the sense

0:15:01.780,0:15:06.494
that you should know the[br]length of a arc of a curve.

0:15:06.494,0:15:09.890
What is the length[br]of an arc of a curve?

0:15:09.890,0:15:15.240
Well, we have to look back[br]at Calculus 2 a little bit

0:15:15.240,0:15:20.900
and remember that the length of[br]an arc of a curve in Calculus 2

0:15:20.900,0:15:24.390
was given by, what?

0:15:24.390,0:15:30.050
So you say, well, yeah.

0:15:30.050,0:15:31.410
That was a long time ago.

0:15:31.410,0:15:33.400
Well, some of you[br]already don't even

0:15:33.400,0:15:39.650
remember that as being integral[br]from a to b of square root of 1

0:15:39.650,0:15:43.480
plus 1 prime of x squared dx.

0:15:43.480,0:15:46.630
And you were freaking[br]out thinking, oh my god,

0:15:46.630,0:15:51.556
I don't see how this[br]formula from Calc 2,

0:15:51.556,0:15:55.170
the arc of a curve, had[br]you travel between time

0:15:55.170,0:16:01.740
equals a and time equals b[br]will relate to this formula.

0:16:01.740,0:16:03.440
So what happened in Calc 2?

0:16:03.440,0:16:07.060
In Calc 2, hopefully,[br]you have a good teacher.

0:16:07.060,0:16:09.660
And hopefully,[br]you've learned a lot.

0:16:09.660,0:16:12.540
This is between a and b, right?

0:16:12.540,0:16:14.210
What did they teach[br]you in Calc 2?

0:16:14.210,0:16:16.600
They taught you that[br]you have to take

0:16:16.600,0:16:18.600
integral from a to b[br]of square root of 1

0:16:18.600,0:16:21.020
plus y prime of x squared ds.

0:16:21.020,0:16:21.810
Why?

0:16:21.810,0:16:23.730
You never asked[br]your teacher why.

0:16:23.730,0:16:24.230
That's bad.

0:16:24.230,0:16:25.922
You should do that.

0:16:25.922,0:16:29.060
You should ask why every time.

0:16:29.060,0:16:32.790
They make you swallow a[br]formula via memorization

0:16:32.790,0:16:35.472
without understanding[br]this is the speed.

0:16:35.472,0:16:37.710
And now I'm coming[br]with the good news.

0:16:37.710,0:16:39.990
I have a proof of that.

0:16:39.990,0:16:42.320
I know what speed[br]means when I'm moving

0:16:42.320,0:16:46.810
along the arc of[br]a curve in plane.

0:16:46.810,0:16:51.145
OK, so what is the distance[br]travelled between time equals A

0:16:51.145,0:16:52.500
and time equals B?

0:16:52.500,0:16:57.070
It's going to be integral form[br]a to be of the speed, right?

0:16:57.070,0:16:59.450
This is the same one I'm[br]driving from-- level two--

0:16:59.450,0:17:01.711
Amarillo or anywhere else.

0:17:01.711,0:17:02.210
There.

0:17:02.210,0:17:05.069
Now, what they showed[br]you and they fooled you

0:17:05.069,0:17:10.618
into memorizing that is just[br]a consequence of this formula

0:17:10.618,0:17:12.530
because of what he said.

0:17:12.530,0:17:13.480
Why?

0:17:13.480,0:17:16.785
The most usual[br]parameterization is

0:17:16.785,0:17:22.680
going to be y of t equals t--[br]I'm sorry, x of t equals vxst

0:17:22.680,0:17:25.910
and y of t equals y of t.

0:17:25.910,0:17:27.940
So, again x is time.

0:17:27.940,0:17:33.130
In many linear curves, you can[br]take x to be time, thank God.

0:17:33.130,0:17:38.560
And then your parametrization[br]will be t comma y of t.

0:17:38.560,0:17:40.640
Because x is t.

0:17:40.640,0:17:43.150
And x prime of t will be 1.

0:17:43.150,0:17:45.930
Y prime of t will[br]be y prime of t.

0:17:45.930,0:17:50.040
When you take them, squish[br]them, square them, sum them up,

0:17:50.040,0:17:51.990
you get exactly this one.

0:17:51.990,0:17:54.402
But you notice[br]this is the speed.

0:17:54.402,0:17:56.070
What is this the speed?

0:17:56.070,0:18:03.288
Of some value over prime[br]of t, which is speed.

0:18:03.288,0:18:07.250
You see that what they forced[br]you to memorize in Calc 2

0:18:07.250,0:18:10.920
is nothing but the speed.

0:18:10.920,0:18:12.920
And I could change[br]the parameterization

0:18:12.920,0:18:14.980
to something more general.

0:18:14.980,0:18:19.560
Now, can I do this[br]parameterization for a circle?

0:18:19.560,0:18:20.230
No.

0:18:20.230,0:18:22.460
Why not?

0:18:22.460,0:18:25.000
I could, but then[br]I'd have to split

0:18:25.000,0:18:26.760
into the upper[br]part and lower part

0:18:26.760,0:18:29.040
because the circle[br]is not a graph.

0:18:29.040,0:18:31.210
So I take t between[br]this and that

0:18:31.210,0:18:35.920
and then I have square root[br]of 1 minus t squared on top.

0:18:35.920,0:18:38.800
And underneath, I have[br]minus square root of 1

0:18:38.800,0:18:39.570
minus t squared.

0:18:39.570,0:18:43.980
So I split the poor circle[br]into a graph and another graph.

0:18:43.980,0:18:45.230
And I do it separately.

0:18:45.230,0:18:47.310
And I can still apply that.

0:18:47.310,0:18:49.430
But only a fool[br]would do that, right?

0:18:49.430,0:18:52.900
So what does a smart[br]mathematician do?

0:18:52.900,0:18:54.670
A smart mathematician[br]will say, OK,

0:18:54.670,0:18:59.740
for the circle, x is[br]cosine t, y is sine t.

0:18:59.740,0:19:01.830
And that is the[br]parameterization I'm

0:19:01.830,0:19:04.170
going to use for this formula.

0:19:04.170,0:19:05.900
And I get speed 1.

0:19:05.900,0:19:08.620
And I'm going to[br]be happy, right?

0:19:08.620,0:19:10.970
So it's a lot easier[br]to understand what

0:19:10.970,0:19:13.280
a general parameterization is.

0:19:13.280,0:19:19.490
What is the length of an arc[br]of a curve for a curving space?

0:19:19.490,0:19:20.910
There's the bug.

0:19:20.910,0:19:22.110
Time equals t0.

0:19:22.110,0:19:23.970
He's buzzing.

0:19:23.970,0:19:26.180
And after 10 seconds,[br]he will be at the end.

0:19:26.180,0:19:30.620
So it goes, [BUZZING] jump.

0:19:30.620,0:19:35.120
OK, how much did he travel?

0:19:35.120,0:19:41.700
Integral from a to b of square[br]root of x prime of t squared

0:19:41.700,0:19:44.430
plus y prime of t[br]squared plus z prime of t

0:19:44.430,0:19:50.430
squared-- no matter what that[br]position vector x of ty of t0

0:19:50.430,0:19:51.360
give us.

0:19:51.360,0:19:56.200
So you take the coordinates[br]of the velocity vector.

0:19:56.200,0:19:57.150
You look at them.

0:19:57.150,0:19:57.900
You square them.

0:19:57.900,0:19:59.280
You add them together.

0:19:59.280,0:20:00.740
You put them under[br]the square root.

0:20:00.740,0:20:02.510
That's going to be the speed.

0:20:02.510,0:20:06.370
And displacement is[br]integral of speed.

0:20:06.370,0:20:09.100
When you guys learned[br]in school, your teacher

0:20:09.100,0:20:10.935
oversimplified the things.

0:20:10.935,0:20:12.950
What did your teacher[br]say in physics?

0:20:12.950,0:20:15.700
Space equals speed times time.

0:20:15.700,0:20:16.680
Say it again.

0:20:16.680,0:20:19.935
He said space traveled[br]is speed times time.

0:20:19.935,0:20:23.635
But he assumed the speed[br]is constant or constant

0:20:23.635,0:20:26.570
on portions-- like,[br]speedswise constant.

0:20:26.570,0:20:28.940
Well, if it's a[br]constant, the speed

0:20:28.940,0:20:30.790
will get the heck out of here.

0:20:30.790,0:20:35.300
And then the space will[br]be speed times b minus a.

0:20:35.300,0:20:37.840
But b minus a is delta t.

0:20:37.840,0:20:41.200
In mathematics, in physics,[br]we say b minus a is delta t.

0:20:41.200,0:20:44.720
That's the interval of time that[br]the bug travels or the particle

0:20:44.720,0:20:45.730
travels.

0:20:45.730,0:20:47.870
So he or she was right.

0:20:47.870,0:20:51.060
Space is speed times[br]time, but it's not like

0:20:51.060,0:20:53.670
that unless the[br]speed is constant.

0:20:53.670,0:20:55.830
So he oversimplified[br]your knowledge

0:20:55.830,0:20:57.520
of mathematics and physics.

0:20:57.520,0:20:59.040
Now you see the truth.

0:20:59.040,0:21:04.500
Space is integral of speed.

0:21:04.500,0:21:06.340
OK, now we understand.

0:21:06.340,0:21:09.830
And I promised you last[br]time that after reviewing,

0:21:09.830,0:21:13.730
I didn't even say I would review[br]anything from 10.2 and 10.4.

0:21:13.730,0:21:14.850
I promised you more.

0:21:14.850,0:21:17.580
I promised you that I'm going[br]to compute something that's

0:21:17.580,0:21:23.960
out of 10.4 which is called[br]a curvature of a helix

0:21:23.960,0:21:25.230
in particular.

0:21:25.230,0:21:29.680
Because we looked at curvature[br]of a parametric curve

0:21:29.680,0:21:31.190
in general.

0:21:31.190,0:21:36.700
I want to organize the material[br]of review from 10.2 and 10.4

0:21:36.700,0:21:40.260
in a big problem just like[br]you will have in the exams,

0:21:40.260,0:21:42.347
in the midterm,[br]and in the final.

0:21:42.347,0:21:43.430
I don't want to scare you.

0:21:43.430,0:21:45.920
I just want to[br]prepare you better

0:21:45.920,0:21:49.844
for the kind of multiple[br]questions we are going to have.

0:21:49.844,0:21:55.240
So let me give you a[br]funny looking curve.

0:21:55.240,0:21:59.430
I want you to think about[br]it and tell me what it is.

0:21:59.430,0:22:01.935
a and b are positive numbers.

0:22:01.935,0:22:07.140
a cosine ba sine t bt will be[br]some sort of funny trajectory.

0:22:07.140,0:22:09.930
You are already[br]familiar to that.

0:22:09.930,0:22:13.414
Last time, I gave you an example[br]where a was 4-- oh my god,

0:22:13.414,0:22:14.550
I don't even remember.

0:22:14.550,0:22:16.460
You'll need to help me.

0:22:16.460,0:22:18.684
[INAUDIBLE]

0:22:18.684,0:22:21.120
STUDENT: 4, 4, 3.

0:22:21.120,0:22:24.760
PROFESSOR: I took those because[br]they are Pythagorean numbers.

0:22:24.760,0:22:26.360
So what does it mean?

0:22:26.360,0:22:28.970
3 squared plus 4 squared[br]equals 5 squared.

0:22:28.970,0:22:31.920
I wanted the sum of them[br]to be a perfect square.

0:22:31.920,0:22:33.230
So I was playing games.

0:22:33.230,0:22:36.590
You can do that for any a and b.

0:22:36.590,0:22:37.710
Now, what do I want?

0:22:37.710,0:22:43.620
A-- like in 10.2 where[br]you write r prime of t,

0:22:43.620,0:22:46.540
rewrite that double prime of t.

0:22:46.540,0:22:49.730
So it's a complex problem.

0:22:49.730,0:22:53.210
In b, I want you to[br]find t and r prime

0:22:53.210,0:22:55.750
of t over-- who[br]remembers the formula?

0:22:55.750,0:22:57.700
I shouldn't have[br]spoon-fed you that.

0:22:57.700,0:22:58.620
STUDENT: Absolute--

0:22:58.620,0:23:00.755
PROFESSOR: Absolute[br]magnitude, actually.

0:23:00.755,0:23:03.520
It's more correct to[br]say magnitude, right?

0:23:03.520,0:23:04.300
Very good.

0:23:04.300,0:23:08.636
And what else did I[br]spoon-feed you last name?

0:23:08.636,0:23:10.280
I spoon-fed you n.

0:23:10.280,0:23:13.970
Let's compute n as well[br]as part of the problem

0:23:13.970,0:23:21.200
t prime t over t[br]prime of t magnitude.

0:23:21.200,0:23:24.140
STUDENT: So you're looking[br]for the tangent unit vector.

0:23:24.140,0:23:25.497
PROFESSOR: Tangent unit vector?

0:23:25.497,0:23:27.080
STUDENT: And then[br]you're looking for--

0:23:27.080,0:23:27.913
PROFESSOR: Yes, sir.

0:23:27.913,0:23:30.830
And-- OK, don't you[br]like me to also give you

0:23:30.830,0:23:34.050
something like a grading[br]grid, how much everything

0:23:34.050,0:23:35.320
would be worth.

0:23:35.320,0:23:36.570
Imagine you're taking an exam.

0:23:36.570,0:23:39.710
Why not put yourself[br]in an exam mode

0:23:39.710,0:23:44.130
so you don't freak out[br]during the actual exam?

0:23:44.130,0:23:47.680
C will be another[br]question, something smart.

0:23:47.680,0:24:02.480
Let's see-- reparameterize an[br]arc length to a plane, a curve,

0:24:02.480,0:24:05.380
rho of s.

0:24:05.380,0:24:08.510
Why not r of s like some[br]people call-- use it

0:24:08.510,0:24:10.160
and some books use it?

0:24:10.160,0:24:11.759
Because if you're[br]reparameterizing s,

0:24:11.759,0:24:13.467
it's going to be the[br]same physical limits

0:24:13.467,0:24:15.870
but a different function.

0:24:15.870,0:24:19.640
So if you remember the[br]diagram I wrote before,

0:24:19.640,0:24:24.480
little r is a function that[br]comes from integral i time

0:24:24.480,0:24:29.450
integral 2r3 and rho would[br]be coming from a j to r3.

0:24:29.450,0:24:32.610
And what is the[br]relationship between them?

0:24:32.610,0:24:36.360
This is t goes to s and[br]this is s goes to t.

0:24:36.360,0:24:39.450
What is d I'm asking you?

0:24:39.450,0:24:41.380
Well, if you're d[br]and c, of course

0:24:41.380,0:24:44.530
you know what the arc[br]length parameter will be.

0:24:44.530,0:24:49.630
It's going to be integral[br]from 0 to t or any t0 here

0:24:49.630,0:24:54.962
of the speed-- of the speed[br]of the original function here

0:24:54.962,0:24:56.310
of t.

0:24:56.310,0:25:01.820
The tau-- maybe tau is better[br]than the dummy variable t.

0:25:01.820,0:25:05.242
And e I want.

0:25:05.242,0:25:06.750
You say, how much[br]more do you want?

0:25:06.750,0:25:07.650
I want a lot.

0:25:07.650,0:25:09.132
I'm a greedy person.

0:25:09.132,0:25:13.840
I want the curvature[br]of the curve.

0:25:13.840,0:25:17.550
And you have to remind me.

0:25:17.550,0:25:19.990
Some of you are very good[br]students, better than me.

0:25:19.990,0:25:23.622
I mean, I'm still behind[br]with a research course

0:25:23.622,0:25:25.080
that I have--[br]research paper i have

0:25:25.080,0:25:29.786
to read in two days in biology.

0:25:29.786,0:25:35.500
But this curvature of the[br]curve had a very simple formula

0:25:35.500,0:25:36.980
that we all love.

0:25:36.980,0:25:40.120
For mathematicians, it's a[br]piece of cake to remember it.

0:25:40.120,0:25:43.310
K-- that's what I like[br]about being a mathematician.

0:25:43.310,0:25:45.350
I don't need a good memory.

0:25:45.350,0:25:47.920
Now I remember why I didn't[br]go to medical school--

0:25:47.920,0:25:51.010
because my father[br]told me, well, you

0:25:51.010,0:25:53.810
should be able to remember all[br]the bones in a person's body.

0:25:53.810,0:25:55.890
And I said, dad, do you[br]know all these names?

0:25:55.890,0:25:56.210
Yes, of course.

0:25:56.210,0:25:57.293
And he started telling me.

0:25:57.293,0:26:01.410
Well, I realized that I[br]would never remember those.

0:26:01.410,0:26:07.030
But I remember this[br]formula which is r rho.

0:26:07.030,0:26:10.330
In this case, if[br]our r is Greek rho,

0:26:10.330,0:26:13.090
it's got to be rho[br]double prime of what?

0:26:13.090,0:26:15.970
of S. Is this[br]correct, what I wrote?

0:26:15.970,0:26:16.470
No.

0:26:16.470,0:26:18.050
What's missing?

0:26:18.050,0:26:22.520
The acceleration and arc length[br]but in magnitude because that's

0:26:22.520,0:26:23.905
a vector, of course.

0:26:23.905,0:26:26.870
This is the scalar function.

0:26:26.870,0:26:28.660
Anything else you[br]want, Magdalena?

0:26:28.660,0:26:30.190
Oh, that's enough.

0:26:30.190,0:26:34.060
All right, so I want[br]to know everything

0:26:34.060,0:26:38.260
that's possible to know about[br]this curve from 10.2 and 10.4

0:26:38.260,0:26:39.890
sections.

0:26:39.890,0:26:41.840
10.3-- skip 10.5.

0:26:41.840,0:26:44.330
Skip-- you're happy about it.

0:26:44.330,0:26:44.940
Yes sir.

0:26:44.940,0:26:48.426
STUDENT: For the[br]parameter on v, is it a t?

0:26:48.426,0:26:49.920
And what's the integral?

0:26:49.920,0:26:51.010
What's on the bottom.

0:26:51.010,0:26:54.230
PROFESSOR: Ah, that value[br]erased when I wrote that one.

0:26:54.230,0:26:56.200
It was there-- t0.

0:26:56.200,0:27:00.510
So I can start with any fixed[br]t0 as my initial moment in time.

0:27:00.510,0:27:02.560
I would like my[br]initial moment in time

0:27:02.560,0:27:05.980
to be 0 just to make[br]my things easier.

0:27:05.980,0:27:07.940
Are we ready to solve[br]this problem together?

0:27:07.940,0:27:11.570
I think we have just[br]about the exact time

0:27:11.570,0:27:14.070
we need to do everything.

0:27:14.070,0:27:17.610
First of all, you have to tell[br]me what kind of curve this is.

0:27:17.610,0:27:20.020
Of course you know because[br]you were here last time.

0:27:20.020,0:27:23.250
Don't skip classes because[br]you are missing everything out

0:27:23.250,0:27:25.380
and then you will have[br]to drop or withdraw.

0:27:25.380,0:27:27.230
So don't skip class.

0:27:27.230,0:27:31.160
What was that from last time?

0:27:31.160,0:27:33.510
It was a helix.

0:27:33.510,0:27:35.280
I'm going to try and redraw it.

0:27:35.280,0:27:38.010
I know I'm wasting[br]my time, but I would

0:27:38.010,0:27:43.750
try to draw a better curve.

0:27:43.750,0:27:46.325
Ah, what's the equation[br]of the cylinder?

0:27:46.325,0:27:49.938
[CLASS MURMURS]

0:27:49.938,0:27:51.327
PROFESSOR: Huh?

0:27:51.327,0:27:53.382
What's the equation[br]of the cylinder?

0:27:53.382,0:27:55.610
That's a quadratic[br]that you are all

0:27:55.610,0:28:01.252
familiar with on which on my[br]beautiful helix is sitting on.

0:28:01.252,0:28:02.980
I taught you the[br]trick last time.

0:28:02.980,0:28:04.350
Don't forget it.

0:28:04.350,0:28:10.100
STUDENT: a over 4 cosine of[br]t squared plus 8 over 4 sine

0:28:10.100,0:28:10.850
of t squared.

0:28:10.850,0:28:13.850


0:28:13.850,0:28:16.250
PROFESSOR: So we do[br]that-- very good.

0:28:16.250,0:28:19.040
X is going to be-- let[br]me right that down.

0:28:19.040,0:28:20.425
X is cosine.

0:28:20.425,0:28:22.940
Y is a sine t.

0:28:22.940,0:28:24.610
And that's exactly[br]what you asked me.

0:28:24.610,0:28:26.060
And z is bt.

0:28:26.060,0:28:29.740
And then what I need to do[br]is square these guys out

0:28:29.740,0:28:31.565
as you said very well.

0:28:31.565,0:28:33.196
I don't care about this 2z.

0:28:33.196,0:28:34.750
He's not in the picture here.

0:28:34.750,0:28:38.820
X squared plus y squared will be[br]a squared, which means I better

0:28:38.820,0:28:42.900
go ahead and draw a circle[br]of radius a on the bottom

0:28:42.900,0:28:44.920
and then build[br]my-- oh my god, it

0:28:44.920,0:28:49.820
looks horrible-- the cylinder[br]based on that circle.

0:28:49.820,0:28:51.050
Guys, it's now straight.

0:28:51.050,0:28:51.780
I'm sorry.

0:28:51.780,0:28:55.380
I mean, I can do[br]better than that.

0:28:55.380,0:28:58.710
OK, good.

0:28:58.710,0:29:02.760
So I'm starting at what point?

0:29:02.760,0:29:06.334
I'm starting at a0[br]0 time t equals 0.

0:29:06.334,0:29:07.500
We discussed that last time.

0:29:07.500,0:29:09.300
I'm not going to repeat.

0:29:09.300,0:29:12.300
I'm starting here,[br]and two of you

0:29:12.300,0:29:14.090
told me that if t[br]equals phi over two,

0:29:14.090,0:29:17.550
I'm going to be here[br]and so on and so forth.

0:29:17.550,0:29:21.842
If I ask you one more thing[br]for extra credit, what

0:29:21.842,0:29:30.970
is the length of the trajectory[br]traveled by the bug, whatever

0:29:30.970,0:29:38.380
that is, between time t equals[br]0 and time t equals phi over 2.

0:29:38.380,0:29:40.080
I'd say that's extra credit.

0:29:40.080,0:29:52.400
So, oh my god, 20%, 20%, 20%,[br]20%, 20%, and 10% for this one.

0:29:52.400,0:29:56.570
And if you think why does she[br]care about the percentages

0:29:56.570,0:29:59.030
and points, you will[br]care and I care.

0:29:59.030,0:30:02.700
Because I want you to see how[br]you are going to be assessed.

0:30:02.700,0:30:05.460
If you have no idea how[br]you're going to assessed,

0:30:05.460,0:30:08.750
then you're going to be[br]happy and i will be unhappy.

0:30:08.750,0:30:12.030
All right, so for 20%[br]credit on this problem,

0:30:12.030,0:30:15.540
we want to see r prime of t[br]will be, r double prime of t

0:30:15.540,0:30:16.040
will be.

0:30:16.040,0:30:18.095
That's going to be[br]a piece of cake.

0:30:18.095,0:30:21.420
And of course, it's maybe the[br]reward is too big for that,

0:30:21.420,0:30:23.200
but that's life.

0:30:23.200,0:30:31.670
Minus a sine t a equals time[br]t and d, d as in infinity.

0:30:31.670,0:30:34.320
So I have an infinite[br]cylinder on which

0:30:34.320,0:30:37.230
I draw an infinite[br]helix coming from hell

0:30:37.230,0:30:39.370
and going to paradise.

0:30:39.370,0:30:44.220
So between minus infinity and[br]plus infinity, there's a guy.

0:30:44.220,0:30:47.790
I'm going to draw a[br]beautiful infinite helix.

0:30:47.790,0:30:50.460
And this is what I posted here.

0:30:50.460,0:30:53.260
What's the acceleration[br]of this helix?

0:30:53.260,0:30:59.600
Minus a cosine t[br]minus 5 sine t and 0.

0:30:59.600,0:31:03.280
Question, quick[br]question for you.

0:31:03.280,0:31:06.840
Will-- you guys are fast.

0:31:06.840,0:31:10.640
Maybe I shouldn't[br]go ahead of myself.

0:31:10.640,0:31:14.530
Nobody's asking me what[br]the speed is right now.

0:31:14.530,0:31:17.760
So why would I do something[br]that's not on the final, right?

0:31:17.760,0:31:19.980
So let's see.

0:31:19.980,0:31:23.130
T, you will have to compute[br]the speed when you get to here.

0:31:23.130,0:31:26.150
But wait until we get there.

0:31:26.150,0:31:27.420
What is mister t?

0:31:27.420,0:31:29.560
Mister t will be[br]the tangent vector.

0:31:29.560,0:31:34.860
So the velocity is going like[br]a crazy guy, long vector.

0:31:34.860,0:31:39.160
The normal unit vector says,[br]I'm the tangent unit vector.

0:31:39.160,0:31:42.750
I'm always perpendicular[br]to the direction.

0:31:42.750,0:31:43.590
I'm of length 1.

0:31:43.590,0:31:47.491
STUDENT: I thought the tangent[br]was parallel to the direction.

0:31:47.491,0:31:49.490
PROFESSOR: Yes, the[br]direction of motion is this.

0:31:49.490,0:31:51.040
Look at me.

0:31:51.040,0:31:53.220
This is my direction of motion.

0:31:53.220,0:31:54.077
And the tangent is--

0:31:54.077,0:31:55.160
STUDENT: You said it was--

0:31:55.160,0:31:57.030
PROFESSOR: --in the[br]direction of motion.

0:31:57.030,0:31:57.690
STUDENT: But you said[br]it was perpendicular.

0:31:57.690,0:31:59.130
PROFESSOR: I said perpendicular?

0:31:59.130,0:32:03.030
Because I was thinking[br]ahead of myself and n.

0:32:03.030,0:32:04.250
And I apologize.

0:32:04.250,0:32:06.255
So thank you for correcting me.

0:32:06.255,0:32:08.342
So t is the tangent unit vector.

0:32:08.342,0:32:12.770


0:32:12.770,0:32:15.230
I'm going along the[br]direction of motion.

0:32:15.230,0:32:17.690
And it's going to be[br]perpendicular to t.

0:32:17.690,0:32:22.263
And that's the principal[br]normal unit vector--

0:32:22.263,0:32:24.227
principal normal unit vector.

0:32:24.227,0:32:26.630
And you're going to tell[br]me what I'm having here.

0:32:26.630,0:32:27.560
Because I don't know.

0:32:27.560,0:32:30.970


0:32:30.970,0:32:37.410
T is minus a sine[br]t a equals sine t

0:32:37.410,0:32:41.120
and v divided by the speed.

0:32:41.120,0:32:43.630
That's why I was[br]getting ahead of myself

0:32:43.630,0:32:46.920
thinking about the speed that[br]you'll need later on anyway.

0:32:46.920,0:32:50.240
But you already[br]need it here, right?

0:32:50.240,0:32:55.180
Because the denominator of this[br]expression will be the speed.

0:32:55.180,0:32:58.290
Magnitude of r[br]prime-- what is that?

0:32:58.290,0:33:01.470
Piece of cake--[br]square root of the sum

0:33:01.470,0:33:05.747
of the squares of square root[br]of a squared plus b squared.

0:33:05.747,0:33:06.330
Piece of cake.

0:33:06.330,0:33:07.290
I love it.

0:33:07.290,0:33:09.004
So what do I notice?

0:33:09.004,0:33:11.830
That although I'm going[br]on a funny curve which

0:33:11.830,0:33:15.650
is a parametrized helix,[br]I expect some-- maybe

0:33:15.650,0:33:18.210
I expected something[br]wild in terms of speed.

0:33:18.210,0:33:19.500
Well, the speed is constant.

0:33:19.500,0:33:26.850
STUDENT: [INAUDIBLE] the square[br]root of negative a sine t

0:33:26.850,0:33:27.350
squared--

0:33:27.350,0:33:29.570
PROFESSOR: And what are those?

0:33:29.570,0:33:33.470
A squared sine squared plus c[br]squared cosine squared plus b

0:33:33.470,0:33:35.512
squared, right?

0:33:35.512,0:33:37.220
And what sine squared[br]plus cosine squared

0:33:37.220,0:33:38.450
is 1 [INAUDIBLE].

0:33:38.450,0:33:41.160
So you get a squared[br]plus b squared.

0:33:41.160,0:33:46.485
Good-- now let's[br]go on and do the n.

0:33:46.485,0:33:53.020
The n will be t prime[br]over magnitude of t prime.

0:33:53.020,0:33:56.350
When you do t prime,[br]you'll say, wait a minute.

0:33:56.350,0:33:59.752
I have square root of a squared[br]plus b squared on the bottom.

0:33:59.752,0:34:04.930
On the top, I have minus equals[br]sine t minus a sine t and 0.

0:34:04.930,0:34:06.310
We have time to finish?

0:34:06.310,0:34:07.036
I think.

0:34:07.036,0:34:08.590
I hope so.

0:34:08.590,0:34:18.110
Divided by-- divided by the[br]magnitude of this fellow.

0:34:18.110,0:34:20.560
I will say, oh, wait a minute.

0:34:20.560,0:34:24.371
The magnitude of this fellow[br]is simply the magnitude

0:34:24.371,0:34:26.364
of this over this magnitude.

0:34:26.364,0:34:29.860


0:34:29.860,0:34:34.449
And we've seen last time this is[br]the magnitude of this vector a,

0:34:34.449,0:34:35.190
right?

0:34:35.190,0:34:35.840
Good.

0:34:35.840,0:34:39.049
Now, so the answer will[br]be n is going to be a unit

0:34:39.049,0:34:41.920
vector, very nice friend[br]of yours, minus cosine t

0:34:41.920,0:34:44.146
minus sine t0.

0:34:44.146,0:34:49.520
Can you draw a conclusion about[br]how I should draw this vector?

0:34:49.520,0:34:51.609
You see the component in k is 0.

0:34:51.609,0:34:55.610
So this vector[br]cannot be like that--

0:34:55.610,0:34:57.721
cannot be inclined with[br]respect to the horizontal.

0:34:57.721,0:34:58.220
Yes sir.

0:34:58.220,0:35:00.487
STUDENT: So what happens[br]to-- down there-- square root

0:35:00.487,0:35:01.854
of a squared plus b squared?

0:35:01.854,0:35:02.895
PROFESSOR: They simplify.

0:35:02.895,0:35:04.414
This is division.

0:35:04.414,0:35:05.080
STUDENT: Oh, OK.

0:35:05.080,0:35:07.730
PROFESSOR: So this simplifies[br]with that and a simplifies

0:35:07.730,0:35:10.000
with a.

0:35:10.000,0:35:12.090
I should leave some[br]things as an exercise,

0:35:12.090,0:35:15.600
but this is an obvious one so I[br]don't have to explain anything.

0:35:15.600,0:35:18.950
Minus cosine t[br]minus sine t-- if do

0:35:18.950,0:35:22.290
you guys imagine what that is?

0:35:22.290,0:35:25.760
Remember your washer and dryer.

0:35:25.760,0:35:32.500
So if you have an acceleration[br]that's pointing inside

0:35:32.500,0:35:36.170
like from a centrifugal force,[br]the corresponding acceleration

0:35:36.170,0:35:39.480
would go pointing[br]inside, not outside.

0:35:39.480,0:35:43.780
That's going to be exactly[br]minus cosine t minus sine t0.

0:35:43.780,0:35:47.520
So the way I should draw the[br]n would not be just any n,

0:35:47.520,0:35:52.660
but should be at every[br]point a beautiful vector

0:35:52.660,0:35:55.770
that's horizontal and is[br]moving along the helix.

0:35:55.770,0:35:57.840
My elbow is moving[br]along the helix.

0:35:57.840,0:35:58.630
See my elbow?

0:35:58.630,0:35:59.965
Where's my elbow moving?

0:35:59.965,0:36:01.200
I'm trying.

0:36:01.200,0:36:03.070
I swear, I won't do it that way.

0:36:03.070,0:36:06.910
So this is the helix and this[br]is the acceleration, which

0:36:06.910,0:36:12.525
is acceleration and the normal[br]unit vector in this case

0:36:12.525,0:36:13.210
are co-linear.

0:36:13.210,0:36:14.980
They are not[br]co-linear in general.

0:36:14.980,0:36:18.970
But if the speed is a[br]constant, they are co-linear.

0:36:18.970,0:36:20.816
The n and the acceleration.

0:36:20.816,0:36:21.316
Yes, sir?

0:36:21.316,0:36:24.774
STUDENT: How do you know it's[br]pointing in the central axis?

0:36:24.774,0:36:25.645
I thought it was--

0:36:25.645,0:36:26.686
PROFESSOR: Good question.

0:36:26.686,0:36:27.956
Good question.

0:36:27.956,0:36:28.890
Well, yeah.

0:36:28.890,0:36:29.590
Let's see now.

0:36:29.590,0:36:31.160
Plug in t equals 0.

0:36:31.160,0:36:32.330
What do you have?

0:36:32.330,0:36:35.820
Minus cosine 0 minus 1 0, 0.

0:36:35.820,0:36:40.380
So you guys would have to[br]draw the vector minus 1, 0, 0.

0:36:40.380,0:36:42.310
That's minus i, right?

0:36:42.310,0:36:47.980
So when I start here, this[br]is my n-- from here to here,

0:36:47.980,0:36:50.910
from the particle to the insid.

0:36:50.910,0:36:52.720
So I go on that.

0:36:52.720,0:36:55.100
All right, so this is the[br]normal principal vector.

0:36:55.100,0:36:57.066
I'm very happy about it.

0:36:57.066,0:36:59.820
STUDENT: Isn't the normal[br]principal vector is the-- is it

0:36:59.820,0:37:01.440
the derivative of[br]t, or is just--

0:37:01.440,0:37:02.815
PROFESSOR: It was[br]by definition--

0:37:02.815,0:37:05.490
it's in your notes-- t prime[br]over the magnitude of the--

0:37:05.490,0:37:09.100
STUDENT: So then did[br]you-- why didn't you

0:37:09.100,0:37:10.982
take a derivative of t prime?

0:37:10.982,0:37:11.690
PROFESSOR: I did.

0:37:11.690,0:37:12.606
STUDENT: Yeah, I know.

0:37:12.606,0:37:15.880
I see you took a[br]derivative of t of--

0:37:15.880,0:37:19.172
PROFESSOR: This is t prime.

0:37:19.172,0:37:20.380
STUDENT: OK.

0:37:20.380,0:37:23.570
PROFESSOR: And this is[br]magnitude of t prime.

0:37:23.570,0:37:26.360
Why don't you try[br]this at home, like,

0:37:26.360,0:37:30.020
slowly until you're sure[br]this is what yo got?

0:37:30.020,0:37:32.410
So I did-- I did[br]the derivative of i.

0:37:32.410,0:37:33.880
STUDENT: I saw that.

0:37:33.880,0:37:36.060
PROFESSOR: This[br]is a [INAUDIBLE].

0:37:36.060,0:37:37.950
STUDENT: You said you were--

0:37:37.950,0:37:40.340
PROFESSOR: So when we[br]have t times a function

0:37:40.340,0:37:43.460
and we prime the[br]product, k goes out.

0:37:43.460,0:37:45.380
Lucky for us--[br]imagine how life would

0:37:45.380,0:37:46.970
be if it weren't like that.

0:37:46.970,0:37:49.180
So the constant[br]that falls out is

0:37:49.180,0:37:51.890
1 over square root[br]of what I derived.

0:37:51.890,0:37:56.000
And then I have to derive[br]this whole function also.

0:37:56.000,0:37:59.050
So I would suggest to[br]everybody, not just to yo--

0:37:59.050,0:38:01.751
go home and see if[br]you can redo this

0:38:01.751,0:38:03.000
without looking in your notes.

0:38:03.000,0:38:05.100
Close the damn notes.

0:38:05.100,0:38:08.780
Open and then you look at--[br]it's line by line, line by line

0:38:08.780,0:38:10.620
all the derivations.

0:38:10.620,0:38:13.970
Because you guys will have to[br]do that yourselves in the exam,

0:38:13.970,0:38:17.468
either midterm or final anyway.

0:38:17.468,0:38:23.710
Reparameterizing arc lengths[br]to obtain a curve-- I

0:38:23.710,0:38:25.930
still have that to[br]finish the problem.

0:38:25.930,0:38:31.660
Reparameterizing arc length[br]to obtain a curve rho of s.

0:38:31.660,0:38:33.100
How do we do that?

0:38:33.100,0:38:34.070
Who is s?

0:38:34.070,0:38:37.300
First of all, you should[br]start with the s and then

0:38:37.300,0:38:38.660
reparameterize.

0:38:38.660,0:38:39.820
So you say, hey, teacher.

0:38:39.820,0:38:42.050
You try to fool me, right?

0:38:42.050,0:38:45.590
I want s to be grabbed[br]as a parameter first.

0:38:45.590,0:38:49.700
And then I will reparameterize[br]the way you want me to do that.

0:38:49.700,0:38:52.990
So s of t will be[br]integral from 0

0:38:52.990,0:38:56.390
to t square root of a[br]prime a squared times

0:38:56.390,0:38:59.825
b squared b tau-- d tau, yes.

0:38:59.825,0:39:01.205
S of t will be, what?

0:39:01.205,0:39:02.740
Who's helping me on that?

0:39:02.740,0:39:05.040
Because I want you to be awake.

0:39:05.040,0:39:05.985
Are you guys awake?

0:39:05.985,0:39:07.334
[CLASS MURMURS]

0:39:07.334,0:39:08.750
PROFESSOR: The[br]square root of that

0:39:08.750,0:39:14.190
is a constant gets out times t.

0:39:14.190,0:39:18.935
So what did I tell you when[br]it comes to these functions?

0:39:18.935,0:39:21.600
I have to turn my[br]head badly like that.

0:39:21.600,0:39:23.960
This was the alpha t or s of t.

0:39:23.960,0:39:31.685
And this was t of s, which[br]is the inverse function.

0:39:31.685,0:39:33.310
I'm not going to[br]write anything stupid.

0:39:33.310,0:39:36.750
But this is practically the[br]inverse function of s of t.

0:39:36.750,0:39:39.190
I told you it was easiest t do.

0:39:39.190,0:39:40.370
Put it here.

0:39:40.370,0:39:43.480
T has to be replaced[br]by, in terms of s,

0:39:43.480,0:39:45.930
by a certain expression.

0:39:45.930,0:39:47.924
So who is t?

0:39:47.924,0:39:51.860
And you will do that[br]in no time in the exam.

0:39:51.860,0:39:55.910
T pulled out from[br]there will be just

0:39:55.910,0:40:00.720
s over square root a[br]squared plus b squared

0:40:00.720,0:40:04.560
s over square root a[br]squared plus b squared

0:40:04.560,0:40:08.668
and s over square root.

0:40:08.668,0:40:10.600
OK?

0:40:10.600,0:40:13.420
So can I keep the[br]notation out of s?

0:40:13.420,0:40:14.590
No.

0:40:14.590,0:40:18.850
It's not mathematically[br]correct to keep r of s.

0:40:18.850,0:40:21.460
Why do the books[br]sometimes by using

0:40:21.460,0:40:23.040
multiplication keep r of s?

0:40:23.040,0:40:27.390
Because the books are[br]not always rigorous.

0:40:27.390,0:40:28.925
But I'm trying to be rigorous.

0:40:28.925,0:40:30.960
This is an honors class.

0:40:30.960,0:40:34.940
So How do I rewrite[br]the whole thing?

0:40:34.940,0:40:39.961
r of t, who is a function[br]of s, t as a function of s

0:40:39.961,0:40:45.770
was again s over square root[br]a squared plus b squared

0:40:45.770,0:40:49.250
will be renamed rho of s.

0:40:49.250,0:40:51.030
And what is that?

0:40:51.030,0:40:54.810
That is a of cosine[br]of parentheses

0:40:54.810,0:41:00.570
s over square root a[br]squared r b squared, comma,

0:41:00.570,0:41:06.240
a sine of s over square root[br]a squared plus b squared

0:41:06.240,0:41:12.894
and b times s over square[br]root a squared plus b squared.

0:41:12.894,0:41:14.510
So what have I done?

0:41:14.510,0:41:16.221
Did I get my 20%?

0:41:16.221,0:41:16.720
Yes.

0:41:16.720,0:41:17.230
Why?

0:41:17.230,0:41:19.480
Because I reparameterized[br]the curve.

0:41:19.480,0:41:21.920
Did I get my other 20%?

0:41:21.920,0:41:25.680
Yes, because I told[br]people who s of t was.

0:41:25.680,0:41:32.980
So 20% for this box and[br]20% for this expression.

0:41:32.980,0:41:34.600
So what have I done?

0:41:34.600,0:41:39.100
On the same physical curve, I[br]have slowed down, thank God.

0:41:39.100,0:41:41.860
You say, finally, she's[br]slowing down, right?

0:41:41.860,0:41:42.860
I've changed this speed.

0:41:42.860,0:41:46.270


0:41:46.270,0:41:51.130
On the contrary, if a would[br]be 4 and be would be 3,

0:41:51.130,0:41:56.170
I increase my speed[br]multiple five times, right?

0:41:56.170,0:41:59.500
So you can go back and[br]forth between s and t.

0:41:59.500,0:42:02.820
What does s do compared to t?

0:42:02.820,0:42:04.601
It increases the[br]speed five times.

0:42:04.601,0:42:05.100
Yes sir.

0:42:05.100,0:42:06.992
STUDENT: So when[br]you reparameterize,

0:42:06.992,0:42:08.825
it's just basically the[br]integral from 0 to t

0:42:08.825,0:42:11.850
of whatever[br][INAUDIBLE] of tau is.

0:42:11.850,0:42:14.210
PROFESSOR: Exactly.

0:42:14.210,0:42:18.680
So my suggestion to all[br]of you-- it took me a year

0:42:18.680,0:42:21.170
to understand how[br]to reparameterize

0:42:21.170,0:42:24.950
because I was not smart enough[br]to get it as a freshman.

0:42:24.950,0:42:26.380
I got an A in that class.

0:42:26.380,0:42:28.390
I didn't understand anything.

0:42:28.390,0:42:31.808
As a sophomore, I really--[br]because sometimes, you know,

0:42:31.808,0:42:36.180
you can get an A without[br]understanding things in there.

0:42:36.180,0:42:38.607
As a sophomore, I[br]said, OK, what the heck

0:42:38.607,0:42:39.860
was that reparameterization?

0:42:39.860,0:42:42.770
I have to understand that[br]because it bothers me.

0:42:42.770,0:42:43.340
I went back.

0:42:43.340,0:42:45.220
I took the book.

0:42:45.220,0:42:48.080
I learned about[br]reparameterization.

0:42:48.080,0:42:50.540
Our book, I think,[br]does a very good job

0:42:50.540,0:42:52.070
when it comes to[br]reparameterizing.

0:42:52.070,0:42:57.540
So if you open the 10.2 and[br]10.4, you have to skip-- well,

0:42:57.540,0:42:59.860
am I telling you to skip 10.3?

0:42:59.860,0:43:01.100
That's about ballistics.

0:43:01.100,0:43:03.970
If you're interested in[br]dancing and all sorts of,

0:43:03.970,0:43:08.440
like, how the bullet[br]will be projected

0:43:08.440,0:43:11.346
in this motion or that[br]motion, you can learn that.

0:43:11.346,0:43:14.084
Those are plane curves that[br]are interested in physics

0:43:14.084,0:43:14.750
and mathematics.

0:43:14.750,0:43:18.912
But 10.3 is not part of[br]them and they are required.

0:43:18.912,0:43:20.317
Read 10.2 and 10.4.

0:43:20.317,0:43:21.650
You understand this much better.

0:43:21.650,0:43:22.683
Yes, ma'am.

0:43:22.683,0:43:24.891
STUDENT: Will the midterm[br]or the final just be, like,

0:43:24.891,0:43:26.682
a series problems, or[br]will it be anything--

0:43:26.682,0:43:29.730
PROFESSOR: This is going to[br]be like that-- 15 problems

0:43:29.730,0:43:30.255
like that.

0:43:30.255,0:43:31.963
STUDENT: Will it be[br]anything, like, super

0:43:31.963,0:43:33.214
in depth like the extra credit?

0:43:33.214,0:43:35.090
PROFESSOR: That-- isn't[br]that in-depth enough?

0:43:35.090,0:43:37.390
OK, this is going[br]to be like that.

0:43:37.390,0:43:40.610
So I would say at this[br]point, the way I feel,

0:43:40.610,0:43:45.490
I feel that I am ready to[br]put extra credit there.

0:43:45.490,0:43:48.890
My policy is that[br]I read everything.

0:43:48.890,0:43:52.960
So even if at this point,[br]you say extra credit.

0:43:52.960,0:43:54.900
And you put it at[br]the end for me.

0:43:54.900,0:43:57.120
Say, look, I'm doing[br]the extra credit here.

0:43:57.120,0:44:00.350
Then I'll be ready and I'll[br]say, OK, what did she mean?

0:44:00.350,0:44:01.920
Length of the arc?

0:44:01.920,0:44:02.420
Which arc?

0:44:02.420,0:44:05.620
From here to here is[br]ready to be computed.

0:44:05.620,0:44:08.430


0:44:08.430,0:44:11.290
And that's going to be-- you[br]can include your extra credit

0:44:11.290,0:44:13.340
inside the actual problem.

0:44:13.340,0:44:14.259
I see it.

0:44:14.259,0:44:14.800
STUDENT: Yes.

0:44:14.800,0:44:15.670
PROFESSOR: Don't worry.

0:44:15.670,0:44:17.336
STUDENT: Would it[br]just be as like-- just

0:44:17.336,0:44:19.720
like the casual problem[br]on the test or midterm

0:44:19.720,0:44:22.534
or whatever-- would it[br]be, like, an extra credit

0:44:22.534,0:44:23.478
problem in itself?

0:44:23.478,0:44:24.894
I know there will[br]be extra credit,

0:44:24.894,0:44:26.310
but the kind of proving--

0:44:26.310,0:44:29.620
PROFESSOR: That is--[br]that is decided together

0:44:29.620,0:44:32.830
with the course coordinator.

0:44:32.830,0:44:35.100
The course coordinator[br]himself said

0:44:35.100,0:44:39.940
that he is encouraging us to[br]set up the scale so that if you

0:44:39.940,0:44:43.150
all the problems that[br]are written on the exam,

0:44:43.150,0:44:46.680
you get something like 120%[br]if everything is perfect.

0:44:46.680,0:44:47.860
STUDENT: OK, if we can--

0:44:47.860,0:44:49.738
PROFESSOR: So it's sort[br]of in-built in that-- yes.

0:44:49.738,0:44:51.220
STUDENT: If we can[br]do the web work,

0:44:51.220,0:44:53.200
is that a good indication of--

0:44:53.200,0:44:54.090
PROFESSOR: Wonderful.

0:44:54.090,0:44:56.160
That's exactly--[br]because the way we

0:44:56.160,0:44:58.290
write those problems[br]for the final,

0:44:58.290,0:45:01.830
we pull them out of the web work[br]problems we do for homework.

0:45:01.830,0:45:03.670
So a square root[br]of a squared times

0:45:03.670,0:45:08.195
b squared times pi over 2--[br]so what have I discovered?

0:45:08.195,0:45:11.480
If I would take a[br]piece of that paper

0:45:11.480,0:45:13.760
and I would measure from[br]this point to this point

0:45:13.760,0:45:17.600
how much I traveled in[br]inches from here to here,

0:45:17.600,0:45:21.380
that's exactly that square root[br]of- this would be like a 5.

0:45:21.380,0:45:24.580
That's 3.1415 divided by 2.

0:45:24.580,0:45:25.290
Yes, sir.

0:45:25.290,0:45:29.340
STUDENT: So in the[br]interval of a squared plus

0:45:29.340,0:45:31.292
b squared, I know[br]that that's supposed

0:45:31.292,0:45:33.770
to be the interval[br]the magnitude of r--

0:45:33.770,0:45:35.855
PROFESSOR: The speed--[br]integral of speed?

0:45:35.855,0:45:36.480
STUDENT: Right.

0:45:36.480,0:45:39.397
So which is the r prime, right?

0:45:39.397,0:45:40.230
PROFESSOR: Yes, sir.

0:45:40.230,0:45:43.667
STUDENT: OK, so r prime was--

0:45:43.667,0:45:44.500
PROFESSOR: Velocity.

0:45:44.500,0:45:46.780
STUDENT: --a sine--[br]or negative a sine t,

0:45:46.780,0:45:49.380
a cosine t, and then b?

0:45:49.380,0:45:51.855
So where did the square root[br]of a squared plus b squared

0:45:51.855,0:45:53.487
come from?

0:45:53.487,0:45:54.570
STUDENT: That's from the--

0:45:54.570,0:45:57.290
PROFESSOR: I just erased it.

0:45:57.290,0:46:02.630
OK, so you have minus i-- minus[br]a sine b equals sine p and d.

0:46:02.630,0:46:04.970
When you squared them,[br]what did you get?

0:46:04.970,0:46:05.938
He has the same thing.

0:46:05.938,0:46:06.979
STUDENT: So that's just--

0:46:06.979,0:46:08.831
PROFESSOR: The square of that[br]plus the square root of that

0:46:08.831,0:46:10.039
plus the square root of that.

0:46:10.039,0:46:14.717
STUDENT: So it's just like a 2D[br]representation of the top one.

0:46:14.717,0:46:15.550
STUDENT: This side--

0:46:15.550,0:46:18.980


0:46:18.980,0:46:21.365
PROFESSOR: I just need the[br]magnitude of r prime, which

0:46:21.365,0:46:22.880
is this p, right?

0:46:22.880,0:46:23.657
STUDENT: Right.

0:46:23.657,0:46:25.115
PROFESSOR: The[br]magnitude of this is

0:46:25.115,0:46:29.214
the speed, which is square root[br]of a squared plus b squared.

0:46:29.214,0:46:30.620
Is that clear?

0:46:30.620,0:46:31.230
STUDENT: Yes.

0:46:31.230,0:46:32.870
PROFESSOR: I can[br]go on if you want.

0:46:32.870,0:46:36.835
So a square root of-- the sum[br]of the squares of this, this,

0:46:36.835,0:46:40.510
and that is exactly[br]square of [INAUDIBLE].

0:46:40.510,0:46:41.850
Keep this in mind as an example.

0:46:41.850,0:46:44.710
It's an extremely important one.

0:46:44.710,0:46:48.710
It appears very frequently[br]in tests-- on tests.

0:46:48.710,0:46:52.560
And it's one of the[br]most beautiful examples

0:46:52.560,0:46:57.900
in applications of[br]mathematics to physics.

0:46:57.900,0:47:02.740
I have something[br]else that was there.

0:47:02.740,0:47:03.570
Yes ma'am

0:47:03.570,0:47:07.866
STUDENT: I was just going to[br]ask if you want to curvature.

0:47:07.866,0:47:08.480
PROFESSOR: Eh?

0:47:08.480,0:47:09.265
STUDENT: The letter--

0:47:09.265,0:47:10.146
PROFESSOR: Curvature?

0:47:10.146,0:47:11.020
STUDENT: Curvature.

0:47:11.020,0:47:12.603
PROFESSOR: That's[br]exactly what I want.

0:47:12.603,0:47:17.970
And when I said I had something[br]else for 20%, what was k?

0:47:17.970,0:47:22.672
K was rho double prime[br]of s in magnitude.

0:47:22.672,0:47:29.784
So I have to be smart enough[br]to look at that and rho of s.

0:47:29.784,0:47:32.742
And rho of s was[br]the thing that had

0:47:32.742,0:47:35.693
here-- that's going to be[br]probably the end of my lesson

0:47:35.693,0:47:36.193
today.

0:47:36.193,0:47:39.650


0:47:39.650,0:47:46.420
Since you have so many[br]questions, I will continue.

0:47:46.420,0:47:50.360
I should consider--[br]the chapter is finished

0:47:50.360,0:47:54.030
but I will continue with a[br]deeper review, how about that,

0:47:54.030,0:47:57.300
on Tuesday with more problems.

0:47:57.300,0:48:00.890
Because I have the feeling that[br]although we covered 10.1, 10.2,

0:48:00.890,0:48:03.640
10.4, you need a[br]lot more examples

0:48:03.640,0:48:05.790
until you feel comfortable.

0:48:05.790,0:48:08.350
Many of you not,[br]maybe 10 people.

0:48:08.350,0:48:09.880
They feel very comfortable.

0:48:09.880,0:48:10.450
They get it.

0:48:10.450,0:48:13.330
But I think nobody will be[br]hurt by more review and more

0:48:13.330,0:48:16.096
examples and more applications.

0:48:16.096,0:48:20.565
Now, who can help me[br]finish my goal for today?

0:48:20.565,0:48:22.640
Is this hard?

0:48:22.640,0:48:25.580
This is rho of s.

0:48:25.580,0:48:30.360
So you have to tell me with[br]the derivation, is it hard?

0:48:30.360,0:48:31.330
No.

0:48:31.330,0:48:37.530
Minus a sine of the[br]whole thing times 1

0:48:37.530,0:48:40.551
over square root of a squared[br]plus b squared because I'm

0:48:40.551,0:48:42.280
applying the chain rule, right?

0:48:42.280,0:48:43.900
Let me change color.

0:48:43.900,0:48:45.070
Who's the next guy?

0:48:45.070,0:48:50.135
A Cosine of s over square[br]root a squared plus b squared.

0:48:50.135,0:48:53.040
I'm now going to leave[br]you this as an exercise

0:48:53.040,0:48:56.250
because you're going to haunt[br]me back ask me why I got this.

0:48:56.250,0:48:58.810
So I want to make it very clear.

0:48:58.810,0:49:03.940
B times 1 over square root[br]a squared by b squared.

0:49:03.940,0:49:06.380
So are we happy with this?

0:49:06.380,0:49:07.460
Is this understood?

0:49:07.460,0:49:10.980
It's a simple derivation[br]of the philosophy.

0:49:10.980,0:49:12.530
We are not done.

0:49:12.530,0:49:15.300
We have to do the acceleration.

0:49:15.300,0:49:18.400
So the acceleration[br]with respect to s

0:49:18.400,0:49:21.610
of this curve where s was[br]the arc length parameter

0:49:21.610,0:49:24.300
is real easy to compute[br]in the same way.

0:49:24.300,0:49:26.140
What is different?

0:49:26.140,0:49:29.880
I'm not going to[br]write more explicitly

0:49:29.880,0:49:32.360
because this should be[br]visible for everybody.

0:49:32.360,0:49:35.450
STUDENT: x [INAUDIBLE].

0:49:35.450,0:49:38.850
PROFESSOR: Good,[br]minus a over-- I'll

0:49:38.850,0:49:41.640
wait for you to[br]simplify because I don't

0:49:41.640,0:49:43.280
want to pull two roots out.

0:49:43.280,0:49:44.160
STUDENT: A squared--

0:49:44.160,0:49:45.659
PROFESSOR: A squared[br]plus b squared.

0:49:45.659,0:49:47.215
And why is that, [INAUDIBLE]?

0:49:47.215,0:49:52.180
Because you have once and[br]twice from the chain rule.

0:49:52.180,0:49:55.590
So again, I hope you guys don't[br]have a problem with the chain

0:49:55.590,0:50:00.750
rule so I don't have to[br]send you back to Calculus 1.

0:50:00.750,0:50:05.630
A over a squared times b[br]squared with a minus-- why

0:50:05.630,0:50:06.280
with a minus?

0:50:06.280,0:50:07.040
Somebody explain.

0:50:07.040,0:50:09.190
STUDENT: Use the[br]derivative of cosine.

0:50:09.190,0:50:13.360
PROFESSOR: There's a cosine[br]and there's a minus sine.

0:50:13.360,0:50:15.844
From deriving, I have[br]a minus and a sine.

0:50:15.844,0:50:21.510


0:50:21.510,0:50:25.470
And finally, thank[br]God, the 0-- why 0?

0:50:25.470,0:50:31.310
Because I have a constant that[br]I'm deriving with respect to s.

0:50:31.310,0:50:33.050
Is it hard to see what's up?

0:50:33.050,0:50:36.110
What's going out?

0:50:36.110,0:50:40.814
What is the curvature[br]of the helix?

0:50:40.814,0:50:45.230
A beautiful, beautiful[br]function that

0:50:45.230,0:50:48.560
is known in most of[br]these math, calculus,

0:50:48.560,0:50:54.070
multivariable calculus and[br]differential geometry classes.

0:50:54.070,0:50:56.640
What did you get?

0:50:56.640,0:51:02.755
Square root of sum of the[br]squares of all these guys.

0:51:02.755,0:51:04.115
You process it.

0:51:04.115,0:51:06.440
That's very easy.

0:51:06.440,0:51:07.395
Shall I write it down?

0:51:07.395,0:51:09.790
Let me write it down[br]like a silly girl--

0:51:09.790,0:51:13.500
square root of a squared,[br]although I hate when I cannot

0:51:13.500,0:51:15.190
go ahead and simplify it.

0:51:15.190,0:51:18.950
But let's say there's[br]this little baby thing.

0:51:18.950,0:51:21.550


0:51:21.550,0:51:25.070
Now I can say it's[br]a over a squared

0:51:25.070,0:51:26.750
plus b squared-- finally.

0:51:26.750,0:51:29.106
So I'm going to ask[br]you a few questions

0:51:29.106,0:51:30.480
and then I'm going[br]to let you go.

0:51:30.480,0:51:32.840
It's a punishment[br]for one minute.

0:51:32.840,0:51:36.580
OK, if I have the[br]curve we had before,

0:51:36.580,0:51:41.590
the beautiful helix with[br]a Pythagorean number

0:51:41.590,0:51:45.440
like 3 cosine t, 3[br]sine t, and 4t, what

0:51:45.440,0:51:48.498
is the curvature of that helix?

0:51:48.498,0:51:49.800
STUDENT: 3 over 5--

0:51:49.800,0:51:52.246
PROFESSOR: 3 over 5, excellent.

0:51:52.246,0:51:53.730
How about my helix?

0:51:53.730,0:51:57.540
What if I changed the numbers[br]in web work or on the midterm

0:51:57.540,0:52:00.600
and I say it's going[br]to be-- it could even

0:52:00.600,0:52:02.150
be with a minus, guys.

0:52:02.150,0:52:05.130
It's just the way you travel[br]it would be different.

0:52:05.130,0:52:08.030
So whether I put[br]plus minus here,

0:52:08.030,0:52:09.950
you will try on[br]different examples.

0:52:09.950,0:52:13.150
Sometimes if we put[br]minus here or minus here,

0:52:13.150,0:52:15.380
it really doesn't matter.

0:52:15.380,0:52:17.880
Let's say we have[br]cosine t sine t and t.

0:52:17.880,0:52:22.044
What's the curvature of[br]that parametrized curve?

0:52:22.044,0:52:23.010
1 over--

0:52:23.010,0:52:23.976
STUDENT: 2.

0:52:23.976,0:52:26.620
PROFESSOR: 1 over 2-- excellent.

0:52:26.620,0:52:27.380
So you got it.

0:52:27.380,0:52:28.740
So I'm proud of you.

0:52:28.740,0:52:32.396
Now, I want to do more examples[br]until you feel confident

0:52:32.396,0:52:32.895
about it.

0:52:32.895,0:52:36.590
I know I got most of you to[br]the point where I want it.

0:52:36.590,0:52:38.690
But you need more[br]reading definitely

0:52:38.690,0:52:41.360
and you need to[br]see more examples.

0:52:41.360,0:52:43.020
Feel free to read[br]the whole chapter.

0:52:43.020,0:52:48.340
I would-- if you don't have[br]time for 10.3, skip it.

0:52:48.340,0:52:50.140
10.5 is not going[br]to be required.

0:52:50.140,0:52:53.090
So if I were a student,[br]I'd go home, open the book,

0:52:53.090,0:52:55.860
read 10.1, 10.2,[br]10.4, close the book.

0:52:55.860,0:52:59.636
It's actually a lot less[br]than you think it is.

0:52:59.636,0:53:01.510
If you go over the most[br]important formulas,

0:53:01.510,0:53:04.230
then you are ready[br]for the homework.

0:53:04.230,0:53:06.470
The second homework is due when?

0:53:06.470,0:53:08.090
February 11.

0:53:08.090,0:53:10.060
You guys have plenty of time.

0:53:10.060,0:53:14.260
Rather than going to the[br]tutors, ask me for Tuesday.

0:53:14.260,0:53:17.420
On Tuesday, you'll have plenty[br]of time for applications.

0:53:17.420,0:53:20.030
OK, have a wonderful weekend.

0:53:20.030,0:53:23.320
Don't forget to email when[br]you get in trouble, OK?

0:53:23.320,0:53:27.696