I love this model.
Again, thank you, Casey.
I'm not going to take any credit for that.
So if you want to imagine the stool
I was talking about as a bamboo object, that
is about the same thing, at the same scale, compared
to the diameter and the height, scaled or dialated five times.
Uniform, no alterations.
And one can sit on it, [? and circle, ?] to sit on it.
Now, as you see this is a doubly ruled surface.
And you say, oh wait a minute.
You said rule surface, why all of a sudden, why doubly ruled
surface?
Because it is a surface that is ruled and generated
by two different one parameter families.
Each of them has a certain parameter
and that gives them continuity.
So you have two families of lines.
One family is in this direction.
Do you see it?
So these lines-- this line is in motion.
It moves to the right, to the right, to the right,
and it generated.
And the other family of lines is this one
in the other direction.
You have a continuity parameter for each of them.
So you have to imagine some real parameter going
along the entire [? infinite real ?] axis.
Or along a circle which would be about the same thing.
But in any case, you have a one parameter family
and another one parameter family.
Both of them are together generating
this beautiful one-sheeted hyperboloid.
It's incredible because you see where these sort of round,
but if you go towards the ends, it's topologically
a cylinder or a tube.
But if you look towards the end, the two ends
will look more straight.
And you will see the straight lines more clearly.
So imagine that you have a continuation
to infinity in this direction, and in the other direction.
And this actually should be an infinite surface in your model.
You're just cutting it between two z planes,
so you have a patch of a one-sheeted hyperboloid.
Yeah, the one-sheeted hyperboloid
that we wrote last time, do you guys remember
x squared over a squared plus y squared
over b squared minus z squared? z should be this [INAUDIBLE].
Minus z squared over c squared minus 1
equals 0 is an infinite surface area.
At both ends you keep going.
Very beautiful.
Thank you so much.
I appreciate.
And keep the brownies.
No, then I have to pay more.
Than I have to pay money.
STUDENT: It's made out of [INAUDIBLE].
PROFESSOR: When is your birthday?
[LAUGHTER]
Really?
When is it?
STUDENT: February 29.
PROFESSOR: Oh, it's coming.
[INTERPOSING VOICES]
STUDENT: It's coming in a year, too.
PROFESSOR: That was a smart one.
Anyway, I'll remember that.
I appreciate the gift very much.
And I will cherish it and I'll use it
with both my undergraduate students
and my graduate students who are just learning about-- some
of them don't know the one-sheeted hyperboloid model,
but they will learn about it.
Coming back to our lesson.
I announced Section 10.1.
Say goodbye to quadrant for a while.
I know you love them, but they will be there
for you in Chapter 11.
They will wait for you.
Now, let's go to Section 10.1 of Chapter 10.
Chapter 10 is a beautiful chapter.
As you know very well, I announced last time,
it is about vector-valued functions.
And you say, oh my god, I've never
heard about vector-valued functions before.
You deal with them every day.
Every time you move, you are dealing
with a vector-valued function, which
is the displacement, which takes values in a subset in R3.
So let's try and see what you should understand
when you start Section 10.1.
Because the book is pretty good, not that I'm a co-author.
But it was meant to be really written for the students
and explain concepts really well.
How many of you took physics?
OK, quite a lot of you took physics.
Now, one of my students in a previous honors class
told me he enjoyed my class greatly in general.
The most [INAUDIBLE] thing he had from my class, he
learned from my class was the motion of the drunken bug.
And I said, did I say that?
Absolutely, you said that.
So apparently I had started one of my lessons
with imagine you have a fly who went into your coffee mug.
I think I did.
He reproduced the whole thing the way I said it.
It was quite spontaneous.
So imagine your coffee mug had some Baileys Irish Creme in it.
And the fly was really happy after she got up.
She managed to get up.
And the trajectory of the fly was something more
like a helix.
And this is how I actually introduced the helix
in my classroom.
And I thought, OK, is that unusual?
Very.
And I said, but that's an honors class.
Everything is supposed to be unusual, right?
So let's think about the position vector or some sort
of vector-valued function that you're familiar with already
from physics.
He is one of your best friends.
You have a function r of t.
And I will point out that r is practically the position vector
measure that time t, or observed at time t in R3.
So he takes values in R3.
How?
As the mathematician, because I like to write mathematically
all the notion I have, r is defined
on I was a sub-interval of R with values in R3.
And he asked me, my student said, what is this I?
Well, this I could be any interval,
but let's assume for the time being it's
just an open interval of the type
a, b, where a and b are real numbers, a less than b.
So this is practically the time for my bug from the moment,
let's say a equals 0 when she or he starts flying up,
until the moment she completely freaks
out or drops from the maximum point she reached.
And she eventually dies.
Or maybe she doesn't die.
Maybe she's just drunk and she will wake up after a while.
OK, so what do I mean by this displacement vector?
I mean, a function--
STUDENT: Is that Tc?
Do you have [INAUDIBLE]?
PROFESSOR: This is r, little r.
STUDENT: I know, but the Tc.
PROFESSOR: Tc?
STUDENT: Or is that an I?
PROFESSOR: No.
This is I interval, which is the same as a,
b open interval, like from 2 to 7, included.
This is inclusion [INAUDIBLE] included in R.
So I mean R is the real number set and a, b is my interval.
OK, so r of t is going to be what?
x of t, y of t, z of t.
The book tells you, hey, guys-- it doesn't say hey, guys,
but it's quite informal-- if you live in Rn, if your image is
in Rn, instead of x of t, y of t, z of t,
you are going to get something like x1 of t, y1 of t.
x1 of t, x2 of t, x3 of t, et cetera.
What do we assume about R?
We have to assume something about it, right?
STUDENT: It's a function [INAUDIBLE].
PROFESSOR: It's a function that is differentiable
most of the times, right?
What does it mean smooth?
I saw that your books before college level
never mention smooth.
A smooth function is a function that is differentiable
and whose first derivative is continuous.
Some mathematicians even assume that you have c infinity, which
means you have a function that's infinitely many differentiable.
So you have first derivative, second derivative, third
derivative, fifth derivative.
Somebody stop me.
All the derivatives exist and they are all continuous.
By smooth, I will assume c1 in this case.
I know it's not accurate, but let's assume c1.
What does it mean?
Differentiable function whose derivative is continuous.
And I will assume one more thing.
That is not enough for me.
I will also assume that r prime of t in this case
is different from 0 for every t in the interval I.
Could somebody tell me in everyday words what that means?
We call that regular function.
[INAUDIBLE]
You have a brownie [INAUDIBLE].
I have no brownies with me.
But if you answer, so what--
STUDENT: So that means you've got no relative mins or maxes,
and you never-- the object never stops moving.
PROFESSOR: Well, actually, you can have relative mins
and maxes in some way.
I'm talking about something like that, r prime.
This is r of t.
And r prime of t is the derivative.
It's never going to stop.
The velocity.
I'm talking about this piece of information.
Velocity [INAUDIBLE] 0 means that drunken bug between time
a and time b never stops.
He stops at the end, but the end is b, is outside [INAUDIBLE].
So he stops at b and he falls.
So I don't stop.
I move on from time a to time b.
I don't stop at all.
Yes, sir.
STUDENT: Wouldn't the derivative of that line at some point
equal 0 where it flattens out?
PROFESSOR: Let me draw very well.
So at time r of t, this is the position vector.
What is the derivative?
The derivative represents the velocity vector.
A beautiful thing about the velocity vector r prime of t
is that it has a beautiful property.
It's always tangent to the trajectory.
So at every point you're going to have
a velocity vector that is tangent to the trajectory.
[INAUDIBLE] in physics.
This r prime of t should never become 0.
So you will never have a point instead of a segment
when it comes to r prime.
So you don't stop.
You are going to say, wait a minute?
But are you always going to consider curves, regular curves
in space?
Regular curves in space.
And by space, I know you guys mean the Euclidean three space.
Actually, many times I will consider curves in plane.
And the plane is part of the space.
And you say, give us an example.
I will give you an example right now.
You're going to laugh how simple that is.
Now, I have another bug who is really happy,
but it's not drunk at all.
And this bug knows how to circle around a certain point
at the same speed.
So very organized bug.
Yes, sir.
STUDENT: Where did you get c prime?
PROFESSOR: What?
STUDENT: You have c prime is differentiable, is [INAUDIBLE].
PROFESSOR: c1.
STUDENT: c1.
PROFESSOR: OK. c1.
This is the notation for any function that is differentiable
and whose derivative is continuous.
So again, give an example of a c1 function.
STUDENT: x squared.
PROFESSOR: Yeah.
On some real interval.
How about absolute value of x over the real line?
What's the problem with that?
[INTERPOSING VOICES]
PROFESSOR: It's not differentiable at 0.
OK, so we'll talk a little bit later about smoothness.
It's a little bit delicate as a notion.
It's really beautiful on the other side.
Let's find the nice picture trajectory for the bug.
This is a ladybug.
I cannot draw her, anyway.
She is moving along this circle.
And I'll give you the law of motion.
And that reminds me of a student who told me, what
do I care about law of motion?
He never had me as a teacher, obviously.
But he was telling me, well, after I graduated,
I always thought, what do I care about the law of motion?
I mean, I took calculus.
Everything was about the law of motion.
I'm sorry, you should care about the law of motion.
Once you're not there anymore, absolutely you don't care.
But why do you want to [INAUDIBLE] doing calculus?
When you bring [INAUDIBLE] to calculus,
when you walk into calculus, it's law of motion
everywhere whether you like it or not.
So let's try cosine t sine t and z to b 1.
Let's make it 1 to make your life easier.
What kind of curve is this and why am I
claiming that the ladybug following this curve
is moving at a constant speed?
Oh my god.
Go ahead, Alexander.
STUDENT: That's a circle.
PROFESSOR: That's the circle.
It's more than a circle.
It's a parametrized circle.
It's a vector-valued function.
Now, like every mathematician I should specify the domain.
I am just winding around one time,
and I stop where I started.
So I better be smart and realize time is not infinity.
It could be.
I'm wrapping around the circle infinitely many times.
They do that in topology actually when
you're going to be-- seniors takes topology.
But I'm not going around in circles only one time.
So my time will start at 0 when I
start my motion and end at 2 pi seconds
if the time is in seconds
So I say r is defined on the interval I which
is-- say it again, Magdalena.
You just said it.
STUDENT: 0.
PROFESSOR: 0 to pi.
If you want to take 0 together, fine.
But for consistency, let's take it like before, 0 to 2 pi.
I'm actually excluding the origin.
And with values in R3.
Although, this is a [? plane ?] curve, z will be constant.
Do I care about that very much?
You will see the beauty of it.
I have the velocity vector being really pretty.
What is the velocity vector?
STUDENT: [INAUDIBLE].
PROFESSOR: Negative sign t.
Thank you.
STUDENT: [INAUDIBLE].
PROFESSOR: Cosine t.
And 0, finally.
Because as you saw very well in the book,
the way we compute the velocity vector
is by taking x of t, y of t, z of t
and differentiating them in terms of time.
Good.
Is this a regular function?
As the bug moves between time 0 and time equals 2 pi,
is the bug ever going to stop between these times?
STUDENT: No.
PROFESSOR: No.
How do you know?
You guys are faster than me, right?
What did you do?
You did the speed.
What's the relationship?
What's the difference between velocity and speed?
STUDENT: Speed is the absolute value [INAUDIBLE].
PROFESSOR: Wonderful.
This is very good.
You should tell everybody that because people
confuse that left and right.
So the velocity is a vector, like you
learned in engineering.
You learned in physics.
Velocity is a vector.
It changes direction.
I'm going to Amarillo this way.
I'm driving.
The velocity will be a vector pointing this way.
As I come back, will point the opposite way.
The speed will be a scalar, not a vector.
It's a magnitude of a velocity vector.
So say it again, Magdalena.
What is the speed?
The speed is the magnitude of the velocity vector.
It's a scalar.
Speed.
I heard that before in cars, in the movie Cars.
Anyway, r prime of t magnitude.
In magnitude.
Remember, there is a big difference between the velocity
as the notion.
Velocity is a vector.
The speed is a magnitude, is a scalar.
I'm going to go ahead and erase that
and I'm going to ask you what the speed is
for my fellow over here.
What is the speed of a trajectory
of the bug who is sober and moves at the constant speed?
OK.
As I already told you, it's constant.
What is that constant?
What's the constant speed I was talking about?
STUDENT: [INAUDIBLE].
PROFESSOR: I say the magnitude of that.
I'm too lazy to write it down.
It's a Tuesday, almost morning.
So I go square root of minus I squared
plus cosine squared plus 0.
I don't need to write that down.
You write it down.
And how much is that?
STUDENT: [INAUDIBLE].
PROFESSOR: 1.
So I love this curve because in mathematician slang,
especially in [? a geometer's ?] slang-- and my area
is differential geometry.
So in a way, I do calculus in R3 every day on a daily basis.
So I have what?
This is a special kind of curve.
It's a curve parameterized in arc length.
So definition, we say that a curve in R3,
or Rn, well anyway, is parameterized in arc length.
When?
Say it again, Magdalena.
Whenever, if and only if, its speed is constantly 1.
So this is an example where the speed is 1.
In such cases, we avoid the notation with t.
You say, oh my god.
Why?
When the curve is parameterized in arc length,
from now on the we will actually try
to use s whatever we know it's an arc length.
We use s instead of t.
So I'm sorry for the people who cannot change that,
but you should all be able t change that.
So everything will be in s because we just
discovered [? Discovery Channel, ?] we
just discovered that speed is 1.
So there is something special about this s.
In this example-- oh, you can rewrite the whole example
if you want in s so you don't have to smudge the paper.
OK, it's beautiful.
So I am already arc length.
And in that case, I'm going to call my time parameter
little s. s comes from special.
No, s comes from speed [INAUDIBLE].
STUDENT: So you use s when it's [INAUDIBLE]?
PROFESSOR: We use s whenever the speed of that curve will be 1.
STUDENT: So [INAUDIBLE].
PROFESSOR: And we call that arc length parameterization.
I'm moving into the duration of your final thoughts.
Yes, sir.
STUDENT: When we get the question, so
before solving [INAUDIBLE].
PROFESSOR: We don't know.
That's why it was our discovery that, hey, at the end
it is an arc length, so I better change [INAUDIBLE] t into s
because that will help me in the future remember to do that.
Every time I have arc length, that it means speed 1.
I will call it s instead of y.
There is a reason for that.
I'm going to erase the definition
and I'm going to give you the-- more or less,
the explanation that my physics professor gave me.
Because as a freshman, my mathematics professor
in that area, in geometry, was not very, very active.
But practically, what my physics professor told me is that,
hey, I would like to have some sort of a uniform tangent
vector, something that is standardized to be in speed 1.
So I would like that tangent vector to be important to us.
And if r is an arc length, then r
prime would be that unit vector that I'm talking about.
So he introduced for any r of t, which is x of t, y of t,
z of t.
My physics professor introduced the following terminology.
The tangent unit vector for a regular curve--
he was very well-organized I might add about him--
is by definition r prime of t as a vector
divided by the speed of the vector.
So what is he doing?
He is unitarizing the velocity.
Say it again, Magdalena.
He has unitarized the velocity in order
to make research more consistent from the viewpoint of Frenet
frame.
So in Frenet frame, you will see-- you probably
learned about the Frenet frame if you
are a mechanics major, or some solid mechanics or physics
major.
The Frenet frame is an orthogonal frame
moving along a line in time where the three components are
t, and the principal normal vector, and b the [INAUDIBLE].
We only know of the first of them, which
is T, which is a unit vector.
Say it again who it was.
It was the velocity vector divided by its magnitude.
So the velocity vector could be any wild, crazy vector
that's tangent to the trajectory at the point where you are.
His magnitude varies from one point to the other.
He's absolutely crazy.
He or she, the velocity vector.
Yes, sir.
STUDENT: [INAUDIBLE].
PROFESSOR: Here?
Here?
STUDENT: Yeah, down there.
PROFESSOR: D-E-F, definition.
That's how a mathematician defines things.
So to define you write def on top of an equality sign
or double dot equal.
That's a formal way a mathematician introduces
a definition.
Well, he was a physicist, but he does math.
So what do we do?
We say all the blue guys that are not equal,
divide yourselves by your magnitude.
And I'm going to have the T here is next one,
the T here is next one.
They are all equal.
So that T changes direction, but its magnitude will always be 1.
Right?
Know that the magnitude-- that's what unit vector means,
the magnitude is 1.
Why am I so happy about that?
Well let me tell you that we can have
another parametrization and another parametrization
of the same curve.
Say what?
The parametrization of a curve is not unique?
No.
There are infinitely many parametrizations
for a physical curve.
There are infinitely many parametrizations
for an even physical curve.
Like [INAUDIBLE] the regular one?
Well let me give you another example that
says that this is currently R of T
equals cosine 5T sine 5T and 1.
Why 1?
I still want to have the same physical curve.
What's different, guys?
Look at that and then say oh OK, is this
the same curve as a physical curve?
What's different in this case?
I'm still here.
It's still the [? red ?] physical curve
I'm moving along.
What is different?
STUDENT: The velocity.
PROFESSOR: The velocity.
The velocity and actually the speed.
I'm moving faster or slower, I don't know, we have to decide.
Now how do I realize how many times
I'm moving along this curve?
I can be smart and say hey, I'm not stupid.
I know how to move only one time and stop where I started.
So if I start with my T in the interval
zero-- I start at zero, where do I stop?
I can hear your brain buzzing.
STUDENT: [INAUDIBLE].
PROFESSOR: 2pi over 5.
Why is that?
Excellent answer.
STUDENT: Because when you plug it in, it's [INAUDIBLE].
PROFESSOR: 5 times 2pi over 5.
That's where I stop.
So this is not the same interval as before.
Are you guys with me?
This is a new guy, which is called J. Oh, all right.
So there is a relationship between the T
and the S. That's why I use different notations.
And I wish my teachers started it just
like that when I took math analysis as a freshman,
or calculus.
That's calculus.
Because what they started with was a diagram.
What kind of diagram?
Say OK, the parametrizations are both
starting from different intervals.
And first I have the parametrization
from I going to our 3.
And that's called-- how did we baptize that?
R. And the other one, from J to R3,
we call that big R. They're both vectors.
And hey guys, we should have some sort
of correspondence functions between I
and J that are both 1 to 1, and they are 1 being [INAUDIBLE]
the other.
I swear to God, when they started
with this theoretical model, I didn't understand
the motivation at all.
At all.
Now with an example, I can get you
closer to the motivation of such a diagram.
So where does our primary S live?
S lives in I, and T lives in J. So I
have to have a correspondence that takes S to T or T to S.
STUDENT: Wait I thought since R of T
is also pretty much [INAUDIBLE] that we should also
use S [INAUDIBLE].
PROFESSOR: It's very-- actually it's very easy.
This is 5T.
And we cannot use S instead of this T,
because if we use S instead of this T,
and we compute the speed, we get 5.
So it cannot be called S. This is very important.
So T is not an arc length parameter.
I wonder what the speed will be for this guy.
So who wants to compute R prime of T?
Nobody, but I'll force you to.
And the magnitude of that will be god knows what.
I claim it's 5.
Maybe I'm wrong.
I did this in my head.
I have to do it on paper, right.
So I have what?
I have to differentiate component-wise.
And I have [INAUDIBLE] that, because I'm running out of gas.
STUDENT: Minus 5--
PROFESSOR: Minus 5, very good.
Sine of 5T.
What have we applied?
In case you don't know that, out.
That was Calc 1.
Chain rule.
Right?
So 5 times cosine 5T.
And finally, 1 prime, which is 0.
Now let's be brave and write the whole thing down.
I know I'm lazy today, but I'm going to have to do something.
Right?
So I'll say minus 5 sine 5T is all squared.
Let me take it and square it.
Because I see one face is confused.
And since one face is confused, it
doesn't matter that the others are not confused.
OK?
So I have square root of this plus square of [INAUDIBLE] plus
[INAUDIBLE] computing the magnitude.
What do I get out of here?
STUDENT: Five.
PROFESSOR: Five.
Excellent.
This is 5 sine squared plus 5 cosine squared.
Now yes, then I have 5 times 1.
So I have square root of 25 here will be 5.
What is 5?
5 is the speed of the [? bug ?] along the same physical curve
the other way around.
The second time around.
Now can you tell me the relationship between T and S?
They are related.
They are like if you're my uncle, then I'm your niece.
It's the same way.
It depends where you look at.
T is a function of S, and S is a function of T.
So it has to be a 1 to 1 correspondence between the two.
Now any ideas of how I what to compute the-- how do I
want to write the relationship between them.
Well, S is a function of T, right?
I just don't know what function of T that is.
And I wish my professor had started like that,
but he started with this diagram.
So simply here you have S equals S of T,
and here you have T equals T of S,
the inverse of that function.
And when you-- when somebody starts that
without an example as a general diagram philosophy,
then it's really, really tough.
All right?
So I'd like to know who S of T-- how
in the world do I want to define that S of T.
He spoonfed us S of T. I don't want to spoonfeed you anything.
Because this is honors class, and you
should be able to figure this out yourselves.
So who is big R of T?
Big R of T should be, what, should
be the same thing in the end as R of S.
But I should say maybe it's R of function T of S, right?
Which is the same thing as R of S. So
what should be the relationship between T and S?
We have to call them-- one of them should be T equals T of S.
How about this function?
Give it a Greek name, what do you want.
Alpha?
Beta?
What?
STUDENT: [INAUDIBLE].
PROFESSOR: Alpha?
Beta?
Alpha?
I don't know.
So S going to T, alpha.
And this is going to be alpha inverse.
Right?
So T equals alpha of S. It's more elegant to call it
like that than T of S. T equals alpha of S. Alpha of S.
So from this thing, I realize that I
get that R composed with alpha equals R. Say what?
Magdalena?
Yeah, yeah, that was pre-calculus.
R composed with alpha equals little r.
So how do I get a little r by composing R with alpha?
How do we say that?
Alpha followed by R. R composed with alpha.
R of alpha of S equals R of S. Say it again.
R of alpha of S, which is T-- this T is alpha of S-- equals
R.
This is the composition that we learned in pre-calc.
Who can find me the definition of S?
Because this may be a little bit hard.
This may be a little bit hard.
STUDENT: S [INAUDIBLE].
PROFESSOR: Eh, yeah, let me write it down.
I want to find out what S of T is.
Equals what in terms of the function R of T. The one
that's given here.
Why is that?
Let's try some sort of chain rule, right?
So what do I know I have?
I have that.
Look at that.
R prime of S, which is the velocity of-- I
erased it-- the velocity of R with respect to the arc length
parameter is going to be what?
R of alpha of S prime with respect to S, right?
So I should put DDS.
Well I'm a little bit lazy.
Let's do it again.
DDS, R of alpha of S.
OK.
And what do I have in this case?
Well, I have R prime of-- who is alpha of S. T, [INAUDIBLE] of T
and alpha of S times R prime of alpha
of S times the prime outside.
How do we prime in the chain rule?
From the outside to the inside, one at a time.
So I differentiated the outer shell, R prime,
and then times what?
Chain rule, guys.
Alpha prime of S. Very good.
Alpha prime of S.
All right.
So I would like to understand how
I want to compute-- how I want to define S of T. If I take
this in absolute value, R prime of S in absolute value
equals R prime of T in absolute value times alpha prime of S
in absolute value.
What do I get?
Who is R prime of S?
This is my original function in arc length,
and that's the speed in arc length.
What was the speed in arc length?
STUDENT: One.
PROFESSOR: One.
And what is the speed in not in arc length?
STUDENT: Five.
PROFESSOR: In that case, this is going to be five.
And so what is this alpha prime of S guy?
STUDENT: [INAUDIBLE].
PROFESSOR: It's going to be 1/5.
OK.
All right.
Actually alpha of S, who is that going to be?
Alpha of S.
Do you notice the correspondence?
We simply have to re-define this as S. That's how it goes.
That five times is nothing but S.
STUDENT: How did you get the [INAUDIBLE]?
PROFESSOR: Because 1 equals 5 times what?
1, which is arc length speed, equals 5 times what?
1/5.
STUDENT: Yeah, but then where'd you get the 1?
PROFESSOR: That's one way to do it.
Oh, this is by definition, because little r means
curve in arc length, and little s is the arc length parameter.
By definition, that means you get speed 1.
This was our assumption.
So we could've gotten that much faster saying
oh, well, forget about this diagram
that you introduced-- and it's also in the book.
Simply take 5T to BS, 5T to BS.
Then I get my old friend, the curve.
The arc length parameter is the curve.
So this is the same as cosine of S, sine of S, and 1.
So what is the correspondence between S and T?
Since S is 5T in this example, I'll
put it-- where shall I put it.
I'll put it here.
S is 5T.
I'll say S of T is 5T.
and T of S, what is T in terms of S?
T in terms of S is S over 5.
So instead of T of S, we call this alpha
of S. So the correspondence between S and T, what is T?
T is exactly S over 5 in this example.
Say it again.
T is exactly S over 5.
So alpha of S would be S over 5.
In this case, alpha prime of S would simply be 1 over 5.
Oh, so that's how I got it.
That's another way to get it.
Much faster.
Much simpler.
So just think of replacing 5T by the S knowing
that you put S here, the whole thing will have speed of 1.
All right.
So what do I do?
I say OK, alpha prime of S is 1 over 5.
The whole chain rule also spit out alpha prime of S
to B1 over 5.
Now I understand the relationship between S and T.
It's very simple.
S is 5T in this example, or T equals S over 5.
OK?
So if somebody gives you a curve that looks like cosine 5T, sine
5T, 1, and that is in speed 5, as we were able to find,
how do you re-parametrize that in arc length?
You just change something inside so
that you make this curve be representative-- representable
as little r of S. This is in arc length.
In arc length.
OK.
Finally, this is just an example.
Can you tell me how that arc length parameter
is introduced in general?
What is S of T by definition?
What if I have something really wild?
How do I get to that S of T by definition?
What is S of T in terms of the function R?
STUDENT: [INAUDIBLE] velocity [? of the ?] [INAUDIBLE]?
PROFESSOR: S prime of T will be one of the [INAUDIBLE].
STUDENT: Yes.
PROFESSOR: OK.
So let's see what we have if we define S of T
as being integral from 0 to T of the speed R prime of T.
And instead of T, we put tau.
Right?
P tau.
STUDENT: What is that?
PROFESSOR: We cannot put T, T, and T.
STUDENT: Oh.
PROFESSOR: OK?
So tau is the Greek T that runs between zero
and T. This is the definition of S
of T. General definition of the arc length parameter
that is according to the chain rule, given by the chain rule.
Can we verify really quickly in our case,
is it easy to see that in our case it's correct?
STUDENT: Yeah.
PROFESSOR: Oh yeah, S of T will be,
in our case, integral from 0 to T.
We are lucky our prime of tau is a constant, which is 5.
So I'm going to have integral from 0
to T absolute value of 5 [INAUDIBLE] d tau.
And what in the world is absolute value of 5?
It's 5 integral from 0 to T [? of the ?] tau.
What is integral from 0 to T of the tau?
T. 5T.
So S is 5T.
And that's what I said before, right?
S is 5T.
S equals 5T, and T equals S over 5.
So this thing, in general, is told to us by who?
It has to match the chain rule.
It matches the chain rule.
OK.
So again, why does that match the chain rule?
We have that-- we have R-- or how
should I start, the little f, the little r, little r of S,
right?
Little r of S is little r of S of T.
How do I differentiate that with respect to T?
Well DDT of R will be R primed with respect to S.
So I'll say DRDS of S of T times DSDT.
Now what is DSDT?
DSDT was the derivative of that.
It's exactly the speed absolute value of R prime of T.
So when you prime here, S prime of T
will be exactly that, with T replacing tau.
We learned that in Calc 1.
I know it's been a long time.
I can feel you're a little bit rusty.
But it doesn't matter.
So S prime of T, DSDT will simply
be absolute value of R prime of T.
That's the speed of the original curve.
This one.
OK?
All right.
So here, when I look at DRDS, this is going to be 1.
And if you think of this as a function of T,
you have DR of S of T. Who is R of S of T?
This is R-- big R-- of T. So this
is the DRDT Which is exactly the same as R prime of T
when you put the absolute values [INAUDIBLE].
It has to fit.
So indeed, you have R prime of T, R prime of T, and 1.
It's an identity.
If I didn't put DSDT to [? P, ?] our prime of T
in absolute value, it wouldn't work out.
DSDT has to be R prime of T in absolute value.
And this is how we got, again-- are
you going to remember this without having
to re-do the whole thing?
Integral from 0 to T of R prime of T or tau d tau.
When you prime this guy with respect to T
as soon as it's positive-- when it is positive-- assume--
why is this positive, S of T?
Because you integrate from time 0 to another time
a positive number.
So it has to be positive derivative.
It's an increasing function.
This function is increasing.
So DSDT again will be the speed.
Say it again, Magdalena?
DSDT will be the speed of the original line.
DSDT in our case was 5.
Right?
DSDT was 5.
S was 5 times T. S was 5 times T.
All right.
That was a simple example, sort of, kind of.
What do we want to remember?
We remember the formula of the arc length.
Formula of arc length.
So the formula of arc length exists
in this form because of the chain rule [INAUDIBLE]
from this diagram.
So always remember, we have a composition of functions.
We use that composition of function for the chain rule
to re-parametrize it.
And finally, the drunken bug.
what did I take [INAUDIBLE] 14?
R of t.
Let's say this is 2 cosine t, 2 sine t.
Let me make it more beautiful.
Let me put 4-- 4, 4, and 3t.
Can anybody tell me why I did that?
Maybe you can guess my mind.
Find the following things.
The unit vector T, by definition R prime over R prime
of t in absolute value.
Find the speed of this motion R of t.
This is a law of motion.
And reparametrize in arclength-- this curve in arclength.
And you go, oh my God, I have a problem with a, b,c.
The is a typical problem for the final exam, by the way.
This problem popped up on many, many final exams.
Is it hard?
Is it easy?
First of all, how did I know what it looked like?
I should give at least an explanation.
If instead of 3t I would have 3, then I
would have the plane z equals 3 constant.
And then I'll say, I'm moving in circles, in circles,
with t as a real parameter,
and I'm not evolving.
But this is like, what, this like in in the avatar OK?
So I'm performing the circular motion, but at the same time
going on a different level.
Assume another life.
I'm starting another life on the next spiritual level.
OK, I have no religious beliefs in that area,
but it's a good physical example to give.
So I go circular.
Instead of going again circular and again circular,
I go, oh, I go up and up and up, and this 3t
tells me I should also evolve on the vertical.
Ah-hah.
So instead of circular motion I get a helicoidal motion.
This is a helix.
Could somebody tell me how I'm going to draw such a helix?
Is it hard?
Is it easy?
This helix-- yes, sir.
Yes.
STUDENT: [INAUDIBLE]
PROFESSOR: It's like a tornado.
It's like a tornado, hurricane, but how
do I draw the cylinder on which this helix exists?
I have to be a smart girl and remember what I learned before.
What is x squared plus y squared?
Suppose that z is not playing in the picture.
If I take Mr. x and Mr. y and I square them and I add
them together, what do I get?
STUDENT: It's the radius.
PROFESSOR: What is the radius squared?
4 squared.
I'm gonna write 4 squared because it's
easier than writing 16.
Thank you for your help.
So I simply have to go ahead and draw the frame first, x, y, z,
and then I'll say, OK, smart.
R is 4.
The radius should be 4.
This is the cylinder where I'm at.
Where do I start my physical motion?
This bug is drunk, but sort of not.
I don't know.
It's a bug that can keep the same radius, which
is quite something.
STUDENT: It's tipsy.
PROFESSOR: Yeah, exactly, tipsy one.
So how about t equals 0.
Where do I start my motion?
At 4, 0, 0.
Where is 4, 0, 0?
Over here.
So that's my first point where the bug
will start at t equals 0.
STUDENT: How'd you get 4, 0, 0?
PROFESSOR: Because I'm-- very good question.
I'm on x, y, z axes.
4, y is 0, z is 0.
I plug in t, would be 0, and I get 4 times 1, 4 times
0, 3 times 0, so I know I'm starting here.
And when I move, I move along the cylinder like that.
Can somebody tell me at what time I'm gonna be here?
Not at 1:50, but what time am I going to be at this point?
And then I continue, and I go up, and I continue and I go up.
STUDENT: [INAUDIBLE]
PROFESSOR: Pi over 2.
Excellent.
And can you-- can you tell me what
point it is in space in R 3?
Plug in pi over 2.
You can do it faster than me.
STUDENT: 0.
PROFESSOR: 0, 4 and 3 pi over 2.
And I keep going.
So this is the helicoidal motion I'm talking about.
The unit vector-- is it easy to write it on the final?
Can do that in no time.
So we get like, let's say, 30%, 30%, 30%, and 10% for drawing.
How about that?
That would be a typical grid for the problem.
So t will be minus 4 sine t.
If I make a mistake, are you gonna shout, please?
4 cosine t and 3 divided by what?
What is the tangent unit vector?
At every point in space, I'm gonna
have this tangent unit vector.
It has to have length 1, and it has
to be tangent to my trajectory.
I'll draw him.
So he gives me a field, a vector field--
this is beautiful-- T of t is a vector field.
At every point of the trajectory,
I have only one such vector.
That's what we mean by vector field.
What's the magnitude?
It's buzzing.
How did you do it?
4, 16 times sine squared plus cosine squared.
16 plus 9 is 25.
Square root of 25 is 5.
Are you guys with me?
Do I have to write this down?
Are you guys sure?
STUDENT: You plugged in 0 for t?
Is that what you did when you [INAUDIBLE]
PROFESSOR: No, I plugged 0 for t when I started.
But when I'm computing, I don't plug anything,
I just do it in general.
I said 16 sine squared plus 16 cosine squared
is 16 times 1 plus 9.
My son would know this one and he's 10, right?
16 plus 9 square root of 25.
And I taught him about square roots.
So square root of 25, he knows that's 5.
And if he knows that's 5, then you
should do that in a minute-- in a second.
All right.
So t will simply be-- if you don't simplify 1/5 minus 4 sine
t 4 cosine t 3 in the final, it wouldn't be a big deal,
I would give you still partial credit,
but what if we raise this as a multiple choice?
Then you have to be able to find where the 5 is.
What is the speed?
Was that hard for you to find?
Where is the speed hiding?
It's exactly the denominator of R.
This is the speed of the curve in t.
And that was 5.
You told me the speed was 5, and I'm very happy.
So you got 30%, 30%, 10% from the picture-- no, this picture.
This picture's no good.
STUDENT: What does the first word of c say?
Question c, what does the first word say?
PROFESSOR: The first what?
STUDENT: The word.
PROFESSOR: Reparametrize.
Reparametrize this curve in arclength.
Oh my God, so according to that chain rule,
could you guys remember-- if you remember, what is the s of t?
If I want to reparametrize in arclength integral from 0
to t of the speed, how is the speed defined?
Absolute value of r prime of t.
dt, but I don't like t, I write-- I write tau.
Like Dr. [? Solinger, ?] you know him,
he's one of my colleagues, calls that-- that's
the dummy dummy variable.
In many books, tau is the dummy variable.
Or you can-- some people even put t by inclusive notation.
All right?
So in my case, what is s of t?
It should be easy.
Because although this not a circular motion,
I still have constant speed.
So who is that special speed?
5.
Integral from 0 to t5 d tau, and that is 5t, am I right?
5t.
So-- so if I want to reparametrize this helix,
keeping in mind that s is simply 5t,
what do I have to do to get 100% on this problem?
All I have to do is say little r of s, which represents actually
big R of t of s.
Are you guys with me?
Do you have to write all this story down?
No.
But that will remind you of the diagram.
So I have R of t of s.
Or alpha of s.
And this is t of s.
t of s.
R of t of s is R of s, right?
Do you have to remind me?
No.
The heck with the diagram.
As long as you understood it was about a composition
of functions.
And then R of s will simply be what?
How do we do that fast?
We replaced t by s over 5.
Where from?
Little s equals 5t, we just computed it.
Little s equals 5t.
That's all you need to do.
To pull out t, replace the third sub s.
So what is the function t in terms of s?
It's s over 5.
What is the function t, what's the parameter t, in terms of s?
s over 5.
And finally, at the end, 3 times what is the stinking t?
s over 5.
I'm done.
I got 100% I don't want to say how much time it's
gonna take me to do it, but I think
I can do it in like, 2 or 3 minutes, 5 minutes.
If I know the problem I'll do it in a few minutes.
If I waste too much time thinking,
I'm not gonna do it at all.
So what do you have to remember?
You have to remember the formula that says s of t,
the arclength parameter-- the arclength parameter
equals integral from 0 to t is 0 to t of the speed.
Does this element of information remind you of something?
Of course, s will be the arclength, practically.
What kind of parameter is that?
Is you're measuring how big-- how much you travel.
s of t is the time you travel-- the distance
you travel in time t.
So it's a space-time continuum.
It's a space-time relationship.
So it's the space you travel in times t.
Now, if I drive to Amarillo at 60 miles an hour,
I'm happy and sassy, and I say OK, it's gonna be s of t.
My displacement to Amarillo is given
by this linear law, 60 times t.
Suppose I'm on cruise control.
But I've never on cruise control.
So this is going to be very variable.
And the only way you can compute this displacement or distance
traveled, it'll be as an integral.
From time 0, when I start driving, to time t of my speed,
and that's it.
That's all you have to remember.
It's actually-- mathematics should not be memorized.
It should be sort of understood, just like physics.
What if you take your first test, quiz,
whatever, on WeBWorK or in person, and you freak out.
You get such a problem, and you blank.
You just blank.
What do you do?
You sort of know this, but you have a blank.
Always tell me, right?
Always email, say I'm freaking out here.
I don't know what's the matter with me.
Don't cut our correspondence, either by speaking or by email.
Very few of you email me.
I'd like you to be more like my friends,
and I would be more like your tutor,
and when you encounter an obstacle,
you email me and I email you back.
This is what I want.
The WeBWorK, this is what I want our model of interaction
to become.
Don't be shy.
Many of you are shy even to ask questions in the classroom.
And I'm not going to let you be shy.
At 2 o'clock I'm going to let you ask all the questions you
have about homework, and we will do
more homework-like questions.
I want to imitate some WeBWorK questions.
And we will work them out.
So any questions right now?
Yes, sir.
STUDENT: You emailed-- did you email us this weekend
the numbers for WeBWorK?
PROFESSOR: I emailed you the WeBWorK assignment completely.
I mean, the link-- you get in and you of see it.
STUDENT: Which email did you send that to?
PROFESSOR: To your TTU.
All the emails go to your TTU.
You have one week starting yesterday until,
was it the 2nd?
I gave you a little bit more time.
So it's due on the 2nd of February at,
I forgot what time.
1 o'clock or something.
Yes, sir.
STUDENT: [INAUDIBLE] I was confused
at the beginning where you got x squared plus y squared equals
4 squared.
Where did you get that?
PROFESSOR: Oh.
OK.
I eliminated the t between the first two guys.
This is called eliminating a parameter, which was the time
parameter between x and y.
When I do that, I get a beautiful equation which
is x squared plus y squared equals 16, which tells me, hey,
your curve sits on the surface x squared
plus y squared equals 16.
It's not the same with the surface,
because you have additional constraints on the z.
So the z is constrained to follow this thing.
Now, could anybody tell me how I'm gonna write eventually--
this is a harder task, OK, but I'm
glad you asked because I wanted to discuss that.
How do I express t in terms of x and y?
I mean, I'm going to have an intersection of two surfaces.
How?
This is just practically differential geometry
or advanced calculus at the same time.
x squared plus y squared equals our first surface
that I'm thinking about, which I'm sitting with my curve.
But I also have my curve to be at the intersection
between the cylinder and something else.
And it's hard to figure out how I'm going to do the other one.
Can anybody figure out how another
surface-- what is the surface?
A surface will have an implicit equation of the type f of x, y,
z equals a constant.
So you have to sort of eliminate your parameter t.
The heck with the time.
We don't care about time, we only care about space.
So is there any other way to eliminate
t between the equations?
I have to use the information that I haven't used yet.
All right.
Now my question is that, how can I do that?
z is beautiful.
3 is beautiful.
t drives me nuts.
How do I get the t out of the first two equations?
[INTERPOSING VOICES]
Yeah, I divide them one to the other one.
So if I-- for example, I go y over x.
What is y over x?
It's tangent of t.
How do I pull Mr. t out?
Say t, get out.
Well, I have to think about if I'm not losing anything.
But in principle, t would be arctangent of y over x.
OK?
So, I'm having two equations of this type.
I'm eliminating t between the two.
I don't care about the other one.
I only cared for you to draw the cylinder.
So we can draw point by point the helix.
I don't draw many points.
I draw only t equals 0, where I'm starting over here,
t equals pi over 2, which [INAUDIBLE] gave me,
then what was it?
At pi I'm here, and so on.
So I move-- when I move one time,
so let's say from 0 to 2 pi, I should be smart.
Pi over 2, pi, 3 pi over 2, 2 pi just on top of that.
It has to be on the same line.
On top of that-- on the cylinder.
They are all on the cylinder.
I'm not good enough to draw them as being on the cylinder.
So I'm coming where I started from, but on the higher
level of intelligence-- no, on a higher level of experience.
Right?
That's kind of the idea of evolving on the helix?
Any other questions?
Yes, sir.
STUDENT: So that capital R of t is
you position vector, but what's little r of t? [INAUDIBLE]
PROFESSOR: It's also a position vector.
So practically it depends on the type of parametrization
you are using.
The dependence of time is crucial.
The dependence of the time parameter is crucial.
So when you draw this diagram, r of s
will practically be the same as R of s of t-- R of t of s,
I'm sorry.
R of t of s.
So practically it's telling me it's a combination.
Physically, it's the same thing, but at a different time.
So you look at one vector at time-- time is t here,
but s was 5t.
So I'm gonna be-- let me give you an example.
So we had s was 5t, right?
I don't remember how it went.
So when I have little r of s, that
means the same as little r of 5t,
which means this kind of guy.
Now assume that I have something like cosine 5t, sine 5t, and 0.
And what does this mean?
It means that R of 2 pi over 5 is the same as little r of 2
pi where R of t is cosine of 5t, and little r of s
is cosine of s, sine s, 0.
So Mr. t says, I'm running, I'm time.
I'm running from 0 to 2 pi over 5, and that's when I stop.
And little s says, I'm running too.
I'm also time, but I'm a special kind of time,
and I'm running from 0 to 2 pi, and I stop at 2 pi
where the circle will stop.
Then physically, the two vectors,
at two different moments in time, are the same.
Where-- why-- why is that?
So I start here.
And I end here.
So physically, these two guys have the same, the red vector,
but they are there at different moments in time.
All right?
So imagine that you have sister.
And she is five times faster than you in a competition.
It's a math competition, athletic, it doesn't matter.
You both get there, but you get there in different times,
in different amounts of time.
And unfortunately, this is-- I will do philosophy still
in mathematics-- this is the situation with many of us
when it comes to understanding a material,
like calculus or advanced calculus or geometry.
We get to the understanding in different times.
In my class-- I was talking to my old--
they are all old now, all in their 40s--
when did you understand this helix
thing being on a cylinder?
Because I think I understood it when
I was in third-- like a junior level, sophomore level,
and I understood nothing of this kind of stuff
in my freshman [INAUDIBLE] And one of my colleagues
who was really smart, had a big background,
was in a Math Olympiad, said, I think
I understood it as a freshman.
So then the other two that I was talking-- actually
I never understood it.
So we all eventually get to that point, that position,
but at a different moment in time.
And it's also unfortunate it happens about relationships.
You are in a relationship with somebody,
and one is faster than the other one.
One grows faster than the other one.
Eventually both get to the same level of understanding,
but since it's at different moments in time,
the relationship could break by the time
both reach that level of understanding.
So physical phenomena, really tricky.
It's-- physically you see where everything is,
but you have to think dynamically, in time.
Everything evolves in time.
Any other questions?
I'm gonna do problems with you next time,
but you need a break because your brain is overheated.
And so, we will take a break of 10-12 minutes.