TTU Math2450 Calculus3 Sec 10.1

Vector Value Functions

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.

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,
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

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,

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.

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.