TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 2

Derivative of a Vector Function, Unit Tangent Vector, Principal Unit Normal Vector, Arc-length Parameterization and Curvature

PROFESSOR: Plus 1.

And next would be
between-- this is where

most people have the problem.

They thought x is
any real number.

No-- no, no, no, no, no.

You wanted a segment.

x has the values
between this value,

whatever value's on this
axis and that value.

So x equals 1, x equals
2 are the end points.

How do you write a
parameterized equation?

And that should help you
very much on the web work

homework on that problem
for such a function.

Well, you say, wait a minute.

Magdalena, this is
a linear function.

It's a piece of cake.

I have just x plus 1.

I know how to deal with that.

Yes, but I'm asking
you something else.

Rather than writing
the explicit equation

in Cartesian coordinates x and
y, tell me what time it is.

And then I'm going
to travel in time.

I want to travel in time, in
space-time, on the segment,

right?

So why if x equals
x plus 1 has what

is that-- what
parameterization has infinitely

many parameterization?

Somebody will say, ha, you told
us that it has infinitely many.

Why do you insist on one?

Which one is the most natural
and the easiest to grasp?

STUDENT: Zero to one.

PROFESSOR: Zero to one is
not a parameterization.

STUDENT: Times zero one.

PROFESSOR: So, so, so what
is the parametric equation

of a curve in general?

If I have a curve, y equals--
oh, I'll start with x.

X equals x of t
and y equals y of t

represent the two
parametric questions that

give that curve's
equation in plane--

in plane where
the i of t belongs

to a certain interval i.

That's the mysterious interval.

I don't really care
about that in general.

In my case, which one is the
most natural parametrization,

guys?

Take x to be time.

Say again, Magdalena.

Take x to be time.

And that will make
your life easier.

I take x to be time.

And then y would be time plus 1.

And I'm happy.

So the way they asked you to
enter your answer in web work

was as r of t equals-- and
it's blinking, blinking,

interactive field for you.

You say, OK, t?

T what?

And I'm not going to
solve your problem.

But your problem is similar.

Why?

Because r of t, which is the
vector equation of your y

or curve would give you the
position vector, which is what?

Wait a second.

Let me finish. x of t
times i plus y of t times

j is the definition
I gave last time.

Go ahead.

STUDENT: Where'd you get
r of t and what is it?

PROFESSOR: I already
discussed it last time.

So since I'm
reviewing today, just

reviewing today
chapter 10, I really

don't mind going over with you.

But please keep in
mind this is the first

and the last time I'm
going to review things

with you last time.

So what did you say a position
vector is for a curve?

When we talked about
the drunken bug,

we say the drunken bug is
following a trajectory.

He or she is struggling in time.

I have a given frame xyz
system of coordinates-- system

of axes of coordinates
with a certain origin.

Thank God for this origin
because you cannot refer

to a position vector
unless you have a frame--

an original frame, a position
frame, initial frame.

So r of t represents the vector
that originates at the origin o

and ends exactly at the position
of your particle at time t.

If you want, if
you hate bugs, this

is just the particle from
physics that travels in time t.

So--

STUDENT: OK, so the r of t
is represented in the parent

equation

PROFESSOR: Yes, sir.

Exactly.

In a plane where z
is 0-- so you imagine

the z-axis coming at z0.

This is the xy plane.

And I'm very happy
I have on the floor.

This bug is on the floor.

He doesn't want to know
what's the dimension.

So what's he going to do?

He's going to say plus 0 times
k that I don't care about

because the position
vector will be given by--

STUDENT: So--

PROFESSOR: --or
for a plane curve.

STUDENT: So if this
was in 3D space

and we had three equations
so it was like z equals--

is equal to 2y plus x plus 1,
then it would be-- then how

would we do that?

PROFESSOR: Let me remind us in
general the way I pointed it

out last.

R of t in general as
a position vector,

we said many things about it.

We said it is a smooth function.

What does it mean
differential role

with derivative continuous?

What did-- actually, that's c1.

What else did they say?

He said it's a regular.

It's a regular vector function.

What does it mean?

It never stops, not
even for a second.

Well, the velocity
of that is zero.

When we introduced
it-- all right,

I cannot teach the whole
thing all over again,

but I'll be happy to
do review just today.

It's going to be x of ti
plus y of tj plus z over k.

That is a way to
write it like that.

Or the simpler way to write
it as x of t, y of t, z of t.

Now, if it involves
using different notation,

I want to warn you about that.

Some people like to put
braces like angular brackets.

Or some people like
because it's a vector.

And that's the way they define
vector Some people like just

round parentheses.

This is more practically.

These are the coordinates
of a position vector

with respect to the ijk frame.

So since we talked
about this already,

some simple examples
have been given.

One of them was
a circling plane,

another circling plane
of a different speed,

a segment of a line.

This is the segment of a line.

What else have we discussed?

We discuss about
something wilder,

which was the helix
at different speeds?

All right, so very good
question for him was-- so

is this x of tt?

Yes.

Is this y of tt plus 1?

Yes.

Is this z of t 0 in my case?

Precisely

STUDENT: So if you
gave value to z,

what would you chose to
make t parameterized?

PROFESSOR: OK, t in
general, if you are moving,

you have an infinite motion
that comes from nowhere,

goes nowhere, right?

OK, then you can say
t is between minus

infinity plus infinity.

And that's your i--

STUDENT: But what I'm saying--

PROFESSOR: But-- but in
your case-- in your case,

you think oh, I know
where I'm starting.

So to that equals
to 1, t must be 1.

So I start my
movement at 1 second

and I end my movement at 2
seconds where x will be 2,

and y will be 3.

STUDENT: Well, I mean--
so you said x equals t.

You took that from
the y equals x plus 1.

If you had the third
variable t, what would you--

PROFESSOR: It's not
a third variable.

It's the time parameter.

So I work in three
variables-- xyz in space.

Those are my space coordinates.

The space coordinates
are function of time.

So it's all about physics.

So mathematics sometimes
becomes physics.

Thank God we are sisters,
even step-sisters.

X is a function of t.

Y is a function of t.

Z is a function of t.

Right?

Am I answering your
question or maybe

I didn't quite understand the--

STUDENT: Well, I understand
how to parameterize

the idea of a plane.

How do you do it
in space though?

PROFESSOR: In space-- in
space, you're already here.

So if you want to ride this
not in plane but in space,

your parametric equation is
ti plus t plus 1j plus 0k,

for this example,
anywhere in r3.

We live in r3.

All righty?

We live in r3.

OK, let me give
you more examples.

Because I think I'm
running out of time.

But I still have to
cover the material,

eventually get somewhere.

However, I want you to see
more examples that will help

you grasp this notion better.

So guys, imagine that
we have space r3-- that

could be rn-- in
which I have an origin

and I have a [INAUDIBLE].

And somebody gives
me a position vector

for a motion that's
a regular curve.

And that's x of tri plus
y is tj plus z of tk.

And since his question
is a very valid one,

let's see what happens
in a later case.

So I'm going to deviate a
little from my lesson plan.

And I say let us be
flexible and compare

that with the inner curve.

Because in the
process of comparison,

you learn a lot more.

If I were to be right above
my [INAUDIBLE] like that.

So this is the spacial curve in
our three imaginary trajectory

run of a bug or a particle.

As we said, this is the
planar curve-- planar,

parametrized curve in r2.

What's different?

What do we know about them?

We clearly know section 10.2.

What I hate in general
about processors

is if they are way
too structured.

Mathematics cannot be talking
sections where you say, oh,

section 10.2 is only about
velocity and acceleration.

But section 10.4 is
about tangent unit vector

and principle normal.

Well, they are related.

So it's only natural when
we talk about section 10.2

acceleration and velocity
that from acceleration, you

have a induced line to tangent
unit vector-- tangent unit

vector.

And later on, you're going
to compare acceleration

with a normal principal vector.

Sometimes, they
are the same thing.

Sometimes, they are
not the same thing.

It's a good idea to see
when they are the same thing

and when they are not.

So in section 10.4, we
will focus practically

or t, n, and v, the Frenet
frame and its consequences

on curvature, we already
talked about that a little bit.

In 10.2, practically,
we didn't cover much.

I only told you about
velocity, acceleration.

However, I would like
to review that for you.

Because I don't want
to risk losing you.

I'm going to lose
some of you anyway.

Two people said this
course is too hard for me.

I'm going to drop.

You are free to drop and I
think it's better for you

to drop than struggle.

But as long as you can still
learn and you can follow,

you shouldn't drop.

So try to see exactly
how much you can handle.

If you can handle just the
regular section of calc three,

go to that regular section.

If you can handle more, if
you are good at mathematics,

if you have always
been considered bright

in mathematics in high
school, let us stay here.

Otherwise, go.

Don't stay.

All right, so the
velocities are prime of t.

The acceleration is
our double prime of t.

We have done that last time.

We were very happy.

What would happen in a
planar curve seen on 2?

The same thing, of course,
except the last component

is not there.

It's part of ti
plus y prime of tj.

And there is a 0k in both cases.

So all these are factors.

At times, I'm not going
to point that out anymore.

The derivation goes
component-wise.

So if you forgot how to derive
or you want to drink and derive

or something, then you
don't belong in this class.

So again, make sure you know
the derivations and integrations

really well.

I'm going to work
some examples out just

to refresh your memory.

But if you have struggled with
differentiation and integration

in Calc 1, then you do not
do belong in this class.

All right, let's
see about speed.

It's about speed.

It's about time.

It's about time to remember
what the speed was.

The speed was the absolute
value or the magnitude.

It's not an absolute
value, but it's a magnitude

of the velocity factor.

This is the speed.

And the same in this case.

If I want to write an explicit
formula because somebody

asked me by email, can I write
an explicit formula, of course.

That's a piece of cake and you
should know that from before.

X prime of t squared plus
y prime of t squared plus z

prime of t squared
under the square root.

I was not going to insist
on the planar curve.

Of course the planar curve will
have a speed that all of you

know about.

And that's going to be
square root of x prime of t

squared plus y root
prime of t squared.

You should do your own thinking
to see what the particular case

will become.

However, I want to
see if you understood

what derives from
that in the sense

that you should know the
length of a arc of a curve.

What is the length
of an arc of a curve?

Well, we have to look back
at Calculus 2 a little bit

and remember that the length of
an arc of a curve in Calculus 2

was given by, what?

So you say, well, yeah.

That was a long time ago.

Well, some of you
already don't even

remember that as being integral
from a to b of square root of 1

plus 1 prime of x squared dx.

And you were freaking
out thinking, oh my god,

I don't see how this
formula from Calc 2,

the arc of a curve, had
you travel between time

equals a and time equals b
will relate to this formula.

So what happened in Calc 2?

In Calc 2, hopefully,
you have a good teacher.

And hopefully,
you've learned a lot.

This is between a and b, right?

What did they teach
you in Calc 2?

They taught you that
you have to take

integral from a to b
of square root of 1

plus y prime of x squared ds.

Why?

You never asked
your teacher why.

That's bad.

You should do that.

You should ask why every time.

They make you swallow a
formula via memorization

without understanding
this is the speed.

And now I'm coming
with the good news.

I have a proof of that.

I know what speed
means when I'm moving

along the arc of
a curve in plane.

OK, so what is the distance
travelled between time equals A

and time equals B?

It's going to be integral form
a to be of the speed, right?

This is the same one I'm
driving from-- level two--

Amarillo or anywhere else.

There.

Now, what they showed
you and they fooled you

into memorizing that is just
a consequence of this formula

because of what he said.

Why?

The most usual
parameterization is

going to be y of t equals t--
I'm sorry, x of t equals vxst

and y of t equals y of t.

So, again x is time.

In many linear curves, you can
take x to be time, thank God.

And then your parametrization
will be t comma y of t.

Because x is t.

And x prime of t will be 1.

Y prime of t will
be y prime of t.

When you take them, squish
them, square them, sum them up,

you get exactly this one.

But you notice
this is the speed.

What is this the speed?

Of some value over prime
of t, which is speed.

You see that what they forced
you to memorize in Calc 2

is nothing but the speed.

And I could change
the parameterization

to something more general.

Now, can I do this
parameterization for a circle?

No.

Why not?

I could, but then
I'd have to split

into the upper
part and lower part

because the circle
is not a graph.

So I take t between
this and that

and then I have square root
of 1 minus t squared on top.

And underneath, I have
minus square root of 1

minus t squared.

So I split the poor circle
into a graph and another graph.

And I do it separately.

And I can still apply that.

But only a fool
would do that, right?

So what does a smart
mathematician do?

A smart mathematician
will say, OK,

for the circle, x is
cosine t, y is sine t.

And that is the
parameterization I'm

going to use for this formula.

And I get speed 1.

And I'm going to
be happy, right?

So it's a lot easier
to understand what

a general parameterization is.

What is the length of an arc
of a curve for a curving space?

There's the bug.

Time equals t0.

He's buzzing.

And after 10 seconds,
he will be at the end.

So it goes, [BUZZING] jump.

OK, how much did he travel?

Integral from a to b of square
root of x prime of t squared

plus y prime of t
squared plus z prime of t

squared-- no matter what that
position vector x of ty of t0

give us.

So you take the coordinates
of the velocity vector.

You look at them.

You square them.

You add them together.

You put them under
the square root.

That's going to be the speed.

And displacement is
integral of speed.

When you guys learned
in school, your teacher

oversimplified the things.

What did your teacher
say in physics?

Space equals speed times time.

Say it again.

He said space traveled
is speed times time.

But he assumed the speed
is constant or constant

on portions-- like,
speedswise constant.

Well, if it's a
constant, the speed

will get the heck out of here.

And then the space will
be speed times b minus a.

But b minus a is delta t.

In mathematics, in physics,
we say b minus a is delta t.

That's the interval of time that
the bug travels or the particle

travels.

So he or she was right.

Space is speed times
time, but it's not like

that unless the
speed is constant.

So he oversimplified
your knowledge

of mathematics and physics.

Now you see the truth.

Space is integral of speed.

OK, now we understand.

And I promised you last
time that after reviewing,

I didn't even say I would review
anything from 10.2 and 10.4.

I promised you more.

I promised you that I'm going
to compute something that's

out of 10.4 which is called
a curvature of a helix

in particular.

Because we looked at curvature
of a parametric curve

in general.

I want to organize the material
of review from 10.2 and 10.4

in a big problem just like
you will have in the exams,

in the midterm,
and in the final.

I don't want to scare you.

I just want to
prepare you better

for the kind of multiple
questions we are going to have.

So let me give you a
funny looking curve.

I want you to think about
it and tell me what it is.

a and b are positive numbers.

a cosine ba sine t bt will be
some sort of funny trajectory.

You are already
familiar to that.

Last time, I gave you an example
where a was 4-- oh my god,

I don't even remember.

You'll need to help me.

[INAUDIBLE]

STUDENT: 4, 4, 3.

PROFESSOR: I took those because
they are Pythagorean numbers.

So what does it mean?

3 squared plus 4 squared
equals 5 squared.

I wanted the sum of them
to be a perfect square.

So I was playing games.

You can do that for any a and b.

Now, what do I want?

A-- like in 10.2 where
you write r prime of t,

rewrite that double prime of t.

So it's a complex problem.

In b, I want you to
find t and r prime

of t over-- who
remembers the formula?

I shouldn't have
spoon-fed you that.

STUDENT: Absolute--

PROFESSOR: Absolute
magnitude, actually.

It's more correct to
say magnitude, right?

Very good.

And what else did I
spoon-feed you last name?

I spoon-fed you n.

Let's compute n as well
as part of the problem

t prime t over t
prime of t magnitude.

STUDENT: So you're looking
for the tangent unit vector.

PROFESSOR: Tangent unit vector?

STUDENT: And then
you're looking for--

PROFESSOR: Yes, sir.

And-- OK, don't you
like me to also give you

something like a grading
grid, how much everything

would be worth.

Imagine you're taking an exam.

Why not put yourself
in an exam mode

so you don't freak out
during the actual exam?

C will be another
question, something smart.

Let's see-- reparameterize an
arc length to a plane, a curve,

rho of s.

Why not r of s like some
people call-- use it

and some books use it?

Because if you're
reparameterizing s,

it's going to be the
same physical limits

but a different function.

So if you remember the
diagram I wrote before,

little r is a function that
comes from integral i time

integral 2r3 and rho would
be coming from a j to r3.

And what is the
relationship between them?

This is t goes to s and
this is s goes to t.

What is d I'm asking you?

Well, if you're d
and c, of course

you know what the arc
length parameter will be.

It's going to be integral
from 0 to t or any t0 here

of the speed-- of the speed
of the original function here

of t.

The tau-- maybe tau is better
than the dummy variable t.

And e I want.

You say, how much
more do you want?

I want a lot.

I'm a greedy person.

I want the curvature
of the curve.

And you have to remind me.

Some of you are very good
students, better than me.

I mean, I'm still behind
with a research course

that I have--
research paper i have

to read in two days in biology.

But this curvature of the
curve had a very simple formula

that we all love.

For mathematicians, it's a
piece of cake to remember it.

K-- that's what I like
about being a mathematician.

I don't need a good memory.

Now I remember why I didn't
go to medical school--

because my father
told me, well, you

should be able to remember all
the bones in a person's body.

And I said, dad, do you
know all these names?

Yes, of course.

And he started telling me.

Well, I realized that I
would never remember those.

But I remember this
formula which is r rho.

In this case, if
our r is Greek rho,

it's got to be rho
double prime of what?

of S. Is this
correct, what I wrote?

No.

What's missing?

The acceleration and arc length
but in magnitude because that's

a vector, of course.

This is the scalar function.

Anything else you
want, Magdalena?

Oh, that's enough.

All right, so I want
to know everything

that's possible to know about
this curve from 10.2 and 10.4

sections.

10.3-- skip 10.5.

Skip-- you're happy about it.

Yes sir.

STUDENT: For the
parameter on v, is it a t?

And what's the integral?

What's on the bottom.

PROFESSOR: Ah, that value
erased when I wrote that one.

It was there-- t0.

So I can start with any fixed
t0 as my initial moment in time.

I would like my
initial moment in time

to be 0 just to make
my things easier.

Are we ready to solve
this problem together?

I think we have just
about the exact time

we need to do everything.

First of all, you have to tell
me what kind of curve this is.

Of course you know because
you were here last time.

Don't skip classes because
you are missing everything out

and then you will have
to drop or withdraw.

So don't skip class.

What was that from last time?

It was a helix.

I'm going to try and redraw it.

I know I'm wasting
my time, but I would

try to draw a better curve.

Ah, what's the equation
of the cylinder?

[CLASS MURMURS]

PROFESSOR: Huh?

What's the equation
of the cylinder?

That's a quadratic
that you are all

familiar with on which on my
beautiful helix is sitting on.

I taught you the
trick last time.

Don't forget it.

STUDENT: a over 4 cosine of
t squared plus 8 over 4 sine

of t squared.

PROFESSOR: So we do
that-- very good.

X is going to be-- let
me right that down.

X is cosine.

Y is a sine t.

And that's exactly
what you asked me.

And z is bt.

And then what I need to do
is square these guys out

as you said very well.

I don't care about this 2z.

He's not in the picture here.

X squared plus y squared will be
a squared, which means I better

go ahead and draw a circle
of radius a on the bottom

and then build
my-- oh my god, it

looks horrible-- the cylinder
based on that circle.

Guys, it's now straight.

I'm sorry.

I mean, I can do
better than that.

OK, good.

So I'm starting at what point?

I'm starting at a0
0 time t equals 0.

We discussed that last time.

I'm not going to repeat.

I'm starting here,
and two of you

told me that if t
equals phi over two,

I'm going to be here
and so on and so forth.

If I ask you one more thing
for extra credit, what

is the length of the trajectory
traveled by the bug, whatever

that is, between time t equals
0 and time t equals phi over 2.

I'd say that's extra credit.

So, oh my god, 20%, 20%, 20%,
20%, 20%, and 10% for this one.

And if you think why does she
care about the percentages

and points, you will
care and I care.

Because I want you to see how
you are going to be assessed.

If you have no idea how
you're going to assessed,

then you're going to be
happy and i will be unhappy.

All right, so for 20%
credit on this problem,

we want to see r prime of t
will be, r double prime of t

will be.

That's going to be
a piece of cake.

And of course, it's maybe the
reward is too big for that,

but that's life.

Minus a sine t a equals time
t and d, d as in infinity.

So I have an infinite
cylinder on which

I draw an infinite
helix coming from hell

and going to paradise.

So between minus infinity and
plus infinity, there's a guy.

I'm going to draw a
beautiful infinite helix.

And this is what I posted here.

What's the acceleration
of this helix?

Minus a cosine t
minus 5 sine t and 0.

Question, quick
question for you.

Will-- you guys are fast.

Maybe I shouldn't
go ahead of myself.

Nobody's asking me what
the speed is right now.

So why would I do something
that's not on the final, right?

So let's see.

T, you will have to compute
the speed when you get to here.

But wait until we get there.

What is mister t?

Mister t will be
the tangent vector.

So the velocity is going like
a crazy guy, long vector.

The normal unit vector says,
I'm the tangent unit vector.

I'm always perpendicular
to the direction.

I'm of length 1.

STUDENT: I thought the tangent
was parallel to the direction.

PROFESSOR: Yes, the
direction of motion is this.

Look at me.

This is my direction of motion.

And the tangent is--

STUDENT: You said it was--

PROFESSOR: --in the
direction of motion.

STUDENT: But you said
it was perpendicular.

PROFESSOR: I said perpendicular?

Because I was thinking
ahead of myself and n.

And I apologize.

So thank you for correcting me.

So t is the tangent unit vector.

I'm going along the
direction of motion.

And it's going to be
perpendicular to t.

And that's the principal
normal unit vector--

principal normal unit vector.

And you're going to tell
me what I'm having here.

Because I don't know.

T is minus a sine
t a equals sine t

and v divided by the speed.

That's why I was
getting ahead of myself

thinking about the speed that
you'll need later on anyway.

But you already
need it here, right?

Because the denominator of this
expression will be the speed.

Magnitude of r
prime-- what is that?

Piece of cake--
square root of the sum

of the squares of square root
of a squared plus b squared.

Piece of cake.

I love it.

So what do I notice?

That although I'm going
on a funny curve which

is a parametrized helix,
I expect some-- maybe

I expected something
wild in terms of speed.

Well, the speed is constant.

STUDENT: [INAUDIBLE] the square
root of negative a sine t

squared--

PROFESSOR: And what are those?

A squared sine squared plus c
squared cosine squared plus b

squared, right?

And what sine squared
plus cosine squared

is 1 [INAUDIBLE].

So you get a squared
plus b squared.

Good-- now let's
go on and do the n.

The n will be t prime
over magnitude of t prime.

When you do t prime,
you'll say, wait a minute.

I have square root of a squared
plus b squared on the bottom.

On the top, I have minus equals
sine t minus a sine t and 0.

We have time to finish?

I think.

I hope so.

Divided by-- divided by the
magnitude of this fellow.

I will say, oh, wait a minute.

The magnitude of this fellow
is simply the magnitude

of this over this magnitude.

And we've seen last time this is
the magnitude of this vector a,

right?

Good.

Now, so the answer will
be n is going to be a unit

vector, very nice friend
of yours, minus cosine t

minus sine t0.

Can you draw a conclusion about
how I should draw this vector?

You see the component in k is 0.

So this vector
cannot be like that--

cannot be inclined with
respect to the horizontal.

Yes sir.

STUDENT: So what happens
to-- down there-- square root

of a squared plus b squared?

PROFESSOR: They simplify.

This is division.

STUDENT: Oh, OK.

PROFESSOR: So this simplifies
with that and a simplifies

with a.

I should leave some
things as an exercise,

but this is an obvious one so I
don't have to explain anything.

Minus cosine t
minus sine t-- if do

you guys imagine what that is?

Remember your washer and dryer.

So if you have an acceleration
that's pointing inside

like from a centrifugal force,
the corresponding acceleration

would go pointing
inside, not outside.

That's going to be exactly
minus cosine t minus sine t0.

So the way I should draw the
n would not be just any n,

but should be at every
point a beautiful vector

that's horizontal and is
moving along the helix.

My elbow is moving
along the helix.

See my elbow?

Where's my elbow moving?

I'm trying.

I swear, I won't do it that way.

So this is the helix and this
is the acceleration, which

is acceleration and the normal
unit vector in this case

are co-linear.

They are not
co-linear in general.

But if the speed is a
constant, they are co-linear.

The n and the acceleration.

Yes, sir?

STUDENT: How do you know it's
pointing in the central axis?

I thought it was--

PROFESSOR: Good question.

Good question.

Well, yeah.

Let's see now.

Plug in t equals 0.

What do you have?

Minus cosine 0 minus 1 0, 0.

So you guys would have to
draw the vector minus 1, 0, 0.

That's minus i, right?

So when I start here, this
is my n-- from here to here,

from the particle to the insid.

So I go on that.

All right, so this is the
normal principal vector.

I'm very happy about it.

STUDENT: Isn't the normal
principal vector is the-- is it

the derivative of
t, or is just--

PROFESSOR: It was
by definition--

it's in your notes-- t prime
over the magnitude of the--

STUDENT: So then did
you-- why didn't you

take a derivative of t prime?

PROFESSOR: I did.

STUDENT: Yeah, I know.

I see you took a
derivative of t of--

PROFESSOR: This is t prime.

STUDENT: OK.

PROFESSOR: And this is
magnitude of t prime.

Why don't you try
this at home, like,

slowly until you're sure
this is what yo got?

So I did-- I did
the derivative of i.

STUDENT: I saw that.

PROFESSOR: This
is a [INAUDIBLE].

STUDENT: You said you were--

PROFESSOR: So when we
have t times a function

and we prime the
product, k goes out.

Lucky for us--
imagine how life would

be if it weren't like that.

So the constant
that falls out is

1 over square root
of what I derived.

And then I have to derive
this whole function also.

So I would suggest to
everybody, not just to yo--

go home and see if
you can redo this

without looking in your notes.

Close the damn notes.

Open and then you look at--
it's line by line, line by line

all the derivations.

Because you guys will have to
do that yourselves in the exam,

either midterm or final anyway.

Reparameterizing arc lengths
to obtain a curve-- I

still have that to
finish the problem.

Reparameterizing arc length
to obtain a curve rho of s.

How do we do that?

Who is s?

First of all, you should
start with the s and then

reparameterize.

So you say, hey, teacher.

You try to fool me, right?

I want s to be grabbed
as a parameter first.

And then I will reparameterize
the way you want me to do that.

So s of t will be
integral from 0

to t square root of a
prime a squared times

b squared b tau-- d tau, yes.

S of t will be, what?

Who's helping me on that?

Because I want you to be awake.

Are you guys awake?

[CLASS MURMURS]

PROFESSOR: The
square root of that

is a constant gets out times t.

So what did I tell you when
it comes to these functions?

I have to turn my
head badly like that.

This was the alpha t or s of t.

And this was t of s, which
is the inverse function.

I'm not going to
write anything stupid.

But this is practically the
inverse function of s of t.

I told you it was easiest t do.

Put it here.

T has to be replaced
by, in terms of s,

by a certain expression.

So who is t?

And you will do that
in no time in the exam.

T pulled out from
there will be just

s over square root a
squared plus b squared

s over square root a
squared plus b squared

and s over square root.

OK?

So can I keep the
notation out of s?

No.

It's not mathematically
correct to keep r of s.

Why do the books
sometimes by using

multiplication keep r of s?

Because the books are
not always rigorous.

But I'm trying to be rigorous.

This is an honors class.

So How do I rewrite
the whole thing?

r of t, who is a function
of s, t as a function of s

was again s over square root
a squared plus b squared

will be renamed rho of s.

And what is that?

That is a of cosine
of parentheses

s over square root a
squared r b squared, comma,

a sine of s over square root
a squared plus b squared

and b times s over square
root a squared plus b squared.

So what have I done?

Did I get my 20%?

Yes.

Why?

Because I reparameterized
the curve.

Did I get my other 20%?

Yes, because I told
people who s of t was.

So 20% for this box and
20% for this expression.

So what have I done?

On the same physical curve, I
have slowed down, thank God.

You say, finally, she's
slowing down, right?

I've changed this speed.

On the contrary, if a would
be 4 and be would be 3,

I increase my speed
multiple five times, right?

So you can go back and
forth between s and t.

What does s do compared to t?

It increases the
speed five times.

Yes sir.

STUDENT: So when
you reparameterize,

it's just basically the
integral from 0 to t

of whatever
[INAUDIBLE] of tau is.

PROFESSOR: Exactly.

So my suggestion to all
of you-- it took me a year

to understand how
to reparameterize

because I was not smart enough
to get it as a freshman.

I got an A in that class.

I didn't understand anything.

As a sophomore, I really--
because sometimes, you know,

you can get an A without
understanding things in there.

As a sophomore, I
said, OK, what the heck

was that reparameterization?

I have to understand that
because it bothers me.

I went back.

I took the book.

I learned about
reparameterization.

Our book, I think,
does a very good job

when it comes to
reparameterizing.

So if you open the 10.2 and
10.4, you have to skip-- well,

am I telling you to skip 10.3?

That's about ballistics.

If you're interested in
dancing and all sorts of,

like, how the bullet
will be projected

in this motion or that
motion, you can learn that.

Those are plane curves that
are interested in physics

and mathematics.

But 10.3 is not part of
them and they are required.

Read 10.2 and 10.4.

You understand this much better.

Yes, ma'am.

STUDENT: Will the midterm
or the final just be, like,

a series problems, or
will it be anything--

PROFESSOR: This is going to
be like that-- 15 problems

like that.

STUDENT: Will it be
anything, like, super

in depth like the extra credit?

PROFESSOR: That-- isn't
that in-depth enough?

OK, this is going
to be like that.

So I would say at this
point, the way I feel,

I feel that I am ready to
put extra credit there.

My policy is that
I read everything.

So even if at this point,
you say extra credit.

And you put it at
the end for me.

Say, look, I'm doing
the extra credit here.

Then I'll be ready and I'll
say, OK, what did she mean?

Length of the arc?

Which arc?

From here to here is
ready to be computed.

And that's going to be-- you
can include your extra credit

inside the actual problem.

I see it.

STUDENT: Yes.

PROFESSOR: Don't worry.

STUDENT: Would it
just be as like-- just

like the casual problem
on the test or midterm

or whatever-- would it
be, like, an extra credit

problem in itself?

I know there will
be extra credit,

but the kind of proving--

PROFESSOR: That is--
that is decided together

with the course coordinator.

The course coordinator
himself said

that he is encouraging us to
set up the scale so that if you

all the problems that
are written on the exam,

you get something like 120%
if everything is perfect.

STUDENT: OK, if we can--

PROFESSOR: So it's sort
of in-built in that-- yes.

STUDENT: If we can
do the web work,

is that a good indication of--

PROFESSOR: Wonderful.

That's exactly--
because the way we

write those problems
for the final,

we pull them out of the web work
problems we do for homework.

So a square root
of a squared times

b squared times pi over 2--
so what have I discovered?

If I would take a
piece of that paper

and I would measure from
this point to this point

how much I traveled in
inches from here to here,

that's exactly that square root
of- this would be like a 5.

That's 3.1415 divided by 2.

Yes, sir.

STUDENT: So in the
interval of a squared plus

b squared, I know
that that's supposed

to be the interval
the magnitude of r--

PROFESSOR: The speed--
integral of speed?

STUDENT: Right.

So which is the r prime, right?

PROFESSOR: Yes, sir.

STUDENT: OK, so r prime was--

PROFESSOR: Velocity.

STUDENT: --a sine--
or negative a sine t,

a cosine t, and then b?

So where did the square root
of a squared plus b squared

come from?

STUDENT: That's from the--

PROFESSOR: I just erased it.

OK, so you have minus i-- minus
a sine b equals sine p and d.

When you squared them,
what did you get?

He has the same thing.

STUDENT: So that's just--

PROFESSOR: The square of that
plus the square root of that

plus the square root of that.

STUDENT: So it's just like a 2D
representation of the top one.

STUDENT: This side--

PROFESSOR: I just need the
magnitude of r prime, which

is this p, right?

STUDENT: Right.

PROFESSOR: The
magnitude of this is

the speed, which is square root
of a squared plus b squared.

Is that clear?

STUDENT: Yes.

PROFESSOR: I can
go on if you want.

So a square root of-- the sum
of the squares of this, this,

and that is exactly
square of [INAUDIBLE].

Keep this in mind as an example.

It's an extremely important one.

It appears very frequently
in tests-- on tests.

And it's one of the
most beautiful examples

in applications of
mathematics to physics.

I have something
else that was there.

Yes ma'am

STUDENT: I was just going to
ask if you want to curvature.

PROFESSOR: Eh?

STUDENT: The letter--

PROFESSOR: Curvature?

STUDENT: Curvature.

PROFESSOR: That's
exactly what I want.

And when I said I had something
else for 20%, what was k?

K was rho double prime
of s in magnitude.

So I have to be smart enough
to look at that and rho of s.

And rho of s was
the thing that had

here-- that's going to be
probably the end of my lesson

today.

Since you have so many
questions, I will continue.

I should consider--
the chapter is finished

but I will continue with a
deeper review, how about that,

on Tuesday with more problems.

Because I have the feeling that
although we covered 10.1, 10.2,

10.4, you need a
lot more examples

until you feel comfortable.

Many of you not,
maybe 10 people.

They feel very comfortable.

They get it.

But I think nobody will be
hurt by more review and more

examples and more applications.

Now, who can help me
finish my goal for today?

Is this hard?

This is rho of s.

So you have to tell me with
the derivation, is it hard?

No.

Minus a sine of the
whole thing times 1

over square root of a squared
plus b squared because I'm

applying the chain rule, right?

Let me change color.

Who's the next guy?

A Cosine of s over square
root a squared plus b squared.

I'm now going to leave
you this as an exercise

because you're going to haunt
me back ask me why I got this.

So I want to make it very clear.

B times 1 over square root
a squared by b squared.

So are we happy with this?

Is this understood?

It's a simple derivation
of the philosophy.

We are not done.

We have to do the acceleration.

So the acceleration
with respect to s

of this curve where s was
the arc length parameter

is real easy to compute
in the same way.

What is different?

I'm not going to
write more explicitly

because this should be
visible for everybody.

STUDENT: x [INAUDIBLE].

PROFESSOR: Good,
minus a over-- I'll

wait for you to
simplify because I don't

want to pull two roots out.

STUDENT: A squared--

PROFESSOR: A squared
plus b squared.

And why is that, [INAUDIBLE]?

Because you have once and
twice from the chain rule.

So again, I hope you guys don't
have a problem with the chain

rule so I don't have to
send you back to Calculus 1.

A over a squared times b
squared with a minus-- why

with a minus?

Somebody explain.

STUDENT: Use the
derivative of cosine.

PROFESSOR: There's a cosine
and there's a minus sine.

From deriving, I have
a minus and a sine.

And finally, thank
God, the 0-- why 0?

Because I have a constant that
I'm deriving with respect to s.

Is it hard to see what's up?

What's going out?

What is the curvature
of the helix?

A beautiful, beautiful
function that

is known in most of
these math, calculus,

multivariable calculus and
differential geometry classes.

What did you get?

Square root of sum of the
squares of all these guys.

You process it.

That's very easy.

Shall I write it down?

Let me write it down
like a silly girl--

square root of a squared,
although I hate when I cannot

go ahead and simplify it.

But let's say there's
this little baby thing.

Now I can say it's
a over a squared

plus b squared-- finally.

So I'm going to ask
you a few questions

and then I'm going
to let you go.

It's a punishment
for one minute.

OK, if I have the
curve we had before,

the beautiful helix with
a Pythagorean number

like 3 cosine t, 3
sine t, and 4t, what

is the curvature of that helix?

STUDENT: 3 over 5--

PROFESSOR: 3 over 5, excellent.

How about my helix?

What if I changed the numbers
in web work or on the midterm

and I say it's going
to be-- it could even

be with a minus, guys.

It's just the way you travel
it would be different.

So whether I put
plus minus here,

you will try on
different examples.

Sometimes if we put
minus here or minus here,

it really doesn't matter.

Let's say we have
cosine t sine t and t.

What's the curvature of
that parametrized curve?

1 over--

STUDENT: 2.

PROFESSOR: 1 over 2-- excellent.

So you got it.

So I'm proud of you.

Now, I want to do more examples
until you feel confident

about it.

I know I got most of you to
the point where I want it.

But you need more
reading definitely

and you need to
see more examples.

Feel free to read
the whole chapter.

I would-- if you don't have
time for 10.3, skip it.

10.5 is not going
to be required.

So if I were a student,
I'd go home, open the book,

read 10.1, 10.2,
10.4, close the book.

It's actually a lot less
than you think it is.

If you go over the most
important formulas,

then you are ready
for the homework.

The second homework is due when?

February 11.

You guys have plenty of time.

Rather than going to the
tutors, ask me for Tuesday.

On Tuesday, you'll have plenty
of time for applications.

OK, have a wonderful weekend.

Don't forget to email when
you get in trouble, OK?