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
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?