-
-
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?
-