## TTU Math2450 Calculus3 Sec 10.1

• 0:00 - 0:01
• 0:01 - 0:05
I love this model.
• 0:05 - 0:06
Again, thank you, Casey.
• 0:06 - 0:10
I'm not going to take
any credit for that.
• 0:10 - 0:12
So if you want to
imagine the stool
• 0:12 - 0:17
I was talking about as
a bamboo object, that
• 0:17 - 0:21
is about the same thing,
at the same scale, compared
• 0:21 - 0:28
to the diameter and the height,
scaled or dialated five times.
• 0:28 - 0:30
Uniform, no alterations.
• 0:30 - 0:35
And one can sit on it,
[? and circle, ?] to sit on it.
• 0:35 - 0:40
Now, as you see this is
a doubly ruled surface.
• 0:40 - 0:41
And you say, oh wait a minute.
• 0:41 - 0:45
You said rule surface, why all
of a sudden, why doubly ruled
• 0:45 - 0:46
surface?
• 0:46 - 0:52
Because it is a surface
that is ruled and generated
• 0:52 - 0:58
by two different one
parameter families.
• 0:58 - 1:00
Each of them has a
certain parameter
• 1:00 - 1:03
and that gives them continuity.
• 1:03 - 1:04
So you have two
families of lines.
• 1:04 - 1:07
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One family is in this direction.
• 1:10 - 1:11
Do you see it?
• 1:11 - 1:15
So these lines-- this
line is in motion.
• 1:15 - 1:17
It moves to the right, to
the right, to the right,
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and it generated.
• 1:20 - 1:23
And the other family
of lines is this one
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in the other direction.
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You have a continuity
parameter for each of them.
• 1:28 - 1:34
So you have to imagine
some real parameter going
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along the entire
[? infinite real ?] axis.
• 1:37 - 1:40
Or along a circle which would
be about the same thing.
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But in any case, you have
a one parameter family
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and another one
parameter family.
• 1:49 - 1:52
Both of them are
together generating
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this beautiful
one-sheeted hyperboloid.
• 1:56 - 2:01
It's incredible because you
see where these sort of round,
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but if you go towards the
ends, it's topologically
• 2:07 - 2:08
a cylinder or a tube.
• 2:08 - 2:15
But if you look towards
the end, the two ends
• 2:15 - 2:19
will look more straight.
• 2:19 - 2:24
And you will see the
straight lines more clearly.
• 2:24 - 2:28
So imagine that you
have a continuation
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to infinity in this direction,
and in the other direction.
• 2:31 - 2:36
And this actually should be an
infinite surface in your model.
• 2:36 - 2:39
You're just cutting it
between two z planes,
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so you have a patch of a
one-sheeted hyperboloid.
• 2:42 - 2:43
Yeah, the one-sheeted
hyperboloid
• 2:43 - 2:47
that we wrote last time,
do you guys remember
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x squared over a
squared plus y squared
• 2:50 - 2:53
over b squared minus z squared?
z should be this [INAUDIBLE].
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Minus z squared over
c squared minus 1
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equals 0 is an
infinite surface area.
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At both ends you keep going.
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Very beautiful.
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Thank you so much.
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I appreciate.
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And keep the brownies.
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No, then I have to pay more.
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Than I have to pay money.
• 3:17 - 3:19
out of [INAUDIBLE].
• 3:19 - 3:20
PROFESSOR: When
• 3:20 - 3:23
[LAUGHTER]
• 3:23 - 3:24
Really?
• 3:24 - 3:24
When is it?
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STUDENT: February 29.
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PROFESSOR: Oh, it's coming.
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[INTERPOSING VOICES]
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STUDENT: It's coming
in a year, too.
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PROFESSOR: That was a smart one.
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Anyway, I'll remember that.
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I appreciate the gift very much.
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And I will cherish
it and I'll use it
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with both my
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and my graduate students who
are just learning about-- some
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of them don't know the
one-sheeted hyperboloid model,
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but they will learn about it.
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Coming back to our lesson.
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I announced Section 10.1.
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Say goodbye to
quadrant for a while.
• 4:07 - 4:10
I know you love them,
but they will be there
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for you in Chapter 11.
• 4:11 - 4:13
They will wait for you.
• 4:13 - 4:21
Now, let's go to Section
10.1 of Chapter 10.
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Chapter 10 is a
beautiful chapter.
• 4:23 - 4:27
As you know very well,
I announced last time,
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vector-valued functions.
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And you say, oh
my god, I've never
• 4:43 - 4:46
functions before.
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You deal with them every day.
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Every time you move,
you are dealing
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with a vector-valued
function, which
• 4:55 - 5:02
is the displacement, which
takes values in a subset in R3.
• 5:02 - 5:07
So let's try and see what
you should understand
• 5:07 - 5:10
when you start Section 10.1.
• 5:10 - 5:14
Because the book is pretty
good, not that I'm a co-author.
• 5:14 - 5:19
But it was meant to be really
written for the students
• 5:19 - 5:23
and explain concepts
really well.
• 5:23 - 5:26
How many of you took physics?
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OK, quite a lot of
you took physics.
• 5:29 - 5:34
Now, one of my students
in a previous honors class
• 5:34 - 5:38
told me he enjoyed my
class greatly in general.
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The most [INAUDIBLE] thing
he had from my class, he
• 5:41 - 5:46
learned from my class was the
motion of the drunken bug.
• 5:46 - 5:48
And I said, did I say that?
• 5:48 - 5:50
Absolutely, you said that.
• 5:50 - 5:54
So apparently I had
started one of my lessons
• 5:54 - 6:00
with imagine you have a fly
who went into your coffee mug.
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I think I did.
• 6:00 - 6:04
He reproduced the whole
thing the way I said it.
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It was quite spontaneous.
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some Baileys Irish Creme in it.
• 6:10 - 6:16
And the fly was really
happy after she got up.
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She managed to get up.
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And the trajectory of the
fly was something more
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like a helix.
• 6:23 - 6:26
And this is how I actually
introduced the helix
• 6:26 - 6:27
in my classroom.
• 6:27 - 6:30
And I thought, OK,
is that unusual?
• 6:30 - 6:30
Very.
• 6:30 - 6:33
And I said, but that's
an honors class.
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Everything is supposed
to be unusual, right?
• 6:36 - 6:50
So let's think about the
position vector or some sort
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of vector-valued function that
you're familiar with already
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from physics.
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He is one of your best friends.
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You have a function r of t.
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And I will point out that r is
practically the position vector
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measure that time t, or
observed at time t in R3.
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So he takes values in R3.
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How?
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As the mathematician, because
I like to write mathematically
• 7:24 - 7:27
all the notion I
have, r is defined
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on I was a sub-interval
of R with values in R3.
• 7:34 - 7:39
And he asked me, my student
said, what is this I?
• 7:39 - 7:41
Well, this I could
be any interval,
• 7:41 - 7:43
but let's assume for
the time being it's
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just an open
interval of the type
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a, b, where a and b are
real numbers, a less than b.
• 7:52 - 7:58
So this is practically the time
for my bug from the moment,
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let's say a equals 0 when
she or he starts flying up,
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until the moment she
completely freaks
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out or drops from the
maximum point she reached.
• 8:12 - 8:13
And she eventually dies.
• 8:13 - 8:15
Or maybe she doesn't die.
• 8:15 - 8:19
Maybe she's just drunk and she
will wake up after a while.
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OK, so what do I mean by
this displacement vector?
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I mean, a function--
• 8:26 - 8:26
STUDENT: Is that Tc?
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Do you have [INAUDIBLE]?
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PROFESSOR: This is r, little r.
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STUDENT: I know, but the Tc.
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PROFESSOR: Tc?
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STUDENT: Or is that an I?
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PROFESSOR: No.
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This is I interval,
which is the same as a,
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b open interval, like
from 2 to 7, included.
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This is inclusion
[INAUDIBLE] included in R.
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So I mean R is the real number
set and a, b is my interval.
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OK, so r of t is
going to be what?
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x of t, y of t, z of t.
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The book tells you, hey, guys--
it doesn't say hey, guys,
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but it's quite informal-- if
you live in Rn, if your image is
• 9:10 - 9:13
in Rn, instead of x
of t, y of t, z of t,
• 9:13 - 9:19
you are going to get something
like x1 of t, y1 of t.
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x1 of t, x2 of t,
x3 of t, et cetera.
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What do we assume about R?
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We have to assume
something about it, right?
• 9:29 - 9:30
STUDENT: It's a
function [INAUDIBLE].
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PROFESSOR: It's a function
that is differentiable
• 9:33 - 9:37
most of the times, right?
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What does it mean smooth?
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I saw that your books
before college level
• 9:43 - 9:44
never mention smooth.
• 9:44 - 9:49
A smooth function is a
function that is differentiable
• 9:49 - 9:51
and whose first
derivative is continuous.
• 9:51 - 9:56
Some mathematicians even assume
that you have c infinity, which
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means you have a function that's
infinitely many differentiable.
• 10:00 - 10:03
So you have first derivative,
second derivative, third
• 10:03 - 10:04
derivative, fifth derivative.
• 10:04 - 10:05
Somebody stop me.
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All the derivatives exist
and they are all continuous.
• 10:09 - 10:13
By smooth, I will
assume c1 in this case.
• 10:13 - 10:16
I know it's not accurate,
but let's assume c1.
• 10:16 - 10:19
What does it mean?
• 10:19 - 10:24
Differentiable function whose
derivative is continuous.
• 10:24 - 10:31
• 10:31 - 10:35
And I will assume
one more thing.
• 10:35 - 10:37
That is not enough for me.
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I will also assume that
r prime of t in this case
• 10:42 - 10:50
is different from 0 for
every t in the interval I.
• 10:50 - 10:54
Could somebody tell me in
everyday words what that means?
• 10:54 - 10:56
We call that regular function.
• 10:56 - 10:56
[INAUDIBLE]
• 10:56 - 11:00
• 11:00 - 11:02
You have a brownie [INAUDIBLE].
• 11:02 - 11:03
I have no brownies with me.
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But if you answer, so what--
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STUDENT: So that means you've
got no relative mins or maxes,
• 11:08 - 11:11
and you never-- the
object never stops moving.
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PROFESSOR: Well, actually,
you can have relative mins
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and maxes in some way.
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I'm talking about something
like that, r prime.
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• 11:28 - 11:30
This is r of t.
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And r prime of t
is the derivative.
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It's never going to stop.
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The velocity.
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piece of information.
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Velocity [INAUDIBLE] 0 means
that drunken bug between time
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a and time b never stops.
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He stops at the end, but the end
is b, is outside [INAUDIBLE].
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So he stops at b and he falls.
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So I don't stop.
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I move on from time a to time b.
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I don't stop at all.
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Yes, sir.
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STUDENT: Wouldn't the derivative
of that line at some point
• 12:07 - 12:08
equal 0 where it flattens out?
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PROFESSOR: Let me
draw very well.
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So at time r of t, this
is the position vector.
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What is the derivative?
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The derivative represents
the velocity vector.
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A beautiful thing about the
velocity vector r prime of t
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is that it has a
beautiful property.
• 12:27 - 12:30
It's always tangent
to the trajectory.
• 12:30 - 12:33
So at every point
you're going to have
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a velocity vector that is
tangent to the trajectory.
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[INAUDIBLE] in physics.
• 12:38 - 12:42
This r prime of t
should never become 0.
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So you will never have a
point instead of a segment
• 12:47 - 12:51
when it comes to r prime.
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So you don't stop.
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• 12:59 - 13:00
You are going to
say, wait a minute?
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But are you always going to
consider curves, regular curves
• 13:04 - 13:06
in space?
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Regular curves in space.
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And by space, I know you guys
mean the Euclidean three space.
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Actually, many times I will
consider curves in plane.
• 13:20 - 13:23
And the plane is
part of the space.
• 13:23 - 13:26
And you say, give us an example.
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I will give you an
example right now.
• 13:28 - 13:30
You're going to laugh
how simple that is.
• 13:30 - 13:33
• 13:33 - 13:37
Now, I have another bug
who is really happy,
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but it's not drunk at all.
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And this bug knows how to
circle around a certain point
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at the same speed.
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So very organized bug.
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Yes, sir.
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STUDENT: Where did
you get c prime?
• 13:55 - 13:56
PROFESSOR: What?
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STUDENT: You have c prime is
differentiable, is [INAUDIBLE].
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PROFESSOR: c1.
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STUDENT: c1.
• 14:03 - 14:04
PROFESSOR: OK. c1.
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This is the notation for any
function that is differentiable
• 14:09 - 14:12
and whose derivative
is continuous.
• 14:12 - 14:17
So again, give an
example of a c1 function.
• 14:17 - 14:18
STUDENT: x squared.
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PROFESSOR: Yeah.
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On some real interval.
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How about absolute value
of x over the real line?
• 14:27 - 14:30
What's the problem with that?
• 14:30 - 14:31
[INTERPOSING VOICES]
• 14:31 - 14:33
PROFESSOR: It's not
differentiable at 0.
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OK, so we'll talk a little
bit later about smoothness.
• 14:37 - 14:40
It's a little bit
delicate as a notion.
• 14:40 - 14:43
It's really beautiful
on the other side.
• 14:43 - 14:50
Let's find the nice picture
trajectory for the bug.
• 14:50 - 14:51
This is a ladybug.
• 14:51 - 14:54
I cannot draw her, anyway.
• 14:54 - 14:57
She is moving along this circle.
• 14:57 - 15:01
And I'll give you
the law of motion.
• 15:01 - 15:07
And that reminds me of a
student who told me, what
• 15:07 - 15:09
do I care about law of motion?
• 15:09 - 15:11
He never had me as a
teacher, obviously.
• 15:11 - 15:14
But he was telling me,
well, after I graduated,
• 15:14 - 15:18
I always thought, what do I
care about the law of motion?
• 15:18 - 15:21
I mean, I took calculus.
• 15:21 - 15:24
the law of motion.
• 15:24 - 15:27
I'm sorry, you should care
about the law of motion.
• 15:27 - 15:30
Once you're not there anymore,
absolutely you don't care.
• 15:30 - 15:33
But why do you want to
[INAUDIBLE] doing calculus?
• 15:33 - 15:35
When you bring
[INAUDIBLE] to calculus,
• 15:35 - 15:37
when you walk into
calculus, it's law of motion
• 15:37 - 15:40
everywhere whether
you like it or not.
• 15:40 - 15:49
So let's try cosine t
sine t and z to b 1.
• 15:49 - 15:52
Let's make it 1 to
make your life easier.
• 15:52 - 15:54
What kind of curve
is this and why am I
• 15:54 - 15:58
claiming that the ladybug
following this curve
• 15:58 - 16:01
is moving at a constant speed?
• 16:01 - 16:01
Oh my god.
• 16:01 - 16:03
• 16:03 - 16:04
STUDENT: That's a circle.
• 16:04 - 16:05
PROFESSOR: That's the circle.
• 16:05 - 16:07
It's more than a circle.
• 16:07 - 16:08
It's a parametrized circle.
• 16:08 - 16:10
It's a vector-valued function.
• 16:10 - 16:15
Now, like every mathematician
I should specify the domain.
• 16:15 - 16:18
I am just winding
around one time,
• 16:18 - 16:21
and I stop where I started.
• 16:21 - 16:24
So I better be smart and
realize time is not infinity.
• 16:24 - 16:26
It could be.
• 16:26 - 16:28
I'm wrapping around the
circle infinitely many times.
• 16:28 - 16:30
They do that in
topology actually when
• 16:30 - 16:34
you're going to be--
seniors takes topology.
• 16:34 - 16:38
But I'm not going around
in circles only one time.
• 16:38 - 16:41
So my time will
start at 0 when I
• 16:41 - 16:45
start my motion and
end at 2 pi seconds
• 16:45 - 16:48
if the time is in seconds
• 16:48 - 16:52
So I say r is defined
on the interval I which
• 16:52 - 16:54
is-- say it again, Magdalena.
• 16:54 - 16:55
You just said it.
• 16:55 - 16:56
STUDENT: 0.
• 16:56 - 16:58
PROFESSOR: 0 to pi.
• 16:58 - 17:01
If you want to take
0 together, fine.
• 17:01 - 17:06
But for consistency, let's
take it like before, 0 to 2 pi.
• 17:06 - 17:08
I'm actually
excluding the origin.
• 17:08 - 17:11
• 17:11 - 17:12
And with values in R3.
• 17:12 - 17:18
Although, this is a [? plane ?]
curve, z will be constant.
• 17:18 - 17:20
Do I care about that very much?
• 17:20 - 17:22
You will see the beauty of it.
• 17:22 - 17:26
I have the velocity vector
being really pretty.
• 17:26 - 17:28
What is the velocity vector?
• 17:28 - 17:30
STUDENT: [INAUDIBLE].
• 17:30 - 17:32
PROFESSOR: Negative sign t.
• 17:32 - 17:32
Thank you.
• 17:32 - 17:33
STUDENT: [INAUDIBLE].
• 17:33 - 17:36
PROFESSOR: Cosine t.
• 17:36 - 17:37
And 0, finally.
• 17:37 - 17:41
Because as you saw
very well in the book,
• 17:41 - 17:44
the way we compute
the velocity vector
• 17:44 - 17:47
is by taking x of
t, y of t, z of t
• 17:47 - 17:50
and differentiating
them in terms of time.
• 17:50 - 17:54
• 17:54 - 17:55
Good.
• 17:55 - 17:58
Is this a regular function?
• 17:58 - 18:03
As the bug moves between
time 0 and time equals 2 pi,
• 18:03 - 18:07
is the bug ever going to
stop between these times?
• 18:07 - 18:07
STUDENT: No.
• 18:07 - 18:08
PROFESSOR: No.
• 18:08 - 18:09
How do you know?
• 18:09 - 18:11
You guys are faster
than me, right?
• 18:11 - 18:11
What did you do?
• 18:11 - 18:13
You did the speed.
• 18:13 - 18:14
What's the relationship?
• 18:14 - 18:17
What's the difference
between velocity and speed?
• 18:17 - 18:19
STUDENT: Speed is the
absolute value [INAUDIBLE].
• 18:19 - 18:19
PROFESSOR: Wonderful.
• 18:19 - 18:20
This is very good.
• 18:20 - 18:22
You should tell everybody
that because people
• 18:22 - 18:24
confuse that left and right.
• 18:24 - 18:27
So the velocity is
a vector, like you
• 18:27 - 18:28
learned in engineering.
• 18:28 - 18:30
You learned in physics.
• 18:30 - 18:31
Velocity is a vector.
• 18:31 - 18:32
It changes direction.
• 18:32 - 18:34
I'm going to Amarillo this way.
• 18:34 - 18:35
I'm driving.
• 18:35 - 18:37
The velocity will be a
vector pointing this way.
• 18:37 - 18:41
As I come back, will
point the opposite way.
• 18:41 - 18:44
The speed will be a
scalar, not a vector.
• 18:44 - 18:46
It's a magnitude of
a velocity vector.
• 18:46 - 18:48
So say it again, Magdalena.
• 18:48 - 18:49
What is the speed?
• 18:49 - 18:56
The speed is the magnitude
of the velocity vector.
• 18:56 - 18:59
It's a scalar.
• 18:59 - 19:01
Speed.
• 19:01 - 19:03
Speed.
• 19:03 - 19:06
I heard that before in
cars, in the movie Cars.
• 19:06 - 19:11
Anyway, r prime of t magnitude.
• 19:11 - 19:12
In magnitude.
• 19:12 - 19:17
Remember, there is a big
difference between the velocity
• 19:17 - 19:19
as the notion.
• 19:19 - 19:23
Velocity is a vector.
• 19:23 - 19:26
The speed is a
magnitude, is a scalar.
• 19:26 - 19:28
I'm going to go
ahead and erase that
• 19:28 - 19:33
and I'm going to ask
you what the speed is
• 19:33 - 19:36
for my fellow over here.
• 19:36 - 19:41
What is the speed
of a trajectory
• 19:41 - 19:46
of the bug who is sober and
moves at the constant speed?
• 19:46 - 19:47
OK.
• 19:47 - 19:49
As I already told
you, it's constant.
• 19:49 - 19:50
What is that constant?
• 19:50 - 19:54
• 19:54 - 19:57
What's the constant speed
I was talking about?
• 19:57 - 19:59
STUDENT: [INAUDIBLE].
• 19:59 - 20:02
PROFESSOR: I say the
magnitude of that.
• 20:02 - 20:04
I'm too lazy to write it down.
• 20:04 - 20:06
It's a Tuesday, almost morning.
• 20:06 - 20:10
So I go square root
of minus I squared
• 20:10 - 20:12
plus cosine squared plus 0.
• 20:12 - 20:14
I don't need to write that down.
• 20:14 - 20:15
You write it down.
• 20:15 - 20:17
And how much is that?
• 20:17 - 20:17
STUDENT: [INAUDIBLE].
• 20:17 - 20:18
PROFESSOR: 1.
• 20:18 - 20:25
So I love this curve because
in mathematician slang,
• 20:25 - 20:29
especially in [? a geometer's ?]
slang-- and my area
• 20:29 - 20:30
is differential geometry.
• 20:30 - 20:35
So in a way, I do calculus in
R3 every day on a daily basis.
• 20:35 - 20:37
So I have what?
• 20:37 - 20:43
This is a special kind of curve.
• 20:43 - 20:46
It's a curve parameterized
in arc length.
• 20:46 - 20:58
So definition, we say
that a curve in R3,
• 20:58 - 21:10
or Rn, well anyway, is
parameterized in arc length.
• 21:10 - 21:13
• 21:13 - 21:13
When?
• 21:13 - 21:14
Say it again, Magdalena.
• 21:14 - 21:32
Whenever, if and only if,
its speed is constantly 1.
• 21:32 - 21:36
• 21:36 - 21:41
So this is an example
where the speed is 1.
• 21:41 - 21:46
In such cases, we avoid
the notation with t.
• 21:46 - 21:46
You say, oh my god.
• 21:46 - 21:47
Why?
• 21:47 - 21:50
When the curve is
parameterized in arc length,
• 21:50 - 21:55
from now on the we
will actually try
• 21:55 - 21:59
to use s whatever we
know it's an arc length.
• 21:59 - 22:01
We use s instead of t.
• 22:01 - 22:05
So I'm sorry for the people
who cannot change that,
• 22:05 - 22:09
but you should all be
able t change that.
• 22:09 - 22:13
So everything will be
in s because we just
• 22:13 - 22:15
discovered
[? Discovery Channel, ?] we
• 22:15 - 22:19
just discovered that speed is 1.
• 22:19 - 22:24
So there is something
• 22:24 - 22:29
• 22:29 - 22:33
In this example-- oh, you
can rewrite the whole example
• 22:33 - 22:37
if you want in s so you don't
have to smudge the paper.
• 22:37 - 22:39
OK, it's beautiful.
• 22:39 - 22:41
So I am already arc length.
• 22:41 - 22:44
And in that case, I'm going
to call my time parameter
• 22:44 - 22:46
little s. s comes from special.
• 22:46 - 22:48
No, s comes from
speed [INAUDIBLE].
• 22:48 - 22:52
STUDENT: So you use s
when it's [INAUDIBLE]?
• 22:52 - 22:57
PROFESSOR: We use s whenever the
speed of that curve will be 1.
• 22:57 - 22:58
STUDENT: So [INAUDIBLE].
• 22:58 - 23:00
PROFESSOR: And we call that
arc length parameterization.
• 23:00 - 23:03
• 23:03 - 23:06
I'm moving into the duration
of your final thoughts.
• 23:06 - 23:08
Yes, sir.
• 23:08 - 23:10
STUDENT: When we
get the question, so
• 23:10 - 23:11
before solving [INAUDIBLE].
• 23:11 - 23:13
• 23:13 - 23:14
PROFESSOR: We don't know.
• 23:14 - 23:18
That's why it was our
discovery that, hey, at the end
• 23:18 - 23:22
it is an arc length, so I better
change [INAUDIBLE] t into s
• 23:22 - 23:27
because that will help me in
the future remember to do that.
• 23:27 - 23:30
Every time I have arc length,
that it means speed 1.
• 23:30 - 23:33
I will call it s instead of y.
• 23:33 - 23:35
There is a reason for that.
• 23:35 - 23:37
I'm going to erase
the definition
• 23:37 - 23:43
and I'm going to give
you the-- more or less,
• 23:43 - 23:46
the explanation that my
physics professor gave me.
• 23:46 - 23:50
Because as a freshman,
my mathematics professor
• 23:50 - 23:54
in that area, in geometry,
was not very, very active.
• 23:54 - 23:57
But practically, what my physics
professor told me is that,
• 23:57 - 24:06
hey, I would like to have
some sort of a uniform tangent
• 24:06 - 24:10
vector, something that is
standardized to be in speed 1.
• 24:10 - 24:16
So I would like that tangent
vector to be important to us.
• 24:16 - 24:20
And if r is an
arc length, then r
• 24:20 - 24:24
prime would be that unit
vector that I'm talking about.
• 24:24 - 24:31
So he introduced for any r of
t, which is x of t, y of t,
• 24:31 - 24:32
z of t.
• 24:32 - 24:37
My physics professor introduced
the following terminology.
• 24:37 - 24:43
The tangent unit vector
for a regular curve--
• 24:43 - 24:47
he was very well-organized
• 24:47 - 24:53
is by definition r
prime of t as a vector
• 24:53 - 24:54
divided by the
speed of the vector.
• 24:54 - 24:56
So what is he doing?
• 24:56 - 24:59
He is unitarizing the velocity.
• 24:59 - 25:00
Say it again, Magdalena.
• 25:00 - 25:03
He has unitarized
the velocity in order
• 25:03 - 25:08
to make research more consistent
from the viewpoint of Frenet
• 25:08 - 25:10
frame.
• 25:10 - 25:13
So in Frenet frame, you
will see-- you probably
• 25:13 - 25:14
Frenet frame if you
• 25:14 - 25:18
are a mechanics major, or some
solid mechanics or physics
• 25:18 - 25:19
major.
• 25:19 - 25:23
The Frenet frame is
an orthogonal frame
• 25:23 - 25:29
moving along a line in time
where the three components are
• 25:29 - 25:33
t, and the principal normal
vector, and b the [INAUDIBLE].
• 25:33 - 25:36
We only know of the
first of them, which
• 25:36 - 25:39
is T, which is a unit vector.
• 25:39 - 25:40
Say it again who it was.
• 25:40 - 25:45
It was the velocity vector
divided by its magnitude.
• 25:45 - 25:47
So the velocity vector could
be any wild, crazy vector
• 25:47 - 25:55
that's tangent to the trajectory
at the point where you are.
• 25:55 - 25:58
His magnitude varies from
one point to the other.
• 25:58 - 26:00
He's absolutely crazy.
• 26:00 - 26:01
He or she, the velocity vector.
• 26:01 - 26:02
Yes, sir.
• 26:02 - 26:03
STUDENT: [INAUDIBLE].
• 26:03 - 26:07
• 26:07 - 26:08
PROFESSOR: Here?
• 26:08 - 26:08
Here?
• 26:08 - 26:09
STUDENT: Yeah, down there.
• 26:09 - 26:11
PROFESSOR: D-E-F, definition.
• 26:11 - 26:13
That's how a mathematician
defines things.
• 26:13 - 26:18
So to define you write def
on top of an equality sign
• 26:18 - 26:21
or double dot equal.
• 26:21 - 26:24
That's a formal way a
mathematician introduces
• 26:24 - 26:25
a definition.
• 26:25 - 26:28
Well, he was a physicist,
but he does math.
• 26:28 - 26:29
So what do we do?
• 26:29 - 26:32
We say all the blue
guys that are not equal,
• 26:32 - 26:35
divide yourselves
• 26:35 - 26:40
And I'm going to have
the T here is next one,
• 26:40 - 26:43
the T here is next one,
the T here is next one.
• 26:43 - 26:44
They are all equal.
• 26:44 - 26:51
So that T changes direction, but
its magnitude will always be 1.
• 26:51 - 26:52
Right?
• 26:52 - 26:55
Know that the magnitude--
that's what unit vector means,
• 26:55 - 26:58
the magnitude is 1.
• 26:58 - 27:01
Why am I so happy about that?
• 27:01 - 27:04
Well let me tell
you that we can have
• 27:04 - 27:07
another parametrization
and another parametrization
• 27:07 - 27:11
and another parametrization
of the same curve.
• 27:11 - 27:12
Say what?
• 27:12 - 27:15
The parametrization of
a curve is not unique?
• 27:15 - 27:16
No.
• 27:16 - 27:19
There are infinitely
many parametrizations
• 27:19 - 27:22
for a physical curve.
• 27:22 - 27:34
There are infinitely
many parametrizations
• 27:34 - 27:40
for an even physical curve.
• 27:40 - 27:43
• 27:43 - 27:45
Like [INAUDIBLE]
the regular one?
• 27:45 - 27:47
Well let me give you
another example that
• 27:47 - 27:51
says that this is
currently R of T
• 27:51 - 27:58
equals cosine 5T sine 5T and 1.
• 27:58 - 27:59
Why 1?
• 27:59 - 28:03
I still want to have
the same physical curve.
• 28:03 - 28:04
What's different, guys?
• 28:04 - 28:07
Look at that and then
say oh OK, is this
• 28:07 - 28:12
the same curve as
a physical curve?
• 28:12 - 28:13
What's different in this case?
• 28:13 - 28:15
I'm still here.
• 28:15 - 28:17
It's still the
[? red ?] physical curve
• 28:17 - 28:18
I'm moving along.
• 28:18 - 28:19
What is different?
• 28:19 - 28:20
STUDENT: The velocity.
• 28:20 - 28:21
PROFESSOR: The velocity.
• 28:21 - 28:24
The velocity and
actually the speed.
• 28:24 - 28:29
I'm moving faster or slower, I
don't know, we have to decide.
• 28:29 - 28:34
Now how do I realize
how many times
• 28:34 - 28:36
I'm moving along this curve?
• 28:36 - 28:40
I can be smart and say
hey, I'm not stupid.
• 28:40 - 28:43
I know how to move only one
time and stop where I started.
• 28:43 - 28:47
my T in the interval
• 28:47 - 28:53
zero-- I start at
zero, where do I stop?
• 28:53 - 28:54
I can hear your brain buzzing.
• 28:54 - 28:55
STUDENT: [INAUDIBLE].
• 28:55 - 28:58
PROFESSOR: 2pi over 5.
• 28:58 - 28:59
Why is that?
• 28:59 - 29:00
• 29:00 - 29:03
STUDENT: Because when you
plug it in, it's [INAUDIBLE].
• 29:03 - 29:05
PROFESSOR: 5 times 2pi over 5.
• 29:05 - 29:06
That's where I stop.
• 29:06 - 29:08
So this is not the same
interval as before.
• 29:08 - 29:10
Are you guys with me?
• 29:10 - 29:17
This is a new guy, which
is called J. Oh, all right.
• 29:17 - 29:19
So there is a
relationship between the T
• 29:19 - 29:23
and the S. That's why I
use different notations.
• 29:23 - 29:27
And I wish my teachers
started it just
• 29:27 - 29:30
like that when I took math
analysis as a freshman,
• 29:30 - 29:31
or calculus.
• 29:31 - 29:32
That's calculus.
• 29:32 - 29:36
Because what they started
with was a diagram.
• 29:36 - 29:37
What kind of diagram?
• 29:37 - 29:42
Say OK, the
parametrizations are both
• 29:42 - 29:45
starting from
different intervals.
• 29:45 - 29:48
And first I have
the parametrization
• 29:48 - 29:50
from I going to our 3.
• 29:50 - 29:53
And that's called-- how
did we baptize that?
• 29:53 - 29:58
R. And the other
one, from J to R3,
• 29:58 - 30:02
we call that big R.
They're both vectors.
• 30:02 - 30:05
And hey guys, we
should have some sort
• 30:05 - 30:09
of correspondence
functions between I
• 30:09 - 30:14
and J that are both 1 to 1, and
they are 1 being [INAUDIBLE]
• 30:14 - 30:16
the other.
• 30:16 - 30:18
I swear to God,
when they started
• 30:18 - 30:21
with this theoretical
model, I didn't understand
• 30:21 - 30:23
the motivation at all.
• 30:23 - 30:25
At all.
• 30:25 - 30:28
Now with an example,
I can get you
• 30:28 - 30:31
closer to the motivation
of such a diagram.
• 30:31 - 30:35
So where does our
primary S live?
• 30:35 - 30:39
S lives in I, and
T lives in J. So I
• 30:39 - 30:43
have to have a correspondence
that takes S to T or T to S.
• 30:43 - 30:46
STUDENT: Wait I
thought since R of T
• 30:46 - 30:48
is also pretty much
[INAUDIBLE] that we should also
• 30:48 - 30:50
use S [INAUDIBLE].
• 30:50 - 30:53
PROFESSOR: It's very--
actually it's very easy.
• 30:53 - 30:56
This is 5T.
• 30:56 - 31:02
And we cannot use S
instead of this T,
• 31:02 - 31:05
because if we use S
instead of this T,
• 31:05 - 31:08
and we compute the
speed, we get 5.
• 31:08 - 31:11
So it cannot be called S.
This is very important.
• 31:11 - 31:15
So T is not an arc
length parameter.
• 31:15 - 31:18
I wonder what the speed
will be for this guy.
• 31:18 - 31:20
So who wants to
compute R prime of T?
• 31:20 - 31:23
Nobody, but I'll force you to.
• 31:23 - 31:27
And the magnitude of that
will be god knows what.
• 31:27 - 31:28
I claim it's 5.
• 31:28 - 31:30
Maybe I'm wrong.
• 31:30 - 31:31
I did this in my head.
• 31:31 - 31:33
I have to do it on paper, right.
• 31:33 - 31:35
So I have what?
• 31:35 - 31:39
I have to differentiate
component-wise.
• 31:39 - 31:42
And I have [INAUDIBLE] that,
because I'm running out of gas.
• 31:42 - 31:43
STUDENT: Minus 5--
• 31:43 - 31:46
PROFESSOR: Minus 5, very good.
• 31:46 - 31:48
Sine of 5T.
• 31:48 - 31:49
What have we applied?
• 31:49 - 31:52
In case you don't
know that, out.
• 31:52 - 31:53
That was Calc 1.
• 31:53 - 31:54
Chain rule.
• 31:54 - 31:55
Right?
• 31:55 - 32:00
So 5 times cosine 5T.
• 32:00 - 32:04
And finally, 1
prime, which is 0.
• 32:04 - 32:10
Now let's be brave and
write the whole thing down.
• 32:10 - 32:13
I know I'm lazy today, but I'm
going to have to do something.
• 32:13 - 32:14
Right?
• 32:14 - 32:18
So I'll say minus 5
sine 5T is all squared.
• 32:18 - 32:21
Let me take it and square it.
• 32:21 - 32:24
Because I see one
face is confused.
• 32:24 - 32:27
And since one face
is confused, it
• 32:27 - 32:30
doesn't matter that the
others are not confused.
• 32:30 - 32:31
OK?
• 32:31 - 32:36
So I have square root of this
plus square of [INAUDIBLE] plus
• 32:36 - 32:39
[INAUDIBLE] computing
the magnitude.
• 32:39 - 32:40
What do I get out of here?
• 32:40 - 32:40
STUDENT: Five.
• 32:40 - 32:41
PROFESSOR: Five.
• 32:41 - 32:42
Excellent.
• 32:42 - 32:45
This is 5 sine squared
plus 5 cosine squared.
• 32:45 - 32:50
Now yes, then I have 5 times 1.
• 32:50 - 32:55
So I have square root
of 25 here will be 5.
• 32:55 - 32:56
What is 5?
• 32:56 - 33:03
5 is the speed of the [? bug ?]
along the same physical curve
• 33:03 - 33:05
the other way around.
• 33:05 - 33:07
The second time around.
• 33:07 - 33:10
Now can you tell me the
relationship between T and S?
• 33:10 - 33:13
They are related.
• 33:13 - 33:19
They are like if you're my
uncle, then I'm your niece.
• 33:19 - 33:21
It's the same way.
• 33:21 - 33:23
It depends where you look at.
• 33:23 - 33:26
T is a function of S,
and S is a function of T.
• 33:26 - 33:32
So it has to be a 1 to 1
correspondence between the two.
• 33:32 - 33:38
Now any ideas of how I what
to compute the-- how do I
• 33:38 - 33:43
want to write the
relationship between them.
• 33:43 - 33:46
Well, S is a
function of T, right?
• 33:46 - 33:51
I just don't know what
function of T that is.
• 33:51 - 33:52
And I wish my professor
had started like that,
• 33:52 - 33:55
but he started
with this diagram.
• 33:55 - 33:59
So simply here you
have S equals S of T,
• 33:59 - 34:01
and here you have
T equals T of S,
• 34:01 - 34:03
the inverse of that function.
• 34:03 - 34:06
And when you-- when
somebody starts that
• 34:06 - 34:10
without an example as a
general diagram philosophy,
• 34:10 - 34:12
then it's really, really tough.
• 34:12 - 34:13
All right?
• 34:13 - 34:16
So I'd like to know
who S of T-- how
• 34:16 - 34:20
in the world do I want
to define that S of T.
• 34:20 - 34:26
He spoonfed us S of T. I don't
want to spoonfeed you anything.
• 34:26 - 34:28
Because this is
honors class, and you
• 34:28 - 34:31
should be able to figure
this out yourselves.
• 34:31 - 34:36
So who is big R of T?
• 34:36 - 34:42
Big R of T should
be, what, should
• 34:42 - 34:45
be the same thing in
the end as R of S.
• 34:45 - 34:57
But I should say maybe it's
R of function T of S, right?
• 34:57 - 35:00
Which is the same
thing as R of S. So
• 35:00 - 35:06
what should be the
relationship between T and S?
• 35:06 - 35:11
We have to call them-- one of
them should be T equals T of S.
• 35:11 - 35:13
• 35:13 - 35:16
Give it a Greek name,
what do you want.
• 35:16 - 35:16
Alpha?
• 35:16 - 35:17
Beta?
• 35:17 - 35:17
What?
• 35:17 - 35:18
STUDENT: [INAUDIBLE].
• 35:18 - 35:19
PROFESSOR: Alpha?
• 35:19 - 35:20
Beta?
• 35:20 - 35:20
Alpha?
• 35:20 - 35:22
I don't know.
• 35:22 - 35:26
So S going to T, alpha.
• 35:26 - 35:27
And this is going
to be alpha inverse.
• 35:27 - 35:31
• 35:31 - 35:32
Right?
• 35:32 - 35:37
So T equals alpha of S.
It's more elegant to call it
• 35:37 - 35:45
like that than T of S. T
equals alpha of S. Alpha of S.
• 35:45 - 35:49
So from this thing,
I realize that I
• 35:49 - 35:54
get that R composed with
alpha equals R. Say what?
• 35:54 - 35:55
Magdalena?
• 35:55 - 35:57
Yeah, yeah, that
was pre-calculus.
• 35:57 - 36:01
R composed with alpha
equals little r.
• 36:01 - 36:09
So how do I get a little r
by composing R with alpha?
• 36:09 - 36:12
How do we say that?
• 36:12 - 36:17
Alpha followed by R.
R composed with alpha.
• 36:17 - 36:22
R of alpha of S equals
R of S. Say it again.
• 36:22 - 36:31
R of alpha of S, which is T--
this T is alpha of S-- equals
• 36:31 - 36:31
R.
• 36:31 - 36:39
This is the composition
that we learned in pre-calc.
• 36:39 - 36:41
Who can find me the
definition of S?
• 36:41 - 36:44
Because this may be
a little bit hard.
• 36:44 - 36:47
This may be a little bit hard.
• 36:47 - 36:49
STUDENT: S [INAUDIBLE].
• 36:49 - 36:52
PROFESSOR: Eh, yeah,
let me write it down.
• 36:52 - 36:57
I want to find out
what S of T is.
• 36:57 - 37:00
• 37:00 - 37:11
Equals what in terms of the
function R of T. The one
• 37:11 - 37:14
that's given here.
• 37:14 - 37:15
Why is that?
• 37:15 - 37:23
• 37:23 - 37:26
Let's try some sort
of chain rule, right?
• 37:26 - 37:29
So what do I know I have?
• 37:29 - 37:30
I have that.
• 37:30 - 37:33
Look at that.
• 37:33 - 37:39
R prime of S, which
is the velocity of-- I
• 37:39 - 37:44
erased it-- the velocity of R
with respect to the arc length
• 37:44 - 37:47
parameter is going to be what?
• 37:47 - 37:52
R of alpha of S prime
with respect to S, right?
• 37:52 - 37:54
So I should put DDS.
• 37:54 - 37:55
Well I'm a little bit lazy.
• 37:55 - 37:58
Let's do it again.
• 37:58 - 38:06
DDS, R of alpha of S.
• 38:06 - 38:08
OK.
• 38:08 - 38:11
And what do I have in this case?
• 38:11 - 38:19
Well, I have R prime of-- who is
alpha of S. T, [INAUDIBLE] of T
• 38:19 - 38:27
and alpha of S times
R prime of alpha
• 38:27 - 38:30
of S times the prime outside.
• 38:30 - 38:32
How do we prime
in the chain rule?
• 38:32 - 38:35
From the outside to the
inside, one at a time.
• 38:35 - 38:39
So I differentiated the
outer shell, R prime,
• 38:39 - 38:40
and then times what?
• 38:40 - 38:41
Chain rule, guys.
• 38:41 - 38:45
Alpha prime of S. Very good.
• 38:45 - 38:50
Alpha prime of S.
• 38:50 - 38:51
All right.
• 38:51 - 38:55
So I would like
to understand how
• 38:55 - 39:03
I want to compute-- how I want
to define S of T. If I take
• 39:03 - 39:07
this in absolute value, R
prime of S in absolute value
• 39:07 - 39:12
equals R prime of T in absolute
value times alpha prime of S
• 39:12 - 39:15
in absolute value.
• 39:15 - 39:15
What do I get?
• 39:15 - 39:21
• 39:21 - 39:22
Who is R prime of S?
• 39:22 - 39:26
This is my original
function in arc length,
• 39:26 - 39:29
and that's the
speed in arc length.
• 39:29 - 39:31
What was the speed
in arc length?
• 39:31 - 39:32
STUDENT: One.
• 39:32 - 39:34
PROFESSOR: One.
• 39:34 - 39:37
And what is the speed
in not in arc length?
• 39:37 - 39:38
STUDENT: Five.
• 39:38 - 39:42
PROFESSOR: In that case,
this is going to be five.
• 39:42 - 39:46
And so what is this
alpha prime of S guy?
• 39:46 - 39:47
STUDENT: [INAUDIBLE].
• 39:47 - 39:51
PROFESSOR: It's going to be 1/5.
• 39:51 - 39:52
OK.
• 39:52 - 39:53
All right.
• 39:53 - 39:56
Actually alpha of S,
who is that going to be?
• 39:56 - 40:04
Alpha of S.
• 40:04 - 40:07
Do you notice the
correspondence?
• 40:07 - 40:12
We simply have to re-define
this as S. That's how it goes.
• 40:12 - 40:15
That five times
is nothing but S.
• 40:15 - 40:17
STUDENT: How did you
get the [INAUDIBLE]?
• 40:17 - 40:21
PROFESSOR: Because 1
equals 5 times what?
• 40:21 - 40:26
1, which is arc length
speed, equals 5 times what?
• 40:26 - 40:27
1/5.
• 40:27 - 40:28
STUDENT: Yeah, but then
where'd you get the 1?
• 40:28 - 40:29
PROFESSOR: That's
one way to do it.
• 40:29 - 40:32
Oh, this is by definition,
because little r means
• 40:32 - 40:36
curve in arc length, and little
s is the arc length parameter.
• 40:36 - 40:39
By definition, that
means you get speed 1.
• 40:39 - 40:41
This was our assumption.
• 40:41 - 40:44
So we could've gotten
that much faster saying
• 40:44 - 40:46
oh, well, forget
• 40:46 - 40:49
that you introduced-- and
it's also in the book.
• 40:49 - 40:53
Simply take 5T to BS, 5T to BS.
• 40:53 - 40:56
Then I get my old
friend, the curve.
• 40:56 - 40:59
The arc length
parameter is the curve.
• 40:59 - 41:05
So this is the same as cosine
of S, sine of S, and 1.
• 41:05 - 41:08
So what is the correspondence
between S and T?
• 41:08 - 41:11
• 41:11 - 41:15
Since S is 5T in
this example, I'll
• 41:15 - 41:16
put it-- where shall I put it.
• 41:16 - 41:20
I'll put it here.
• 41:20 - 41:23
S is 5T.
• 41:23 - 41:25
I'll say S of T is 5T.
• 41:25 - 41:28
• 41:28 - 41:32
and T of S, what
is T in terms of S?
• 41:32 - 41:37
T in terms of S is S over 5.
• 41:37 - 41:40
So instead of T of
S, we call this alpha
• 41:40 - 41:48
of S. So the correspondence
between S and T, what is T?
• 41:48 - 41:52
T is exactly S over
5 in this example.
• 41:52 - 41:53
Say it again.
• 41:53 - 41:55
T is exactly S over 5.
• 41:55 - 41:58
So alpha of S would be S over 5.
• 41:58 - 42:02
In this case, alpha prime of
S would simply be 1 over 5.
• 42:02 - 42:04
Oh, so that's how I got it.
• 42:04 - 42:06
That's another way to get it.
• 42:06 - 42:08
Much faster.
• 42:08 - 42:09
Much simpler.
• 42:09 - 42:14
So just think of replacing
5T by the S knowing
• 42:14 - 42:19
that you put S here, the whole
thing will have speed of 1.
• 42:19 - 42:20
All right.
• 42:20 - 42:22
So what do I do?
• 42:22 - 42:25
I say OK, alpha prime
of S is 1 over 5.
• 42:25 - 42:28
The whole chain rule also
spit out alpha prime of S
• 42:28 - 42:30
to B1 over 5.
• 42:30 - 42:33
Now I understand the
relationship between S and T.
• 42:33 - 42:34
It's very simple.
• 42:34 - 42:40
S is 5T in this example,
or T equals S over 5.
• 42:40 - 42:40
OK?
• 42:40 - 42:46
So if somebody gives you a curve
that looks like cosine 5T, sine
• 42:46 - 42:52
5T, 1, and that is in speed
5, as we were able to find,
• 42:52 - 42:57
how do you re-parametrize
that in arc length?
• 42:57 - 43:01
You just change
something inside so
• 43:01 - 43:08
that you make this curve be
representative-- representable
• 43:08 - 43:12
as little r of S.
This is in arc length.
• 43:12 - 43:14
In arc length.
• 43:14 - 43:18
• 43:18 - 43:18
OK.
• 43:18 - 43:20
Finally, this is
just an example.
• 43:20 - 43:24
Can you tell me how that
arc length parameter
• 43:24 - 43:26
is introduced in general?
• 43:26 - 43:30
What is S of T by definition?
• 43:30 - 43:34
What if I have
something really wild?
• 43:34 - 43:36
How do I get to that
S of T by definition?
• 43:36 - 43:39
• 43:39 - 43:41
What is S of T in terms
of the function R?
• 43:41 - 43:45
STUDENT: [INAUDIBLE] velocity
[? of the ?] [INAUDIBLE]?
• 43:45 - 43:48
PROFESSOR: S prime of T will
be one of the [INAUDIBLE].
• 43:48 - 43:49
STUDENT: Yes.
• 43:49 - 43:49
PROFESSOR: OK.
• 43:49 - 43:59
So let's see what we
have if we define S of T
• 43:59 - 44:12
as being integral from 0 to
T of the speed R prime of T.
• 44:12 - 44:14
And instead of T, we put tau.
• 44:14 - 44:15
Right?
• 44:15 - 44:16
P tau.
• 44:16 - 44:18
STUDENT: What is that?
• 44:18 - 44:20
PROFESSOR: We cannot
put T, T, and T.
• 44:20 - 44:21
STUDENT: Oh.
• 44:21 - 44:22
PROFESSOR: OK?
• 44:22 - 44:26
So tau is the Greek T
that runs between zero
• 44:26 - 44:29
and T. This is the
definition of S
• 44:29 - 44:44
of T. General definition
of the arc length parameter
• 44:44 - 44:50
that is according to the chain
rule, given by the chain rule.
• 44:50 - 44:57
• 44:57 - 45:00
Can we verify really
quickly in our case,
• 45:00 - 45:02
is it easy to see that
in our case it's correct?
• 45:02 - 45:03
STUDENT: Yeah.
• 45:03 - 45:06
PROFESSOR: Oh yeah,
S of T will be,
• 45:06 - 45:08
in our case,
integral from 0 to T.
• 45:08 - 45:14
We are lucky our prime of tau
is a constant, which is 5.
• 45:14 - 45:16
So I'm going to
have integral from 0
• 45:16 - 45:21
to T absolute value of
5 [INAUDIBLE] d tau.
• 45:21 - 45:23
And what in the world
is absolute value of 5?
• 45:23 - 45:28
It's 5 integral from 0
to T [? of the ?] tau.
• 45:28 - 45:31
What is integral from
0 to T of the tau?
• 45:31 - 45:34
T. 5T.
• 45:34 - 45:37
So S is 5T.
• 45:37 - 45:40
And that's what I
said before, right?
• 45:40 - 45:42
S is 5T.
• 45:42 - 45:47
S equals 5T, and
T equals S over 5.
• 45:47 - 45:51
So this thing, in general,
is told to us by who?
• 45:51 - 45:53
It has to match the chain rule.
• 45:53 - 45:55
It matches the chain rule.
• 45:55 - 46:20
• 46:20 - 46:20
OK.
• 46:20 - 46:25
So again, why does that
match the chain rule?
• 46:25 - 46:31
We have that-- we
have R-- or how
• 46:31 - 46:35
should I start, the little f,
the little r, little r of S,
• 46:35 - 46:36
right?
• 46:36 - 46:41
Little r of S is
little r of S of T.
• 46:41 - 46:45
How do I differentiate
that with respect to T?
• 46:45 - 46:53
Well DDT of R will be R
primed with respect to S.
• 46:53 - 47:02
So I'll say DRDS of
S of T times DSDT.
• 47:02 - 47:05
• 47:05 - 47:06
Now what is DSDT?
• 47:06 - 47:09
DSDT was the derivative of that.
• 47:09 - 47:16
It's exactly the speed
absolute value of R prime of T.
• 47:16 - 47:18
So when you prime
here, S prime of T
• 47:18 - 47:23
will be exactly that,
with T replacing tau.
• 47:23 - 47:24
We learned that in Calc 1.
• 47:24 - 47:27
I know it's been a long time.
• 47:27 - 47:29
I can feel you're
a little bit rusty.
• 47:29 - 47:30
But it doesn't matter.
• 47:30 - 47:33
So S prime of T,
DSDT will simply
• 47:33 - 47:36
be absolute value
of R prime of T.
• 47:36 - 47:41
That's the speed of
the original curve.
• 47:41 - 47:44
This one.
• 47:44 - 47:46
OK?
• 47:46 - 47:47
All right.
• 47:47 - 47:59
So here, when I look at
DRDS, this is going to be 1.
• 47:59 - 48:02
• 48:02 - 48:06
And if you think of
this as a function of T,
• 48:06 - 48:12
you have DR of S of
T. Who is R of S of T?
• 48:12 - 48:15
This is R-- big
R-- of T. So this
• 48:15 - 48:22
is the DRDT Which is exactly
the same as R prime of T
• 48:22 - 48:25
when you put the absolute
values [INAUDIBLE].
• 48:25 - 48:26
It has to fit.
• 48:26 - 48:33
So indeed, you have R prime
of T, R prime of T, and 1.
• 48:33 - 48:35
It's an identity.
• 48:35 - 48:39
If I didn't put DSDT to
[? P, ?] our prime of T
• 48:39 - 48:42
in absolute value,
it wouldn't work out.
• 48:42 - 48:48
DSDT has to be R prime
of T in absolute value.
• 48:48 - 48:51
And this is how we
got, again-- are
• 48:51 - 48:54
you going to remember
this without having
• 48:54 - 48:56
to re-do the whole thing?
• 48:56 - 49:11
Integral from 0 to T of R
prime of T or tau d tau.
• 49:11 - 49:14
When you prime this
guy with respect to T
• 49:14 - 49:18
as soon as it's positive--
when it is positive-- assume--
• 49:18 - 49:20
why is this positive, S of T?
• 49:20 - 49:24
Because you integrate from
time 0 to another time
• 49:24 - 49:25
a positive number.
• 49:25 - 49:29
So it has to be
positive derivative.
• 49:29 - 49:30
It's an increasing function.
• 49:30 - 49:34
This function is increasing.
• 49:34 - 49:37
So DSDT again will be the speed.
• 49:37 - 49:39
Say it again, Magdalena?
• 49:39 - 49:44
DSDT will be the speed
of the original line.
• 49:44 - 49:47
DSDT in our case was 5.
• 49:47 - 49:48
Right?
• 49:48 - 49:50
DSDT was 5.
• 49:50 - 49:55
S was 5 times T.
S was 5 times T.
• 49:55 - 49:55
All right.
• 49:55 - 49:58
That was a simple
example, sort of, kind of.
• 49:58 - 50:00
What do we want to remember?
• 50:00 - 50:04
We remember the formula
of the arc length.
• 50:04 - 50:06
Formula of arc length.
• 50:06 - 50:09
• 50:09 - 50:11
So the formula of
arc length exists
• 50:11 - 50:15
in this form because of
the chain rule [INAUDIBLE]
• 50:15 - 50:19
from this diagram.
• 50:19 - 50:25
So always remember, we have
a composition of functions.
• 50:25 - 50:28
We use that composition of
function for the chain rule
• 50:28 - 50:29
to re-parametrize it.
• 50:29 - 50:31
And finally, the drunken bug.
• 50:31 - 50:34
• 50:34 - 50:35
what did I take [INAUDIBLE] 14?
• 50:35 - 50:37
R of t.
• 50:37 - 50:44
Let's say this is 2
cosine t, 2 sine t.
• 50:44 - 50:46
Let me make it more beautiful.
• 50:46 - 50:54
Let me put 4-- 4, 4, and 3t.
• 50:54 - 50:57
Can anybody tell
me why I did that?
• 50:57 - 51:00
Maybe you can guess my mind.
• 51:00 - 51:04
Find the following things.
• 51:04 - 51:11
The unit vector T, by
definition R prime over R prime
• 51:11 - 51:16
of t in absolute value.
• 51:16 - 51:22
Find the speed of
this motion R of t.
• 51:22 - 51:25
This is a law of motion.
• 51:25 - 51:32
And reparametrize in arclength--
this curve in arclength.
• 51:32 - 51:37
• 51:37 - 51:40
And you go, oh my God, I
have a problem with a, b,c.
• 51:40 - 51:43
The is a typical problem for
the final exam, by the way.
• 51:43 - 51:46
This problem popped up on
many, many final exams.
• 51:46 - 51:47
Is it hard?
• 51:47 - 51:49
Is it easy?
• 51:49 - 51:53
First of all, how did I
know what it looked like?
• 51:53 - 51:57
I should give at
least an explanation.
• 51:57 - 52:01
If instead of 3t I
would have 3, then I
• 52:01 - 52:05
would have the plane
z equals 3 constant.
• 52:05 - 52:08
And then I'll say, I'm moving
in circles, in circles,
• 52:08 - 52:11
in circles, in circles,
with t as a real parameter,
• 52:11 - 52:14
and I'm not evolving.
• 52:14 - 52:17
But this is like, what, this
like in in the avatar OK?
• 52:17 - 52:22
So I'm performing the circular
motion, but at the same time
• 52:22 - 52:25
going on a different level.
• 52:25 - 52:27
Assume another life.
• 52:27 - 52:31
I'm starting another life
on the next spiritual level.
• 52:31 - 52:34
OK, I have no religious
beliefs in that area,
• 52:34 - 52:36
but it's a good physical
example to give.
• 52:36 - 52:38
So I go circular.
• 52:38 - 52:42
Instead of going again
circular and again circular,
• 52:42 - 52:45
I go, oh, I go up and
up and up, and this 3t
• 52:45 - 52:49
tells me I should also
evolve on the vertical.
• 52:49 - 52:50
Ah-hah.
• 52:50 - 52:55
So instead of circular motion
I get a helicoidal motion.
• 52:55 - 52:56
This is a helix.
• 52:56 - 52:59
• 52:59 - 53:02
Could somebody tell me how I'm
going to draw such a helix?
• 53:02 - 53:03
Is it hard?
• 53:03 - 53:04
Is it easy?
• 53:04 - 53:05
This helix-- yes, sir.
• 53:05 - 53:08
• 53:08 - 53:09
Yes.
• 53:09 - 53:11
STUDENT: [INAUDIBLE]
• 53:11 - 53:12
PROFESSOR: It's like a tornado.
• 53:12 - 53:14
It's like a tornado,
hurricane, but how
• 53:14 - 53:18
do I draw the cylinder on
which this helix exists?
• 53:18 - 53:22
I have to be a smart girl and
remember what I learned before.
• 53:22 - 53:25
What is x squared
plus y squared?
• 53:25 - 53:29
Suppose that z is not
playing in the picture.
• 53:29 - 53:33
If I take Mr. x and Mr. y
and I square them and I add
• 53:33 - 53:35
them together, what do I get?
• 53:35 - 53:36
STUDENT: It's the radius.
• 53:36 - 53:38
PROFESSOR: What is
• 53:38 - 53:39
4 squared.
• 53:39 - 53:41
I'm gonna write 4
squared because it's
• 53:41 - 53:43
easier than writing 16.
• 53:43 - 53:44
Thank you for your help.
• 53:44 - 53:51
So I simply have to go ahead and
draw the frame first, x, y, z,
• 53:51 - 53:55
and then I'll say, OK, smart.
• 53:55 - 53:58
R is 4.
• 53:58 - 54:00
The radius should be 4.
• 54:00 - 54:02
This is the cylinder
where I'm at.
• 54:02 - 54:07
Where do I start
my physical motion?
• 54:07 - 54:10
This bug is drunk,
but sort of not.
• 54:10 - 54:12
I don't know.
• 54:12 - 54:16
It's a bug that can keep
the same radius, which
• 54:16 - 54:17
is quite something.
• 54:17 - 54:18
STUDENT: It's tipsy.
• 54:18 - 54:20
PROFESSOR: Yeah,
exactly, tipsy one.
• 54:20 - 54:23
So how about t equals 0.
• 54:23 - 54:25
Where do I start my motion?
• 54:25 - 54:27
At 4, 0, 0.
• 54:27 - 54:29
Where is 4, 0, 0?
• 54:29 - 54:29
Over here.
• 54:29 - 54:32
So that's my first
point where the bug
• 54:32 - 54:33
will start at t equals 0.
• 54:33 - 54:34
STUDENT: How'd you get 4, 0, 0?
• 54:34 - 54:36
PROFESSOR: Because I'm--
very good question.
• 54:36 - 54:39
I'm on x, y, z axes.
• 54:39 - 54:42
4, y is 0, z is 0.
• 54:42 - 54:47
I plug in t, would be 0,
and I get 4 times 1, 4 times
• 54:47 - 54:51
0, 3 times 0, so I
know I'm starting here.
• 54:51 - 54:56
And when I move, I move
along the cylinder like that.
• 54:56 - 55:00
Can somebody tell me at
what time I'm gonna be here?
• 55:00 - 55:04
Not at 1:50, but what time am
I going to be at this point?
• 55:04 - 55:08
And then I continue, and I go
up, and I continue and I go up.
• 55:08 - 55:10
STUDENT: [INAUDIBLE]
• 55:10 - 55:11
PROFESSOR: Pi over 2.
• 55:11 - 55:13
Excellent.
• 55:13 - 55:14
And can you-- can
you tell me what
• 55:14 - 55:17
point it is in space in R 3?
• 55:17 - 55:18
Plug in pi over 2.
• 55:18 - 55:20
You can do it faster than me.
• 55:20 - 55:20
STUDENT: 0.
• 55:20 - 55:24
PROFESSOR: 0, 4 and 3 pi over 2.
• 55:24 - 55:26
And I keep going.
• 55:26 - 55:29
So this is the helicoidal
motion I'm talking about.
• 55:29 - 55:32
The unit vector-- is it easy
to write it on the final?
• 55:32 - 55:33
Can do that in no time.
• 55:33 - 55:39
So we get like, let's say, 30%,
30%, 30%, and 10% for drawing.
• 55:39 - 55:41
• 55:41 - 55:44
That would be a typical
grid for the problem.
• 55:44 - 55:50
So t will be minus 4 sine t.
• 55:50 - 55:54
If I make a mistake, are
you gonna shout, please?
• 55:54 - 55:59
4 cosine t and 3
divided by what?
• 55:59 - 56:01
What is the tangent unit vector?
• 56:01 - 56:04
At every point in
space, I'm gonna
• 56:04 - 56:06
have this tangent unit vector.
• 56:06 - 56:08
It has to have
length 1, and it has
• 56:08 - 56:11
to be tangent to my trajectory.
• 56:11 - 56:12
I'll draw him.
• 56:12 - 56:16
So he gives me a
field, a vector field--
• 56:16 - 56:19
this is beautiful-- T
of t is a vector field.
• 56:19 - 56:21
At every point of
the trajectory,
• 56:21 - 56:23
I have only one such vector.
• 56:23 - 56:27
That's what we mean
by vector field.
• 56:27 - 56:30
What's the magnitude?
• 56:30 - 56:31
It's buzzing.
• 56:31 - 56:33
It's buzzing.
• 56:33 - 56:35
How did you do it?
• 56:35 - 56:40
4, 16 times sine squared
plus cosine squared.
• 56:40 - 56:42
16 plus 9 is 25.
• 56:42 - 56:46
Square root of 25 is 5.
• 56:46 - 56:48
Are you guys with me?
• 56:48 - 56:50
Do I have to write this down?
• 56:50 - 56:52
Are you guys sure?
• 56:52 - 56:53
STUDENT: You plugged in 0 for t?
• 56:53 - 56:56
Is that what you did
when you [INAUDIBLE]
• 56:56 - 56:59
PROFESSOR: No, I plugged
0 for t when I started.
• 56:59 - 57:02
But when I'm computing,
I don't plug anything,
• 57:02 - 57:04
I just do it in general.
• 57:04 - 57:08
I said 16 sine squared
plus 16 cosine squared
• 57:08 - 57:10
is 16 times 1 plus 9.
• 57:10 - 57:13
My son would know this
one and he's 10, right?
• 57:13 - 57:16
16 plus 9 square root of 25.
• 57:16 - 57:18
And I taught him
• 57:18 - 57:21
So square root of 25,
he knows that's 5.
• 57:21 - 57:22
And if he knows
that's 5, then you
• 57:22 - 57:24
should do that in a
minute-- in a second.
• 57:24 - 57:25
All right.
• 57:25 - 57:32
So t will simply be-- if you
don't simplify 1/5 minus 4 sine
• 57:32 - 57:37
t 4 cosine t 3 in the final,
it wouldn't be a big deal,
• 57:37 - 57:39
I would give you
still partial credit,
• 57:39 - 57:42
but what if we raise this
as a multiple choice?
• 57:42 - 57:47
Then you have to be able
to find where the 5 is.
• 57:47 - 57:47
What is the speed?
• 57:47 - 57:49
Was that hard for you to find?
• 57:49 - 57:51
Where is the speed hiding?
• 57:51 - 57:54
It's exactly the
denominator of R.
• 57:54 - 57:57
This is the speed
of the curve in t.
• 57:57 - 57:59
And that was 5.
• 57:59 - 58:01
You told me the speed was
5, and I'm very happy.
• 58:01 - 58:08
So you got 30%, 30%, 10% from
the picture-- no, this picture.
• 58:08 - 58:09
This picture's no good.
• 58:09 - 58:13
STUDENT: What does the
first word of c say?
• 58:13 - 58:15
Question c, what does
the first word say?
• 58:15 - 58:16
PROFESSOR: The first what?
• 58:16 - 58:18
STUDENT: The word.
• 58:18 - 58:19
PROFESSOR: Reparametrize.
• 58:19 - 58:23
Reparametrize this
curve in arclength.
• 58:23 - 58:26
Oh my God, so according
to that chain rule,
• 58:26 - 58:31
could you guys remember-- if you
remember, what is the s of t?
• 58:31 - 58:39
If I want to reparametrize
in arclength integral from 0
• 58:39 - 58:46
to t of the speed, how
is the speed defined?
• 58:46 - 58:49
Absolute value of r prime of t.
• 58:49 - 58:54
dt, but I don't like t,
I write-- I write tau.
• 58:54 - 58:57
Like Dr. [? Solinger, ?]
you know him,
• 58:57 - 58:59
he's one of my colleagues,
calls that-- that's
• 58:59 - 59:01
the dummy dummy variable.
• 59:01 - 59:04
In many books, tau is
the dummy variable.
• 59:04 - 59:08
Or you can-- some people even
put t by inclusive notation.
• 59:08 - 59:10
All right?
• 59:10 - 59:13
So in my case, what is s of t?
• 59:13 - 59:14
It should be easy.
• 59:14 - 59:19
Because although this
not a circular motion,
• 59:19 - 59:21
I still have constant speed.
• 59:21 - 59:24
So who is that special speed?
• 59:24 - 59:24
5.
• 59:24 - 59:31
Integral from 0 to t5 d tau,
and that is 5t, am I right?
• 59:31 - 59:32
5t.
• 59:32 - 59:37
So-- so if I want to
reparametrize this helix,
• 59:37 - 59:42
keeping in mind
that s is simply 5t,
• 59:42 - 59:47
what do I have to do to
get 100% on this problem?
• 59:47 - 59:58
All I have to do is say little r
of s, which represents actually
• 59:58 - 60:01
big R of t of s.
• 60:01 - 60:02
Are you guys with me?
• 60:02 - 60:04
Do you have to write
all this story down?
• 60:04 - 60:05
No.
• 60:05 - 60:08
But that will remind
you of the diagram.
• 60:08 - 60:12
So I have R of t of s.
• 60:12 - 60:13
Or alpha of s.
• 60:13 - 60:15
And this is t of s.
• 60:15 - 60:16
t of s.
• 60:16 - 60:20
R of t of s is R of s, right?
• 60:20 - 60:21
Do you have to remind me?
• 60:21 - 60:22
No.
• 60:22 - 60:23
The heck with the diagram.
• 60:23 - 60:27
As long as you understood
it was about a composition
• 60:27 - 60:28
of functions.
• 60:28 - 60:31
And then R of s
will simply be what?
• 60:31 - 60:33
How do we do that fast?
• 60:33 - 60:37
We replaced t by s over 5.
• 60:37 - 60:39
Where from?
• 60:39 - 60:42
Little s equals 5t,
we just computed it.
• 60:42 - 60:44
Little s equals 5t.
• 60:44 - 60:45
That's all you need to do.
• 60:45 - 60:49
To pull out t, replace
the third sub s.
• 60:49 - 60:53
So what is the function
t in terms of s?
• 60:53 - 60:55
It's s over 5.
• 60:55 - 61:00
What is the function t, what's
the parameter t, in terms of s?
• 61:00 - 61:01
s over 5.
• 61:01 - 61:07
And finally, at the end, 3
times what is the stinking t?
• 61:07 - 61:09
s over 5.
• 61:09 - 61:11
I'm done.
• 61:11 - 61:16
I got 100% I don't want
to say how much time it's
• 61:16 - 61:18
gonna take me to
do it, but I think
• 61:18 - 61:20
I can do it in like, 2
or 3 minutes, 5 minutes.
• 61:20 - 61:24
If I know the problem I'll
do it in a few minutes.
• 61:24 - 61:27
If I waste too
much time thinking,
• 61:27 - 61:29
I'm not gonna do it at all.
• 61:29 - 61:30
So what do you have to remember?
• 61:30 - 61:35
You have to remember the
formula that says s of t,
• 61:35 - 61:41
the arclength parameter--
the arclength parameter
• 61:41 - 61:47
equals integral from 0 to
t is 0 to t of the speed.
• 61:47 - 61:53
Does this element of information
remind you of something?
• 61:53 - 61:56
Of course, s will be the
arclength, practically.
• 61:56 - 61:58
What kind of parameter is that?
• 61:58 - 62:04
Is you're measuring how
big-- how much you travel.
• 62:04 - 62:07
s of t is the time you
travel-- the distance
• 62:07 - 62:11
you travel in time t.
• 62:11 - 62:16
• 62:16 - 62:20
So it's a space-time continuum.
• 62:20 - 62:24
It's a space-time relationship.
• 62:24 - 62:27
So it's the space you
travel in times t.
• 62:27 - 62:30
Now, if I drive to Amarillo
at 60 miles an hour,
• 62:30 - 62:35
I'm happy and sassy, and I
say OK, it's gonna be s of t.
• 62:35 - 62:38
My displacement to
Amarillo is given
• 62:38 - 62:42
by this linear law, 60 times t.
• 62:42 - 62:43
Suppose I'm on cruise control.
• 62:43 - 62:44
But I've never on
cruise control.
• 62:44 - 62:47
• 62:47 - 62:51
So this is going to
be very variable.
• 62:51 - 62:55
And the only way you can compute
this displacement or distance
• 62:55 - 62:57
traveled, it'll
be as an integral.
• 62:57 - 63:01
From time 0, when I start
driving, to time t of my speed,
• 63:01 - 63:02
and that's it.
• 63:02 - 63:04
That's all you have to remember.
• 63:04 - 63:08
It's actually-- mathematics
should not be memorized.
• 63:08 - 63:12
It should be sort of
understood, just like physics.
• 63:12 - 63:15
What if you take your
first test, quiz,
• 63:15 - 63:19
whatever, on WeBWorK or in
person, and you freak out.
• 63:19 - 63:23
You get such a
problem, and you blank.
• 63:23 - 63:25
You just blank.
• 63:25 - 63:28
What do you do?
• 63:28 - 63:32
You sort of know this,
but you have a blank.
• 63:32 - 63:34
Always tell me, right?
• 63:34 - 63:36
Always email, say I'm
freaking out here.
• 63:36 - 63:39
I don't know what's
the matter with me.
• 63:39 - 63:46
Don't cut our correspondence,
either by speaking or by email.
• 63:46 - 63:49
Very few of you email me.
• 63:49 - 63:52
I'd like you to be
more like my friends,
• 63:52 - 63:53
and I would be more
• 63:53 - 63:55
and when you
encounter an obstacle,
• 63:55 - 63:58
you email me and
I email you back.
• 63:58 - 64:01
This is what I want.
• 64:01 - 64:04
The WeBWorK, this is what I
want our model of interaction
• 64:04 - 64:06
to become.
• 64:06 - 64:07
Don't be shy.
• 64:07 - 64:11
Many of you are shy even to
ask questions in the classroom.
• 64:11 - 64:12
And I'm not going
to let you be shy.
• 64:12 - 64:17
At 2 o'clock I'm going to let
you ask all the questions you
• 64:17 - 64:20
and we will do
• 64:20 - 64:21
more homework-like questions.
• 64:21 - 64:24
I want to imitate some
WeBWorK questions.
• 64:24 - 64:28
And we will work them out.
• 64:28 - 64:32
So any questions right now?
• 64:32 - 64:33
Yes, sir.
• 64:33 - 64:36
STUDENT: You emailed-- did
you email us this weekend
• 64:36 - 64:38
the numbers for WeBWorK?
• 64:38 - 64:41
PROFESSOR: I emailed you the
WeBWorK assignment completely.
• 64:41 - 64:45
I mean, the link-- you
get in and you of see it.
• 64:45 - 64:48
STUDENT: Which email
did you send that to?
• 64:48 - 64:50
PROFESSOR: To your TTU.
• 64:50 - 64:52
All the emails go to your TTU.
• 64:52 - 64:56
You have one week
starting yesterday until,
• 64:56 - 64:58
was it the 2nd?
• 64:58 - 65:00
I gave you a little
bit more time.
• 65:00 - 65:03
So it's due on the
2nd of February at,
• 65:03 - 65:04
I forgot what time.
• 65:04 - 65:05
1 o'clock or something.
• 65:05 - 65:06
Yes, sir.
• 65:06 - 65:08
STUDENT: [INAUDIBLE]
I was confused
• 65:08 - 65:10
at the beginning where you got
x squared plus y squared equals
• 65:10 - 65:11
4 squared.
• 65:11 - 65:14
Where did you get that?
• 65:14 - 65:14
PROFESSOR: Oh.
• 65:14 - 65:15
OK.
• 65:15 - 65:19
I eliminated the t between
the first two guys.
• 65:19 - 65:25
This is called eliminating a
parameter, which was the time
• 65:25 - 65:28
parameter between x and y.
• 65:28 - 65:32
When I do that, I get a
beautiful equation which
• 65:32 - 65:37
is x squared plus y squared
equals 16, which tells me, hey,
• 65:37 - 65:40
your curve sits on
the surface x squared
• 65:40 - 65:42
plus y squared equals 16.
• 65:42 - 65:44
It's not the same
with the surface,
• 65:44 - 65:48
because you have additional
constraints on the z.
• 65:48 - 65:52
So the z is constrained
to follow this thing.
• 65:52 - 66:00
Now, could anybody tell me how
I'm gonna write eventually--
• 66:00 - 66:02
this is a harder
task, OK, but I'm
• 66:02 - 66:09
wanted to discuss that.
• 66:09 - 66:13
How do I express t
in terms of x and y?
• 66:13 - 66:17
I mean, I'm going to have an
intersection of two surfaces.
• 66:17 - 66:18
How?
• 66:18 - 66:21
This is just practically
differential geometry
• 66:21 - 66:24
at the same time.
• 66:24 - 66:28
x squared plus y squared
equals our first surface
• 66:28 - 66:32
that I'm thinking about, which
I'm sitting with my curve.
• 66:32 - 66:35
But I also have my curve
to be at the intersection
• 66:35 - 66:39
between the cylinder
and something else.
• 66:39 - 66:45
And it's hard to figure out how
I'm going to do the other one.
• 66:45 - 66:49
Can anybody figure
out how another
• 66:49 - 66:51
surface-- what is the surface?
• 66:51 - 66:56
A surface will have an implicit
equation of the type f of x, y,
• 66:56 - 66:58
z equals a constant.
• 66:58 - 67:01
So you have to sort of
eliminate your parameter t.
• 67:01 - 67:02
The heck with the time.
• 67:02 - 67:05
We don't care about time,
we only care about space.
• 67:05 - 67:07
So is there any other
way to eliminate
• 67:07 - 67:10
t between the equations?
• 67:10 - 67:14
I have to use the information
that I haven't used yet.
• 67:14 - 67:15
All right.
• 67:15 - 67:20
Now my question is
that, how can I do that?
• 67:20 - 67:23
z is beautiful.
• 67:23 - 67:24
3 is beautiful.
• 67:24 - 67:26
t drives me nuts.
• 67:26 - 67:30
How do I get the t out of
the first two equations?
• 67:30 - 67:33
[INTERPOSING VOICES]
• 67:33 - 67:36
Yeah, I divide them
one to the other one.
• 67:36 - 67:40
So if I-- for example,
I go y over x.
• 67:40 - 67:43
What is y over x?
• 67:43 - 67:45
It's tangent of t.
• 67:45 - 67:49
How do I pull Mr. t out?
• 67:49 - 67:52
Say t, get out.
• 67:52 - 67:55
Well, I have to think about
if I'm not losing anything.
• 67:55 - 67:58
But in principle, t would
be arctangent of y over x.
• 67:58 - 68:02
• 68:02 - 68:02
OK?
• 68:02 - 68:06
So, I'm having two
equations of this type.
• 68:06 - 68:08
I'm eliminating t
between the two.
• 68:08 - 68:10
I don't care about
the other one.
• 68:10 - 68:14
I only cared for you
to draw the cylinder.
• 68:14 - 68:17
So we can draw point
by point the helix.
• 68:17 - 68:19
I don't draw many points.
• 68:19 - 68:23
I draw only t equals 0,
where I'm starting over here,
• 68:23 - 68:25
t equals pi over 2, which
[INAUDIBLE] gave me,
• 68:25 - 68:27
then what was it?
• 68:27 - 68:30
At pi I'm here, and so on.
• 68:30 - 68:34
So I move-- when
I move one time,
• 68:34 - 68:39
so let's say from 0 to
2 pi, I should be smart.
• 68:39 - 68:49
Pi over 2, pi, 3 pi over 2,
2 pi just on top of that.
• 68:49 - 68:52
It has to be on the same line.
• 68:52 - 68:55
On top of that--
on the cylinder.
• 68:55 - 68:56
They are all on the cylinder.
• 68:56 - 68:59
I'm not good enough to draw
them as being on the cylinder.
• 68:59 - 69:03
So I'm coming where I started
from, but on the higher
• 69:03 - 69:08
level of intelligence-- no, on
a higher level of experience.
• 69:08 - 69:09
Right?
• 69:09 - 69:14
That's kind of the idea
of evolving on the helix?
• 69:14 - 69:17
Any other questions?
• 69:17 - 69:18
Yes, sir.
• 69:18 - 69:20
STUDENT: So that
capital R of t is
• 69:20 - 69:24
you position vector, but what's
little r of t? [INAUDIBLE]
• 69:24 - 69:26
PROFESSOR: It's also
a position vector.
• 69:26 - 69:32
So practically it depends on
the type of parametrization
• 69:32 - 69:33
you are using.
• 69:33 - 69:36
• 69:36 - 69:40
The dependence of
time is crucial.
• 69:40 - 69:43
The dependence of the
time parameter is crucial.
• 69:43 - 69:51
So when you draw
this diagram, r of s
• 69:51 - 69:59
will practically be the same
as R of s of t-- R of t of s,
• 69:59 - 70:00
I'm sorry.
• 70:00 - 70:02
R of t of s.
• 70:02 - 70:06
So practically it's telling
me it's a combination.
• 70:06 - 70:12
Physically, it's the same
thing, but at a different time.
• 70:12 - 70:20
So you look at one vector
at time-- time is t here,
• 70:20 - 70:23
but s was 5t.
• 70:23 - 70:26
So I'm gonna be-- let
me give you an example.
• 70:26 - 70:30
So we had s was 5t, right?
• 70:30 - 70:33
I don't remember how it went.
• 70:33 - 70:36
So when I have
little r of s, that
• 70:36 - 70:43
means the same as
little r of 5t,
• 70:43 - 70:47
which means this kind of guy.
• 70:47 - 70:58
Now assume that I have something
like cosine 5t, sine 5t, and 0.
• 70:58 - 71:01
And what does this mean?
• 71:01 - 71:10
It means that R of 2 pi over
5 is the same as little r of 2
• 71:10 - 71:16
pi where R of t is cosine
of 5t, and little r of s
• 71:16 - 71:21
is cosine of s, sine s, 0.
• 71:21 - 71:24
So Mr. t says, I'm
running, I'm time.
• 71:24 - 71:29
I'm running from 0 to 2 pi
over 5, and that's when I stop.
• 71:29 - 71:31
And little s says,
I'm running too.
• 71:31 - 71:35
I'm also time, but I'm
a special kind of time,
• 71:35 - 71:38
and I'm running from 0 to
2 pi, and I stop at 2 pi
• 71:38 - 71:41
where the circle will stop.
• 71:41 - 71:44
Then physically,
the two vectors,
• 71:44 - 71:49
at two different moments
in time, are the same.
• 71:49 - 71:51
Where-- why-- why is that?
• 71:51 - 71:53
So I start here.
• 71:53 - 71:56
And I end here.
• 71:56 - 72:01
So physically, these two guys
have the same, the red vector,
• 72:01 - 72:05
but they are there at
different moments in time.
• 72:05 - 72:07
All right?
• 72:07 - 72:13
So imagine that you have sister.
• 72:13 - 72:18
And she is five times faster
than you in a competition.
• 72:18 - 72:21
It's a math competition,
athletic, it doesn't matter.
• 72:21 - 72:26
You both get there, but you
get there in different times,
• 72:26 - 72:27
in different amounts of time.
• 72:27 - 72:31
And unfortunately, this is--
I will do philosophy still
• 72:31 - 72:36
in mathematics-- this is the
situation with many of us
• 72:36 - 72:39
when it comes to
understanding a material,
• 72:39 - 72:42
like calculus or advanced
calculus or geometry.
• 72:42 - 72:47
We get to the understanding
in different times.
• 72:47 - 72:51
In my class-- I was
talking to my old--
• 72:51 - 72:56
they are all old now,
all in their 40s--
• 72:56 - 72:59
when did you
understand this helix
• 72:59 - 73:02
thing being on a cylinder?
• 73:02 - 73:04
Because I think I
understood it when
• 73:04 - 73:08
I was in third-- like a
junior level, sophomore level,
• 73:08 - 73:10
and I understood nothing
of this kind of stuff
• 73:10 - 73:15
in my freshman [INAUDIBLE]
And one of my colleagues
• 73:15 - 73:18
who was really smart,
had a big background,
• 73:18 - 73:21
was in a Math
Olympiad, said, I think
• 73:21 - 73:23
I understood it as a freshman.
• 73:23 - 73:25
So then the other two that
I was talking-- actually
• 73:25 - 73:28
I never understood it.
• 73:28 - 73:32
So we all eventually get to
that point, that position,
• 73:32 - 73:35
but at a different
moment in time.
• 73:35 - 73:39
And it's also unfortunate it
• 73:39 - 73:42
You are in a relationship
with somebody,
• 73:42 - 73:45
and one is faster
than the other one.
• 73:45 - 73:47
One grows faster
than the other one.
• 73:47 - 73:50
Eventually both get to the
same level of understanding,
• 73:50 - 73:53
but since it's at
different moments in time,
• 73:53 - 73:56
the relationship could
break by the time
• 73:56 - 73:59
both reach that level
of understanding.
• 73:59 - 74:03
So physical phenomena,
really tricky.
• 74:03 - 74:05
It's-- physically you
see where everything is,
• 74:05 - 74:08
but you have to think
dynamically, in time.
• 74:08 - 74:11
Everything evolves in time.
• 74:11 - 74:15
Any other questions?
• 74:15 - 74:18
I'm gonna do problems
with you next time,
• 74:18 - 74:23
but you need a break because
your brain is overheated.
• 74:23 - 74:28
And so, we will take a
break of 10-12 minutes.
• 74:28 - 74:31
Title:
TTU Math2450 Calculus3 Sec 10.1
Description:

Vector Value Functions

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Video Language:
English
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Sec 10.1