WEBVTT

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PROFESSOR TODA: And Calc II.

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And I will go ahead and
solve some problems today out

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of chapter 10 as a review.

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Meaning what?

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Meaning, that you have
section 10.1 followed by 10.2

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followed by 10.4.

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These ones are
required sections,

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but I'm putting the material
all together as a compact set.

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So, if we cannot officially
cut between, as I told you,

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cut between the sections.

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One thing that I did
not work examples on,

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trusting that you'd
remember it was integration.

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In particular, I didn't
cover integration

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of vector valued functions
and examples that

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are very very important.

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Now, do you need to learn
something special for that?

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No.

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But just like you cannot learn
organic chemistry without

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knowing inorganic chemistry,
then you could not know how

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to integrate a vector value
function r prime of d to get r

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of d unless you know calculus
one and caluculus two, right?

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So let's say first
a bunch of formulas

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that you use going back
to last week's knowledge

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what have we learned?

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We work with regular
curves in r3.

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And in particular if
they are part of R2,

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they are plain curves.

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I want to encourage
you to ask questions

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about the example
[INAUDIBLE] now.

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In the review session we
have applications [INAUDIBLE]

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from 2 2 3.

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What was a regular curve?

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Is anybody willing to tell
me what a regular curve was?

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Was it vector value function?

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Do you like big r or little r?

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STUDENT: Doesn't matter.

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PROFESSOR TODA: Big r of t.

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Vector value function.

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x of t [INAUDIBLE] You know,
I told you that sometimes we

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use brackets here.

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Sometimes we use round
parentheses depending

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how you represent a vector in r3
in our book they use brackets,

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but in other calculus books,
they use round parentheses

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around it.

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So these are the coordinates
of the moving particle in time.

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Doesn't have to be a specific
object, could be a fly,

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could be just a
particle, anything

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in physical motion between this
point a of b equals a and b

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of t equals b.

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So at time a and
time b you are there.

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What have we learned?

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We've learned that a regular
curve means its differentiable

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and the derivative is
continuous, it's a c1 function.

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And what else?

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The derivative of
the position vector

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called velocity never vanishes.

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So it's different from 0
for every t in the interval

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that you take, like ab.

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That's a regular curve.

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Regular curve was something we
talked about at least 5 times.

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The point is how do we
see the backwards process?

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That means if somebody gives you
the velocity of a vector curve,

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they ask you for
the position vector.

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So let's see an example.

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Integration example
1 says I gave you

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the veclocity vector or
a certain law of motion

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that I don't know.

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I just know the velocity
vector is being 1 over 1

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plus t squared.

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Should I put the brace here?

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An angular bracket?

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One over one plus t squared.

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And I'm gonna put a cosign
on 2t, and t squared

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plus equal to minus t.

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And somebody says,
that's all I know for P

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on an arbitrary real integral.

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And we know via the
0 as being even.

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Let's say it's even
as 0 0 and that

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takes a little bit of thinking.

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I don't know.

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How about a 1, which
would be just k.

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Using this velocity vector
find me being normal,

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which means find
the position vector

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corresponding to this velocity.

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What is this?

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It's actually initial value

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STUDENT: [INAUDIBLE]
1, 1, and 1?

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PROFESSOR TODA: 0, what is it?

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When place 0 in?

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STUDENT: Yeah.

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[INTERPOSING VOICES]

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STUDENT: Are these
the initial conditions

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for the location, or--

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PROFESSOR TODA: I'm sorry.

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I wrote r the intial
condition for the location.

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Thank you so much, OK?

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I probably would've realized
it as soon as possible.

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Not the initial velocity
I wanted to give you,

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but the initial position.

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All right, so how do
I get to the r of d?

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I would say integrate,
and when I integrate,

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I have to keep in mind that
I have to add the constants.

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Right?

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OK.

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So from v, v is our priority.

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It follows that r will
be-- who tells me?

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Do you guys remember the
integral of 1 plus t squared?

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STUDENT: [INAUDIBLE]

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PROFESSOR TODA: So
that's the inverse.

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Or, I'll write it [? arc tan, ?]
and I'm very happy that you

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remember that, but there
are many students who don't.

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If you feel you don't, that
means that you have to open

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the -- where? -- Between
chapters 5 and chapter 7.

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You have all these
integration chapters--

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the main ones over there.

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It's a function definted
on the whole real interval,

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so I don't care
to worry about it.

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This what we call an IVP,
initial value problem.

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So what kind of problem is that?

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It's a problem
like somebody would

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give you knowing that f
prime of t is the little f,

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and knowing that big f
of 0 is the initial value

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for your function of find f.

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So you have actually an initial
value problem of the calc

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that you've seen
in previous class.

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arctangent of t plus c1 and then
if you miss the c1 in general,

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this can mess up the whole thing
because-- see, in your case,

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you're really lucky.

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If you plug in the 0 here,
what are you gonna have?

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You're gonna have arctangent
of 0, and that is 0.

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So in that case c1 is just 0.

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And [? three ?] [? not ?] and
if you forgot it would not be

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the end of the world, but
if you forgot it in general,

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it would be a big problem.

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So don't forget
about the constant.

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When you integrate-- the
familiar of antiderivatives

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is cosine 2t.

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I know you know it.

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1/2 sine of t.

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Am I done?

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No, I should say plus C2.

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And finally the familiar
of antiderivatives of t

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squared plus e to minus t.

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STUDENT: 2t minus e
to the negative t.

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PROFESSOR TODA: No, integral of.

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So what's the integral of--

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STUDENT: t 2 squared.

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PROFESSOR TODA: t cubed
over 3-- minus, excellent.

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Now, do you want one
of you guys almost

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kill me during the weekend.

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But that's OK.

00:09:04.010 --> 00:09:06.340
I mean, this problem
had something

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to do with integral minus.

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He put that integral of e to the
minus t was equal to minus t.

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So pay attention to the sign.

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Remember that integral
of e to the at,

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the t is to the at over a plus.

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Right?

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OK, so this is what you
have, a minus plus C3.

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Pay attention also to the exam.

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Because in the
exams, when you rush,

00:09:30.708 --> 00:09:33.030
you make lots of
mistakes like that.

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R of 0 is even.

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So the initial position
is given as C1.

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I'm replacing in my formula.

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It's going to be
C1, C2, and what?

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When I replace the 0 here,
what am I going to get?

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STUDENT: You're going
to get negative 1.

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PROFESSOR TODA: Minus 1 plus C3.

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Note that I fabricated this
example, so that C3 is not

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going to be 0.

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I wanted some customs to
be zero and some customs

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to not be 0, just for
you to realize it's

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important to pay attention.

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OK, minus 1 plus C3.

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And then I have 0, 0, 1 as
given as initial position.

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So what do you get by solving
this linear system that's

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very simple?

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In general, you can get
more complicated stuff.

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C1 is 0, C2 is 0, C3 is a--

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STUDENT: 2.

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PROFESSOR TODA: 2.

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And so it was a piece of cake.

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What is my formula?

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If you leave it like
that, generally you're

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going to get full credit.

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What would you need to
do to get full credit?

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STUDENT: Rt is equal to R10
plus 1/2 sine of 2t plus tq--

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PROFESSOR TODA: Precisely,
and thank you so much

00:10:55.424 --> 00:10:56.810
for your help.

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So you have R10 of
t, 1/2 sine of 2t

00:11:01.550 --> 00:11:09.905
and t cubed over 3 minus
e to the minus e plus 2.

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And close, and that's it.

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And box your answer.

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So I got the long motion back.

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Similarly, you could find,
if somebody gives you

00:11:22.000 --> 00:11:27.840
the acceleration of a
long motion and asks you

00:11:27.840 --> 00:11:29.830
this is the acceleration.

00:11:29.830 --> 00:11:31.910
And I give you some
initial values.

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And you have to find
first the velocity,

00:11:34.520 --> 00:11:36.270
going backwards one step.

00:11:36.270 --> 00:11:39.680
And from the velocity,
backwards a second step,

00:11:39.680 --> 00:11:42.050
get the position vector.

00:11:42.050 --> 00:11:44.050
And that sounds a little
bit more elaborate.

00:11:44.050 --> 00:11:47.110
But it doesn't have to
be a long computation.

00:11:47.110 --> 00:11:50.960
In general, we do not
focus on giving you

00:11:50.960 --> 00:11:54.100
an awfully long computation.

00:11:54.100 --> 00:11:58.870
We just want to test your
understanding of the concepts.

00:11:58.870 --> 00:12:04.490
And having this in mind,
I picked another example.

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I would like to
see what that is.

00:12:08.570 --> 00:12:14.377
And the initial velocity
will be given in this case.

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This is what I was thinking
a little bit ahead of that.

00:12:16.812 --> 00:12:22.890
So somebody gives you the
acceleration in the velocity

00:12:22.890 --> 00:12:31.130
vector at 0 and is asking you
to find the velocity vector So

00:12:31.130 --> 00:12:36.300
let me give it to you
for t between 0 and 2 pi.

00:12:36.300 --> 00:12:38.050
I give you the
acceleration vector,

00:12:38.050 --> 00:12:40.370
it will be nice and sassy.

00:12:40.370 --> 00:12:45.690
Let's see, that's going to be
cosine of t, sine of t and 0.

00:12:45.690 --> 00:12:48.340
And you'll say, oh, I
know how to do those.

00:12:48.340 --> 00:12:49.770
Of course you know.

00:12:49.770 --> 00:12:52.370
But I want you to pay
attention to the constraints

00:12:52.370 --> 00:12:53.140
of integration.

00:12:53.140 --> 00:12:58.250
This is why I do this
kind of exercise again.

00:12:58.250 --> 00:13:02.810
So what do we have for V of t.

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V of 0 is-- somebody will say,
let's give something nice,

00:13:09.980 --> 00:13:15.530
and let's say this would be--
I have no idea what I want.

00:13:15.530 --> 00:13:21.770
Let's say i, j, and that's it.

00:13:21.770 --> 00:13:24.590


00:13:24.590 --> 00:13:26.950
How do you do that?

00:13:26.950 --> 00:13:27.640
V of t.

00:13:27.640 --> 00:13:30.006
Let's integrate together.

00:13:30.006 --> 00:13:31.530
You don't like this?

00:13:31.530 --> 00:13:35.270
I hope that by now,
you've got used to it.

00:13:35.270 --> 00:13:38.560
A bracket, I'm doing a
bracket, like in the book.

00:13:38.560 --> 00:13:42.700
So sine t plus a constant.

00:13:42.700 --> 00:13:45.090
What's the integral
of sine, class?

00:13:45.090 --> 00:13:48.200
V equals sine t plus a constant.

00:13:48.200 --> 00:13:51.380
And C3 is a constant.

00:13:51.380 --> 00:13:52.790
And there I go.

00:13:52.790 --> 00:13:54.670
You say, oh my god,
what am I having?

00:13:54.670 --> 00:13:58.360
V of 0-- is as a
vector, I presented it

00:13:58.360 --> 00:14:03.810
in the canonical standard
basis as 1, 1, and 0.

00:14:03.810 --> 00:14:07.350
So from that one, you
can jump to this one

00:14:07.350 --> 00:14:10.740
and say, yes, I'm going to
plug in 0, see what I get.

00:14:10.740 --> 00:14:13.376
In the general formula,
when you plug in 0,

00:14:13.376 --> 00:14:19.070
you get C1-- what
is cosine of 0?

00:14:19.070 --> 00:14:22.470
Minus 1, I have here, plus C2.

00:14:22.470 --> 00:14:27.900
And C3, that is always there.

00:14:27.900 --> 00:14:34.530
And then V of 0 is
what I got here.

00:14:34.530 --> 00:14:39.730
V of 0 has to be compared to
what your initial data was.

00:14:39.730 --> 00:14:48.380
So C1 is 1, C2 is 2, and C3 is--

00:14:48.380 --> 00:14:51.460
So let me replace it.

00:14:51.460 --> 00:15:05.427
I say the answer will be--
cosine t plus 1, sine t plus 2,

00:15:05.427 --> 00:15:13.315
and the constants.

00:15:13.315 --> 00:15:19.240


00:15:19.240 --> 00:15:24.740
But then somebody, who is
really an experimental guy,

00:15:24.740 --> 00:15:25.539
says well--

00:15:25.539 --> 00:15:26.830
STUDENT: You have it backwards.

00:15:26.830 --> 00:15:28.971
It's sine of t plus
1, and then you

00:15:28.971 --> 00:15:31.054
have the cosine of t plus 2.

00:15:31.054 --> 00:15:32.095
PROFESSOR TODA: Oh, yeah.

00:15:32.095 --> 00:15:35.470


00:15:35.470 --> 00:15:36.490
Wait a minute.

00:15:36.490 --> 00:15:41.550
This is-- I
miscopied looking up.

00:15:41.550 --> 00:15:52.208
So I have sine t, I was
supposed to-- minus cosine t

00:15:52.208 --> 00:15:56.550
and I'm done.

00:15:56.550 --> 00:15:58.380
So thank you for telling me.

00:15:58.380 --> 00:16:04.030
So sum t plus 1 minus
cosine t plus 2 and 0

00:16:04.030 --> 00:16:13.070
are the functions that I put
here by replacing C1, C2, C3.

00:16:13.070 --> 00:16:15.260
And then, somebody
says, wait a minute,

00:16:15.260 --> 00:16:18.500
now let me give you V of 0.

00:16:18.500 --> 00:16:20.860
Let me give you R of 0.

00:16:20.860 --> 00:16:22.495
We have zeroes already there.

00:16:22.495 --> 00:16:25.290


00:16:25.290 --> 00:16:28.800
And you were supposed
to get R from here.

00:16:28.800 --> 00:16:36.320
So what is R of t, the
position vector, find it.

00:16:36.320 --> 00:16:38.000
V of t is given.

00:16:38.000 --> 00:16:40.230
Actually, it's given by
you, because you found it

00:16:40.230 --> 00:16:41.940
at the previous step.

00:16:41.940 --> 00:16:46.300
And R of 0 is given as well.

00:16:46.300 --> 00:16:59.218
And let's say that would
be-- let's say 1, 1, and 1.

00:16:59.218 --> 00:17:02.630


00:17:02.630 --> 00:17:05.220
So what do you need to do next?

00:17:05.220 --> 00:17:14.390


00:17:14.390 --> 00:17:18.069
You have R prime given.

00:17:18.069 --> 00:17:22.140
That leaves you to
integrate to get R t.

00:17:22.140 --> 00:17:24.520
And R of t is going to be what?

00:17:24.520 --> 00:17:29.030
Who is going to tell me
what I have to write down?

00:17:29.030 --> 00:17:39.475
Minus cosine t plus t plus--
let's use the constant K1

00:17:39.475 --> 00:17:40.948
integration.

00:17:40.948 --> 00:17:42.421
And then what?

00:17:42.421 --> 00:17:43.410
STUDENT: Sine of t.

00:17:43.410 --> 00:17:45.380
PROFESSOR TODA: I think
it's minus sine, right?

00:17:45.380 --> 00:17:56.120
Minus sine of t plus 2t
plus K2 and K3, right?

00:17:56.120 --> 00:18:04.200
So R of 0 is going to be what?

00:18:04.200 --> 00:18:07.930
First of all, we use this
piece of information.

00:18:07.930 --> 00:18:12.480
Second of all, we identify
from the formula we got.

00:18:12.480 --> 00:18:16.305
So from the formula I
got, just plugging in 0,

00:18:16.305 --> 00:18:22.690
it should come out straight
as minus 1 plus K1.

00:18:22.690 --> 00:18:28.070
0 for this guy, 0 for the
second term, K2 and K3.

00:18:28.070 --> 00:18:31.830


00:18:31.830 --> 00:18:36.510
So who is helping me solve
the system really quickly?

00:18:36.510 --> 00:18:39.750
K1 is 2.

00:18:39.750 --> 00:18:41.030
K2 is--

00:18:41.030 --> 00:18:41.720
STUDENT: 1.

00:18:41.720 --> 00:18:44.010
PROFESSOR TODA: K3 is 1.

00:18:44.010 --> 00:18:50.610
And I'm going back
to R and replace it.

00:18:50.610 --> 00:18:55.470
And that's my final answer
for this two-step problem.

00:18:55.470 --> 00:18:57.870
So I have a two-step integration
from the acceleration

00:18:57.870 --> 00:19:00.270
to the velocity,
from the velocity

00:19:00.270 --> 00:19:05.080
to the position vector.

00:19:05.080 --> 00:19:08.250
Minus cosine t plus t plus 2.

00:19:08.250 --> 00:19:11.630
Remind me, because I have
a tendency to miscopy,

00:19:11.630 --> 00:19:13.280
an I looking in the right place?

00:19:13.280 --> 00:19:14.340
Yes.

00:19:14.340 --> 00:19:24.930
So I have minus sine t plus
2t plus 1 and K3 is one.

00:19:24.930 --> 00:19:29.751
So this is the process you
are supposed to remember

00:19:29.751 --> 00:19:32.160
for the rest of the semester.

00:19:32.160 --> 00:19:33.420
It's not a hard one.

00:19:33.420 --> 00:19:36.960
It's something that
everybody should master.

00:19:36.960 --> 00:19:38.060
Is it hard?

00:19:38.060 --> 00:19:39.810
How many of you understood this?

00:19:39.810 --> 00:19:41.820
Please raise hands.

00:19:41.820 --> 00:19:45.110
Oh, no problem, good.

00:19:45.110 --> 00:19:52.160
Now would you tell me--
I'm not going to ask you

00:19:52.160 --> 00:19:53.800
what kind of motion this is.

00:19:53.800 --> 00:19:57.030
It's a little bit close to
a circular motion but not

00:19:57.030 --> 00:19:58.376
a circular motion.

00:19:58.376 --> 00:20:00.670
However, can you tell
me anything interesting

00:20:00.670 --> 00:20:04.800
about the type of trajectory
that I have, in terms

00:20:04.800 --> 00:20:06.380
of the acceleration vector?

00:20:06.380 --> 00:20:10.820
The acceleration
vector is beautiful,

00:20:10.820 --> 00:20:13.840
just like in the
case of the washer.

00:20:13.840 --> 00:20:18.870
That was a vector
that-- like this

00:20:18.870 --> 00:20:20.860
would be the circular motion.

00:20:20.860 --> 00:20:23.040
The acceleration would
be this unique vector

00:20:23.040 --> 00:20:25.050
that comes inside.

00:20:25.050 --> 00:20:26.920
Is this going outside
or coming inside?

00:20:26.920 --> 00:20:29.600
Is it a unit vector?

00:20:29.600 --> 00:20:32.720
Yes, it is a unit vector.

00:20:32.720 --> 00:20:37.430
So suppose that I'm
looking at the trajectory,

00:20:37.430 --> 00:20:40.290
if it were more or
less a motion that has

00:20:40.290 --> 00:20:44.760
to do with mixing into a bowl.

00:20:44.760 --> 00:20:48.830
Would this go inside or outside?

00:20:48.830 --> 00:20:51.960
Towards the outside
or towards the inside?

00:20:51.960 --> 00:20:57.650
I plugged j-- depends on
what I'm looking at, in terms

00:20:57.650 --> 00:21:00.150
of surface that I'm on, right?

00:21:00.150 --> 00:21:01.560
Do you remember
from last time we

00:21:01.560 --> 00:21:04.055
had that helix that
was on a cylinder.

00:21:04.055 --> 00:21:07.920
And we asked ourselves, how
is that [INAUDIBLE] pointing?

00:21:07.920 --> 00:21:11.780
And it was pointing
outside of the cylinder,

00:21:11.780 --> 00:21:16.052
in the direction
towards the outside.

00:21:16.052 --> 00:21:26.930
Coming back to the
review, there are

00:21:26.930 --> 00:21:31.440
several things I'd like to
review but not all of them.

00:21:31.440 --> 00:21:34.439
Because some of the
examples we have there,

00:21:34.439 --> 00:21:38.150
you understood them really well.

00:21:38.150 --> 00:21:40.300
I was very proud
of you, and I saw

00:21:40.300 --> 00:21:43.758
that you finished--
almost all of you

00:21:43.758 --> 00:21:45.746
finished the
homework number one.

00:21:45.746 --> 00:21:49.100
So I was looking outside
at homework number

00:21:49.100 --> 00:21:53.180
two that is over
these three sections.

00:21:53.180 --> 00:21:58.471
So I was hoping you would ask
me today, between two and three,

00:21:58.471 --> 00:22:00.856
if you have any difficulties
with homework two.

00:22:00.856 --> 00:22:03.730
That's due February 11.

00:22:03.730 --> 00:22:12.730
And then the latest homework
that I posted yesterday, I

00:22:12.730 --> 00:22:14.980
don't know how many
of you logged in.

00:22:14.980 --> 00:22:18.620
But last night I
posted a homework

00:22:18.620 --> 00:22:21.800
that is getting a huge
extended deadline, which

00:22:21.800 --> 00:22:23.370
is the 28th of February.

00:22:23.370 --> 00:22:29.010
Because somebody's
birthday is February 29.

00:22:29.010 --> 00:22:34.740
I was just thinking why would
somebody need be a whole month?

00:22:34.740 --> 00:22:37.300
You would need the whole
month to have a good view

00:22:37.300 --> 00:22:39.020
of the whole chapter 11.

00:22:39.020 --> 00:22:40.970
I sent you the videos
for chapter 11.

00:22:40.970 --> 00:22:43.540
And for chapter 11, you
have this huge homework

00:22:43.540 --> 00:22:46.770
which is 49 problems.

00:22:46.770 --> 00:22:50.430
So please do not,
do not leave it

00:22:50.430 --> 00:22:52.260
to the last five
days or six days,

00:22:52.260 --> 00:22:55.710
because it's going to kill you.

00:22:55.710 --> 00:22:57.495
There are people who
say, I can finish

00:22:57.495 --> 00:22:58.620
this in the next five days.

00:22:58.620 --> 00:23:00.320
I know you can.

00:23:00.320 --> 00:23:01.950
I know you can,
I don't doubt it.

00:23:01.950 --> 00:23:04.420
That's why I left
you so much freedom.

00:23:04.420 --> 00:23:07.610
But you have-- today is
the second or the third?

00:23:07.610 --> 00:23:10.860
So practically you have
25 days to work on this.

00:23:10.860 --> 00:23:15.200
On the 28th at 11 PM
it's going to close.

00:23:15.200 --> 00:23:18.820
I would work a few
problems every other day.

00:23:18.820 --> 00:23:22.050
Because I need a break,
so I would alternate.

00:23:22.050 --> 00:23:25.030
But don't leave it--
even if you have help,

00:23:25.030 --> 00:23:27.690
especially if you have help,
like a tutor or tutoring

00:23:27.690 --> 00:23:30.280
services here that are
free in the department.

00:23:30.280 --> 00:23:32.220
Do not leave it
to the last days.

00:23:32.220 --> 00:23:35.160
Because you're putting pressure
on yourself, on your brain,

00:23:35.160 --> 00:23:37.221
on your tutor, on everybody.

00:23:37.221 --> 00:23:37.720
Yes sir.

00:23:37.720 --> 00:23:38.640
STUDENT: So that's
homework three?

00:23:38.640 --> 00:23:40.223
PROFESSOR TODA:
That's homework three,

00:23:40.223 --> 00:23:43.305
and it's a huge homework
over chapter 11.

00:23:43.305 --> 00:23:45.610
STUDENT: You said
there are 49 problems?

00:23:45.610 --> 00:23:49.157
PROFESSOR TODA: I don't
remember exactly but 47, 49.

00:23:49.157 --> 00:23:50.240
I don't remember how many.

00:23:50.240 --> 00:23:52.850
STUDENT: Between 45 and 50.

00:23:52.850 --> 00:23:55.910
PROFESSOR TODA:
Between 45 and 50, yes.

00:23:55.910 --> 00:23:59.210
If you encounter any bug--
although there shouldn't

00:23:59.210 --> 00:24:02.020
be bugs, maybe 1 in 1,000.

00:24:02.020 --> 00:24:04.530
If you encounter any
bug that the programmer

00:24:04.530 --> 00:24:09.710
of those problems may
have accidentally put in,

00:24:09.710 --> 00:24:11.370
you let me know.

00:24:11.370 --> 00:24:13.580
So I can contact them.

00:24:13.580 --> 00:24:17.390
If there is a problem that I
consider shouldn't be there,

00:24:17.390 --> 00:24:19.790
I will eliminate that later on.

00:24:19.790 --> 00:24:23.450
But hopefully, everything
will be doable,

00:24:23.450 --> 00:24:28.096
everything will be fair and
you will be able to solve it.

00:24:28.096 --> 00:24:32.140


00:24:32.140 --> 00:24:34.590
Any questions?

00:24:34.590 --> 00:24:37.015
Particular questions
from the homework?

00:24:37.015 --> 00:24:39.925


00:24:39.925 --> 00:24:43.805
STUDENT: [INAUDIBLE] is it to
parametrize a circle of a set,

00:24:43.805 --> 00:24:47.860
like of a certain
radius on the xy-plane?

00:24:47.860 --> 00:24:49.260
PROFESSOR TODA:
Shall we do that?

00:24:49.260 --> 00:24:53.228
Do you want me to do that
in general, in xy-plane, OK.

00:24:53.228 --> 00:24:55.224
STUDENT: [INAUDIBLE]
in the xy-plane.

00:24:55.224 --> 00:24:59.220


00:24:59.220 --> 00:25:04.660
PROFESSOR TODA: xy-plane and
then what was the equation?

00:25:04.660 --> 00:25:10.162
Was it like a equals sine
of t or a equals sine of bt?

00:25:10.162 --> 00:25:12.432
Because it's a
little bit different,

00:25:12.432 --> 00:25:15.790
depending on how the
parametrization was given.

00:25:15.790 --> 00:25:17.165
What's your name
again, I forgot.

00:25:17.165 --> 00:25:18.915
I don't know what to refer you.

00:25:18.915 --> 00:25:19.540
STUDENT: Ryder.

00:25:19.540 --> 00:25:22.397


00:25:22.397 --> 00:25:24.730
PROFESSOR TODA: Was that part
of what's due on the 11th?

00:25:24.730 --> 00:25:27.890
STUDENT: It doesn't-- yes, it
doesn't give a revision set.

00:25:27.890 --> 00:25:29.050
It says--

00:25:29.050 --> 00:25:33.000
PROFESSOR TODA: Let me quickly
read-- find parametrization

00:25:33.000 --> 00:25:38.440
of the circle of radius 7 in
the xy-plane, centered at 3, 1,

00:25:38.440 --> 00:25:40.615
oriented counterclockwise.

00:25:40.615 --> 00:25:43.238
The point 10, 1
should be connected--

00:25:43.238 --> 00:25:44.675
STUDENT: Just one more second.

00:25:44.675 --> 00:25:45.633
PROFESSOR TODA: Do
you mind if I put it.

00:25:45.633 --> 00:25:47.070
I'll take good care of it.

00:25:47.070 --> 00:25:48.028
I won't drop it.

00:25:48.028 --> 00:25:51.860


00:25:51.860 --> 00:25:57.880
So the point-- parametrization
of the circle of radius

00:25:57.880 --> 00:26:01.980
7 in the xy-plane,
centered at 3, 1.

00:26:01.980 --> 00:26:12.090
So circle centered at-- and
I'll say it x0, 1 0, being 3, 1.

00:26:12.090 --> 00:26:16.340


00:26:16.340 --> 00:26:19.040
No, because then I'm
solving your problem.

00:26:19.040 --> 00:26:20.710
But I'm solving
your problem anyway,

00:26:20.710 --> 00:26:23.480
even if I change
change the numbers.

00:26:23.480 --> 00:26:26.060


00:26:26.060 --> 00:26:27.960
Why don't I change
the numbers, and then

00:26:27.960 --> 00:26:30.850
you do it for the given numbers.

00:26:30.850 --> 00:26:33.880
Let's say 1, 0.

00:26:33.880 --> 00:26:39.610
And it's the same type
of problem, right?

00:26:39.610 --> 00:26:42.662
Oriented counterclockwise.

00:26:42.662 --> 00:26:43.370
That's important.

00:26:43.370 --> 00:26:51.720


00:26:51.720 --> 00:26:54.290
So you have circle radius 7.

00:26:54.290 --> 00:26:56.850
I think people could
have any other,

00:26:56.850 --> 00:27:01.370
because problems are-- sometimes
you get a random assignment.

00:27:01.370 --> 00:27:05.472
So you have R
equals 2, let's say.

00:27:05.472 --> 00:27:08.330


00:27:08.330 --> 00:27:14.230
And you have the point,
how to make up something.

00:27:14.230 --> 00:27:21.292
The point corresponding
to t equals

00:27:21.292 --> 00:27:29.794
0 will be given as you have
[INAUDIBLE], 1, 0, whatever.

00:27:29.794 --> 00:27:32.000
OK?

00:27:32.000 --> 00:27:36.850
Use the t as the parameter
for all your answers.

00:27:36.850 --> 00:27:39.180
So use t as a parameter
for all your answers,

00:27:39.180 --> 00:27:42.915
and the answers are written in
the interactive field as x of t

00:27:42.915 --> 00:27:45.310
equals what and y
of t equals what,

00:27:45.310 --> 00:27:47.414
and it's waiting for
you to fill them in.

00:27:47.414 --> 00:27:49.230
You know.

00:27:49.230 --> 00:27:54.470
OK, now I was talking
to [INAUDIBLE].

00:27:54.470 --> 00:27:56.890
I'm going to give
this back to you.

00:27:56.890 --> 00:27:57.680
Thank you, Ryan.

00:27:57.680 --> 00:28:02.760
So when you said it's a
little bit frustrating,

00:28:02.760 --> 00:28:07.800
and I agree wit you, that
in this variant of webwork

00:28:07.800 --> 00:28:10.760
problems you have to enter
both of them correctly

00:28:10.760 --> 00:28:14.640
in order to say yes, correct.

00:28:14.640 --> 00:28:18.070
I was used to another library--
the library was outdated

00:28:18.070 --> 00:28:22.480
[INAUDIBLE]-- where if I
enter this correctly I get 50%

00:28:22.480 --> 00:28:25.665
credit, and if I enter this
incorrectly it's not going

00:28:25.665 --> 00:28:26.810
to penalize me.

00:28:26.810 --> 00:28:29.915
So I a little bit
complained about it,

00:28:29.915 --> 00:28:32.310
and I was shown the
old library where

00:28:32.310 --> 00:28:35.550
I can go ahead and go
back and assign problems

00:28:35.550 --> 00:28:38.870
where you get the answer
correct for this one

00:28:38.870 --> 00:28:42.130
and incorrect for this one,
and you get partial credit.

00:28:42.130 --> 00:28:46.590
So I'm probably going
to switch to that.

00:28:46.590 --> 00:28:47.310
Let's do that.

00:28:47.310 --> 00:28:48.530
This is a very good problem.

00:28:48.530 --> 00:28:51.755
I'm glad you brought it up.

00:28:51.755 --> 00:28:56.850
What have you learned about
conics in high school?

00:28:56.850 --> 00:28:59.770
You've learned about--
well, it depends.

00:28:59.770 --> 00:29:01.380
You've learned about ellipse.

00:29:01.380 --> 00:29:03.000
You've learned about hyperbola.

00:29:03.000 --> 00:29:04.500
You've learned about parabola.

00:29:04.500 --> 00:29:07.190
Some of you put them down
for me for extra credit.

00:29:07.190 --> 00:29:08.980
I was very happy you did that.

00:29:08.980 --> 00:29:10.490
It's a good exercise.

00:29:10.490 --> 00:29:12.169
If you have-- Alex, yes?

00:29:12.169 --> 00:29:14.210
STUDENT: I was just
thinking, does that say 1, 0?

00:29:14.210 --> 00:29:17.600


00:29:17.600 --> 00:29:19.410
The point corresponding
to t0 [INAUDIBLE]?

00:29:19.410 --> 00:29:20.450
PROFESSOR TODA: I think
that's what I meant.

00:29:20.450 --> 00:29:22.060
I don't know, I just
came up with it.

00:29:22.060 --> 00:29:22.620
I made it.

00:29:22.620 --> 00:29:23.227
1, 0.

00:29:23.227 --> 00:29:24.310
I make up all my problems.

00:29:24.310 --> 00:29:26.120
STUDENT: But the center
of the circle isn't 1, 0.

00:29:26.120 --> 00:29:27.190
PROFESSOR TODA: Oh, oops.

00:29:27.190 --> 00:29:29.742
Yes.

00:29:29.742 --> 00:29:31.590
Sorry.

00:29:31.590 --> 00:29:33.330
So 2, 0.

00:29:33.330 --> 00:29:34.100
No--

00:29:34.100 --> 00:29:35.290
[INTERPOSING VOICES]

00:29:35.290 --> 00:29:38.304
PROFESSOR TODA:
--because the radius.

00:29:38.304 --> 00:29:40.720
This is the problem when you
don't think very [INAUDIBLE].

00:29:40.720 --> 00:29:44.240
I always like to make
up my own problems.

00:29:44.240 --> 00:29:48.400
When an author, when we came up
with the problems in the book,

00:29:48.400 --> 00:29:51.700
of course we had to think, draw,
and make sure they made sense.

00:29:51.700 --> 00:29:55.185
But when you just come up with
a problem out of the middle

00:29:55.185 --> 00:29:57.475
of nowhere-- thank you so much.

00:29:57.475 --> 00:29:58.850
Of course, we
would have realized

00:29:58.850 --> 00:30:01.090
that was nonsense
in just a minute.

00:30:01.090 --> 00:30:04.470
But it's good that you told me.

00:30:04.470 --> 00:30:06.526
So x of t, y of t.

00:30:06.526 --> 00:30:10.810


00:30:10.810 --> 00:30:11.900
Let's find it.

00:30:11.900 --> 00:30:13.353
Based on what?

00:30:13.353 --> 00:30:15.560
What is the general
equation of a circle?

00:30:15.560 --> 00:30:22.340
x minus x0 squared plus y minus
y0 squared equals R squared.

00:30:22.340 --> 00:30:24.530
And you have learned
that in high school.

00:30:24.530 --> 00:30:26.590
Am I right or not?

00:30:26.590 --> 00:30:27.390
You have.

00:30:27.390 --> 00:30:28.240
OK.

00:30:28.240 --> 00:30:29.130
Good.

00:30:29.130 --> 00:30:36.190
Now, in our case what
is x0 and what is y0?

00:30:36.190 --> 00:30:40.000
x0 is 1 and y0 is 0.

00:30:40.000 --> 00:30:42.860
Because that's
why-- I don't know.

00:30:42.860 --> 00:30:44.060
I just made it up.

00:30:44.060 --> 00:30:47.070
And I said that's the center.

00:30:47.070 --> 00:30:49.144
I'll draw.

00:30:49.144 --> 00:30:50.810
I should have drawn
it in the beginning,

00:30:50.810 --> 00:30:54.400
and that would have
helped me not come up

00:30:54.400 --> 00:31:00.430
with some nonsensical data.

00:31:00.430 --> 00:31:01.610
c is 1, 0.

00:31:01.610 --> 00:31:03.010
Radius is 2.

00:31:03.010 --> 00:31:04.640
So I'm going this way.

00:31:04.640 --> 00:31:06.790
What point is this way, guys?

00:31:06.790 --> 00:31:08.710
Just by the way.

00:31:08.710 --> 00:31:10.240
Because [INAUDIBLE]
is 1, 0, right?

00:31:10.240 --> 00:31:16.240
And this way the other
extreme, the antipode is 3, 0.

00:31:16.240 --> 00:31:20.088
So that's exactly what
Alexander was saying.

00:31:20.088 --> 00:31:22.020
And now it makes sense.

00:31:22.020 --> 00:31:25.401


00:31:25.401 --> 00:31:26.500
Well, I cannot draw today.

00:31:26.500 --> 00:31:27.333
STUDENT: [INAUDIBLE]

00:31:27.333 --> 00:31:30.354


00:31:30.354 --> 00:31:31.770
PROFESSOR TODA:
It looks horrible.

00:31:31.770 --> 00:31:37.065
It looks like an egg that
is shaped-- disabled egg.

00:31:37.065 --> 00:31:41.470


00:31:41.470 --> 00:31:42.480
OK.

00:31:42.480 --> 00:31:43.060
All right.

00:31:43.060 --> 00:31:49.640
So the motion of-- the
motion will come like that.

00:31:49.640 --> 00:31:53.736
From t equals 0, when I'm
here, counterclockwise,

00:31:53.736 --> 00:31:57.312
I have to draw-- any kind of
circle you have in the homework

00:31:57.312 --> 00:32:00.610
should be drawn on the board.

00:32:00.610 --> 00:32:06.210
If you have a general, you
don't know what the data is.

00:32:06.210 --> 00:32:08.800
I want to help you solve
the general problem.

00:32:08.800 --> 00:32:10.775
For the original problem,
which is a circle

00:32:10.775 --> 00:32:15.470
of center x, 0, y, 0 and
radius R, generic one,

00:32:15.470 --> 00:32:19.574
what is the parametrization
without data?

00:32:19.574 --> 00:32:20.490
Without specific data.

00:32:20.490 --> 00:32:23.240
What is the parametrization?

00:32:23.240 --> 00:32:25.644
And I want you to pay
attention very well.

00:32:25.644 --> 00:32:26.930
You are paying attention.

00:32:26.930 --> 00:32:29.730
You are very careful today.

00:32:29.730 --> 00:32:31.190
[INAUDIBLE]

00:32:31.190 --> 00:32:33.940
So what do you have?

00:32:33.940 --> 00:32:35.740
STUDENT: Cosine.

00:32:35.740 --> 00:32:37.520
PROFESSOR TODA:
Before that cosine

00:32:37.520 --> 00:32:39.580
there is an R, excellent.

00:32:39.580 --> 00:32:43.980
So [INAUDIBLE]
there R cosine of t.

00:32:43.980 --> 00:32:46.550
I'm not done.

00:32:46.550 --> 00:32:47.475
What do I put here?

00:32:47.475 --> 00:32:48.342
STUDENT: Over d.

00:32:48.342 --> 00:32:49.300
PROFESSOR TODA: No, no.

00:32:49.300 --> 00:32:51.276
I'm continuing.

00:32:51.276 --> 00:32:52.240
STUDENT: Plus x0.

00:32:52.240 --> 00:32:53.690
PROFESSOR TODA: Plus x0.

00:32:53.690 --> 00:32:57.460
And R sine t plus y0.

00:32:57.460 --> 00:32:59.560
Who taught me that?

00:32:59.560 --> 00:33:02.640
First of all, this
is not unique.

00:33:02.640 --> 00:33:03.650
It's not unique.

00:33:03.650 --> 00:33:05.920
I could put sine t
here and cosine t here

00:33:05.920 --> 00:33:08.650
and it would be the same
type of parametrization.

00:33:08.650 --> 00:33:11.020
But we usually put
the cosine first

00:33:11.020 --> 00:33:13.500
because we look at the
x-axis corresponding

00:33:13.500 --> 00:33:17.766
to the cosine and the y-axis
corresponding to the sine.

00:33:17.766 --> 00:33:20.845
If I don't know that,
because I happen to know that

00:33:20.845 --> 00:33:24.170
from when I was 16 in high
school, if I don't know that,

00:33:24.170 --> 00:33:25.380
what do I know?

00:33:25.380 --> 00:33:27.550
I cook up my own
parametrization.

00:33:27.550 --> 00:33:29.130
And that's a very good thing.

00:33:29.130 --> 00:33:31.160
And I'm glad Ryan
asked about that.

00:33:31.160 --> 00:33:33.360
How does one come up with this?

00:33:33.360 --> 00:33:34.360
Do we have to memorize?

00:33:34.360 --> 00:33:38.190
In mathematics, thank god,
we don't memorize much.

00:33:38.190 --> 00:33:41.730
The way we cook up things
is just from, in this case,

00:33:41.730 --> 00:33:44.930
from the Pythagorean
theorem of-- no.

00:33:44.930 --> 00:33:47.140
Pythagorean theorem
of trigonometry?

00:33:47.140 --> 00:33:49.170
The fundamental identity
of trigonometry,

00:33:49.170 --> 00:33:52.840
which is the same thing as
the Pythagorean theorem.

00:33:52.840 --> 00:33:55.300
What's the fundamental
identity of trigonometry?

00:33:55.300 --> 00:33:58.210
Cosine squared plus
sin squared equals 1.

00:33:58.210 --> 00:34:03.681
If I have a problem
like that, I must

00:34:03.681 --> 00:34:08.810
have that this is R cosine
t and this is R sine t.

00:34:08.810 --> 00:34:11.310
Because when I take
the red guys and I

00:34:11.310 --> 00:34:13.818
square them and I
add them together,

00:34:13.818 --> 00:34:17.650
I'm going to have R squared.

00:34:17.650 --> 00:34:18.940
All righty, good.

00:34:18.940 --> 00:34:23.370
So no matter what
kind of data you have,

00:34:23.370 --> 00:34:27.774
you should be able to come
up with this on your own.

00:34:27.774 --> 00:34:33.510
And what else is
going to be happening?

00:34:33.510 --> 00:34:38.139
When I solve for x of-- the
point corresponding to t

00:34:38.139 --> 00:34:39.389
equals 0.

00:34:39.389 --> 00:34:43.920
x of 0 and y of 0 will
therefore be what?

00:34:43.920 --> 00:34:49.350
It will be R plus x0.

00:34:49.350 --> 00:34:51.429
This is going to be what?

00:34:51.429 --> 00:34:53.710
Just y0.

00:34:53.710 --> 00:34:56.194
Does anybody give them to me?

00:34:56.194 --> 00:34:59.110
STUDENT: 3, 0.

00:34:59.110 --> 00:35:01.630
PROFESSOR TODA: Alexander
gave me the correct ones.

00:35:01.630 --> 00:35:05.670
They will be 3 and 0.

00:35:05.670 --> 00:35:06.670
Are you guys with me?

00:35:06.670 --> 00:35:10.920
They could be anything,
anything that makes sense.

00:35:10.920 --> 00:35:15.100
All right, for example somebody
would say, I'm starting here.

00:35:15.100 --> 00:35:16.580
I give you other points.

00:35:16.580 --> 00:35:19.910
Then you put them in, you
plug in that initial point,

00:35:19.910 --> 00:35:22.980
meaning that you're
starting your motion here.

00:35:22.980 --> 00:35:26.120
And you do go around
the circle one

00:35:26.120 --> 00:35:31.688
because, you take [INAUDIBLE]
only between 0 and 2 pi.

00:35:31.688 --> 00:35:32.852
Alexander.

00:35:32.852 --> 00:35:34.018
STUDENT: I have [INAUDIBLE].

00:35:34.018 --> 00:35:34.950
PROFESSOR TODA: OK.

00:35:34.950 --> 00:35:35.882
STUDENT: [INAUDIBLE]

00:35:35.882 --> 00:35:38.310
PROFESSOR TODA: No, I thought
that I misprinted something

00:35:38.310 --> 00:35:38.610
again.

00:35:38.610 --> 00:35:40.784
STUDENT: No, I was about to
say something really dumb.

00:35:40.784 --> 00:35:41.575
PROFESSOR TODA: OK.

00:35:41.575 --> 00:35:43.840


00:35:43.840 --> 00:35:48.692
So how do we make sense
of what we have here?

00:35:48.692 --> 00:35:52.280
Well, y0 corresponds
to what I said.

00:35:52.280 --> 00:35:55.680
So this is a
superfluous equation.

00:35:55.680 --> 00:35:57.620
I don't need that.

00:35:57.620 --> 00:36:01.200
What do I know from that?

00:36:01.200 --> 00:36:05.824
R will be 2.

00:36:05.824 --> 00:36:07.690
x1 is 1.

00:36:07.690 --> 00:36:10.000
I have a superfluous equation.

00:36:10.000 --> 00:36:13.890
I have to get identities
in that case, right?

00:36:13.890 --> 00:36:14.700
OK, now.

00:36:14.700 --> 00:36:19.640


00:36:19.640 --> 00:36:26.640
What is going to be my--
my bunch of equations

00:36:26.640 --> 00:36:47.640
will be x of t equals 2
cosine t plus 1 and y of t

00:36:47.640 --> 00:36:49.230
equals-- I don't
like this marker.

00:36:49.230 --> 00:36:49.920
I hate it.

00:36:49.920 --> 00:36:50.772
Where did I get it?

00:36:50.772 --> 00:36:51.730
In the math department.

00:36:51.730 --> 00:36:53.460
And it's a new one.

00:36:53.460 --> 00:36:54.820
I got it as a new one.

00:36:54.820 --> 00:36:56.740
It's not working.

00:36:56.740 --> 00:36:58.180
OK, y of t.

00:36:58.180 --> 00:37:01.150


00:37:01.150 --> 00:37:03.830
The blue contrast is invisible.

00:37:03.830 --> 00:37:07.897
I have 2 sine t.

00:37:07.897 --> 00:37:08.396
Okey dokey.

00:37:08.396 --> 00:37:11.590
When you finish a
problem, always quickly

00:37:11.590 --> 00:37:15.742
verify if what you
got makes sense.

00:37:15.742 --> 00:37:19.500
And obviously if I
look at everything,

00:37:19.500 --> 00:37:21.130
it's matching the whole point.

00:37:21.130 --> 00:37:21.906
Right?

00:37:21.906 --> 00:37:22.680
OK.

00:37:22.680 --> 00:37:29.843
Now going back to-- this is
reminding me of something in 3d

00:37:29.843 --> 00:37:34.673
that I wanted to talk
to you today about.

00:37:34.673 --> 00:37:36.605
This is [INAUDIBLE].

00:37:36.605 --> 00:37:42.900


00:37:42.900 --> 00:37:45.120
I'm going to give
you, in a similar way

00:37:45.120 --> 00:37:47.501
with this simple
problem, I'm going

00:37:47.501 --> 00:37:49.886
to give you something
more complicated

00:37:49.886 --> 00:38:16.700
and say find the
parametrization of a helix.

00:38:16.700 --> 00:38:19.610
And you say, well,
I'm happy that this

00:38:19.610 --> 00:38:21.824
isn't a made-up problem again.

00:38:21.824 --> 00:38:23.918
I have to be a little
bit more careful

00:38:23.918 --> 00:38:27.430
with these made-up problems
so that they make sense.

00:38:27.430 --> 00:38:44.380
Of a helix R of t such that
it is contained or it lies,

00:38:44.380 --> 00:38:59.770
it lies on the circular
cylinder x squared

00:38:59.770 --> 00:39:03.822
plus y squared equals 4.

00:39:03.822 --> 00:39:04.780
Why is that a cylinder?

00:39:04.780 --> 00:39:07.910
The z's missing, so it's
going to be a cylinder whose

00:39:07.910 --> 00:39:09.470
main axis is the z axis.

00:39:09.470 --> 00:39:09.970
Right?

00:39:09.970 --> 00:39:11.450
Are you guys with me?

00:39:11.450 --> 00:39:14.750
I think we are on the same page.

00:39:14.750 --> 00:39:19.449
And you cannot solve the
problem just with this data.

00:39:19.449 --> 00:39:22.160
Do you agree with me?

00:39:22.160 --> 00:39:47.190
And knowing that, the
curvature of the helix is k

00:39:47.190 --> 00:40:04.030
equals 2/5 at every point.

00:40:04.030 --> 00:40:06.080
And of course it's an oxymoron.

00:40:06.080 --> 00:40:08.240
Because what I
proved last time is

00:40:08.240 --> 00:40:12.530
that the curvature of
a helix is a constant.

00:40:12.530 --> 00:40:27.220
So remember, we got the
curvature of a helix

00:40:27.220 --> 00:40:30.056
as being a constant.

00:40:30.056 --> 00:40:34.455


00:40:34.455 --> 00:40:36.413
STUDENT: What's that last
word of the sentence?

00:40:36.413 --> 00:40:38.589
It's "the curvature
is at every" what?

00:40:38.589 --> 00:40:39.880
PROFESSOR TODA: At every point.

00:40:39.880 --> 00:40:45.070
I'm sorry I said, it very--
I abbreviated [INAUDIBLE].

00:40:45.070 --> 00:40:48.100
So at every point you
have the same curvature.

00:40:48.100 --> 00:40:50.920
When you draw a
helix you say, wait,

00:40:50.920 --> 00:40:53.820
the helix is bent uniformly.

00:40:53.820 --> 00:40:58.760
If you were to play with a
spring taken from am old bed,

00:40:58.760 --> 00:41:01.910
you would go with your
hands along the spring.

00:41:01.910 --> 00:41:04.700
And then you say, oh,
it bends about the same.

00:41:04.700 --> 00:41:06.010
Yes, it does.

00:41:06.010 --> 00:41:08.800
And that means the
curvature is the same.

00:41:08.800 --> 00:41:11.720
How would you
solve this problem?

00:41:11.720 --> 00:41:16.380
This problem is hard,
because you cannot integrate

00:41:16.380 --> 00:41:17.450
the curvature.

00:41:17.450 --> 00:41:19.110
Well, what is the curvature?

00:41:19.110 --> 00:41:21.040
The curvature would be--

00:41:21.040 --> 00:41:22.040
STUDENT: Absolute value.

00:41:22.040 --> 00:41:23.910
PROFESSOR TODA: Just
absolute value of R

00:41:23.910 --> 00:41:28.410
double prime if it were in s.

00:41:28.410 --> 00:41:31.030
And you cannot integrate.

00:41:31.030 --> 00:41:34.000
If somebody gave you
the vector equation

00:41:34.000 --> 00:41:36.570
of double prime of
this, them you say,

00:41:36.570 --> 00:41:38.730
yes, I can integrate
one step going back.

00:41:38.730 --> 00:41:40.330
I get R prime of s.

00:41:40.330 --> 00:41:41.729
Then I go back to R of s.

00:41:41.729 --> 00:41:43.270
But this is a little
bit complicated.

00:41:43.270 --> 00:41:45.492
I'm giving you a scalar.

00:41:45.492 --> 00:41:50.760
You have to be a little bit
aware of what you did last time

00:41:50.760 --> 00:41:54.596
and try to remember
what we did last time.

00:41:54.596 --> 00:41:56.330
What did we do last time?

00:41:56.330 --> 00:41:58.110
I would not give you
a problem like that

00:41:58.110 --> 00:42:03.290
on the final, because it would
assume that you have solved

00:42:03.290 --> 00:42:06.282
the problem we did last
time in terms of R of t

00:42:06.282 --> 00:42:09.950
equals A equals sine t.

00:42:09.950 --> 00:42:11.400
A sine t and [? vt. ?]

00:42:11.400 --> 00:42:15.770
And we said, this is the
standard parametrized helix

00:42:15.770 --> 00:42:22.330
that sits on a cylinder of
radius A and has the phb.

00:42:22.330 --> 00:42:27.510
So the distance between
consecutive spirals

00:42:27.510 --> 00:42:28.770
really matters.

00:42:28.770 --> 00:42:30.160
That really makes
the difference.

00:42:30.160 --> 00:42:30.620
STUDENT: I have a question.

00:42:30.620 --> 00:42:32.578
PROFESSOR TODA: You wanted
to ask me something.

00:42:32.578 --> 00:42:34.350
STUDENT: Is s always
the reciprocal of t?

00:42:34.350 --> 00:42:35.952
Are they always--

00:42:35.952 --> 00:42:37.410
PROFESSOR TODA:
No, not reciprocal.

00:42:37.410 --> 00:42:45.800
You mean s of t is a function
is from t0 to t of the speed.

00:42:45.800 --> 00:42:50.430
R prime and t-- d tau, right?

00:42:50.430 --> 00:42:51.630
Tau not t. [INAUDIBLE].

00:42:51.630 --> 00:42:54.220


00:42:54.220 --> 00:43:00.650
t and s are
different parameters.

00:43:00.650 --> 00:43:01.993
Different times.

00:43:01.993 --> 00:43:04.369
Different parameter times.

00:43:04.369 --> 00:43:04.910
And you say--

00:43:04.910 --> 00:43:06.701
STUDENT: Isn't s
the parameter time

00:43:06.701 --> 00:43:08.807
when [INAUDIBLE] parametrized?

00:43:08.807 --> 00:43:09.890
PROFESSOR TODA: Very good.

00:43:09.890 --> 00:43:12.365
So what is the magic s?

00:43:12.365 --> 00:43:13.850
I'm proud of you.

00:43:13.850 --> 00:43:15.940
This is the important
thing to remember.

00:43:15.940 --> 00:43:17.530
t could be any time.

00:43:17.530 --> 00:43:19.960
I start measuring
wherever I want.

00:43:19.960 --> 00:43:23.690
I can set my watch to start now.

00:43:23.690 --> 00:43:24.950
It could be crazy.

00:43:24.950 --> 00:43:26.640
Doesn't have to be uniform.

00:43:26.640 --> 00:43:27.603
Motion, I don't care.

00:43:27.603 --> 00:43:30.760


00:43:30.760 --> 00:43:33.330
s is a friend of
yours that says,

00:43:33.330 --> 00:43:38.290
I am that special time
so that according to me

00:43:38.290 --> 00:43:40.590
the speed will become one.

00:43:40.590 --> 00:43:45.650
So for a physicist to measure
the speed with respect to this,

00:43:45.650 --> 00:43:49.410
parameter s time, the speed
will always become one.

00:43:49.410 --> 00:43:51.660
That is the arclength
time and position.

00:43:51.660 --> 00:43:54.480
How you get from one
another, I told you last time

00:43:54.480 --> 00:43:57.376
that for both of them
you have-- this is R of t

00:43:57.376 --> 00:43:59.230
and this is little r of s.

00:43:59.230 --> 00:44:01.240
And there is a composition.

00:44:01.240 --> 00:44:03.460
s can be viewed as
a function of t,

00:44:03.460 --> 00:44:06.340
and t can be viewed
as a function of s.

00:44:06.340 --> 00:44:09.580
As functions they are
inverse to one another.

00:44:09.580 --> 00:44:12.650
So going back to who they
are, a very good question,

00:44:12.650 --> 00:44:15.580
because this is a review
anyway, [? who wants ?]

00:44:15.580 --> 00:44:19.180
s as a function of t for
this particular problem?

00:44:19.180 --> 00:44:23.949
I hope you remember, we were
like-- have you seen this movie

00:44:23.949 --> 00:44:28.395
with Mickey Mouse going
on a mountain that

00:44:28.395 --> 00:44:32.100
was more like a cylinder.

00:44:32.100 --> 00:44:35.320
And this is the train
going at a constant slope.

00:44:35.320 --> 00:44:42.590
And one of my colleagues,
actually, he's at Stanford,

00:44:42.590 --> 00:44:47.300
was telling me that he
gave his students in Calc 1

00:44:47.300 --> 00:44:51.860
to prove, formally prove,
that the angle formed

00:44:51.860 --> 00:44:56.590
by the law of motion
by the velocity vector,

00:44:56.590 --> 00:45:01.990
with the horizontal plane
passing through the particle,

00:45:01.990 --> 00:45:04.050
is always a constant.

00:45:04.050 --> 00:45:07.165
I didn't think about doing
in now, but of course we can.

00:45:07.165 --> 00:45:08.520
We could do that.

00:45:08.520 --> 00:45:10.964
So maybe the next
thing would be, like,

00:45:10.964 --> 00:45:12.630
if you [INAUDIBLE]
an extra problem, can

00:45:12.630 --> 00:45:17.280
we show that angle between the
velocity vector on the helix

00:45:17.280 --> 00:45:20.702
and the horizontal plane through
that point is a constant.

00:45:20.702 --> 00:45:22.535
STUDENT: Wouldn't it
just be, because B of t

00:45:22.535 --> 00:45:23.935
is just a constant times t?

00:45:23.935 --> 00:45:24.810
PROFESSOR TODA: Yeah.

00:45:24.810 --> 00:45:25.590
We'll get to that.

00:45:25.590 --> 00:45:27.170
We'll get to that in a second.

00:45:27.170 --> 00:45:32.487
So he reminded me of an old
movie from like 70 years ago,

00:45:32.487 --> 00:45:33.820
with Mickey Mouse and the train.

00:45:33.820 --> 00:45:38.650
And the train going
up at the same speed.

00:45:38.650 --> 00:45:41.160
You have to maintain
the same speed.

00:45:41.160 --> 00:45:44.810
Because if you risk it
not, then you sort of

00:45:44.810 --> 00:45:46.160
are getting trouble.

00:45:46.160 --> 00:45:47.760
So you never stop.

00:45:47.760 --> 00:45:49.290
If you stop you go back.

00:45:49.290 --> 00:45:50.740
So it's a regular curve.

00:45:50.740 --> 00:45:52.875
What I have here is
that such a curve.

00:45:52.875 --> 00:45:54.786
Regular curve, never stop.

00:45:54.786 --> 00:45:56.800
Get up with a constant speed.

00:45:56.800 --> 00:45:58.826
Do you guys remember the
speed from last time?

00:45:58.826 --> 00:46:01.077
We'll square root the a
squared plus b squared.

00:46:01.077 --> 00:46:04.430
When we did the
velocity thingie.

00:46:04.430 --> 00:46:10.730
And I get square root a
squared plus b squared times t.

00:46:10.730 --> 00:46:19.040
Now, today I would like
to ask you one question.

00:46:19.040 --> 00:46:21.520
What if-- Ryan brought this up.

00:46:21.520 --> 00:46:22.460
It's very good.

00:46:22.460 --> 00:46:23.660
b is a constant.

00:46:23.660 --> 00:46:26.550
What if b would
not be a constant,

00:46:26.550 --> 00:46:28.610
or maybe could be worse?

00:46:28.610 --> 00:46:32.710
For example, instead of having
another linear function with t,

00:46:32.710 --> 00:46:36.178
but something that contains
higher powers of t.

00:46:36.178 --> 00:46:39.360


00:46:39.360 --> 00:46:43.410
Then you don't go at the
constant speed anymore.

00:46:43.410 --> 00:46:45.370
You can say goodbye
to the cartoon.

00:46:45.370 --> 00:46:45.880
Yes, sir?

00:46:45.880 --> 00:46:49.017
STUDENT: And then
it's [INAUDIBLE].

00:46:49.017 --> 00:46:50.100
One that goes [INAUDIBLE].

00:46:50.100 --> 00:46:51.600
PROFESSOR TODA: I
mean, it's still--

00:46:51.600 --> 00:46:54.826
STUDENT: s is not
multiplied by a constant.

00:46:54.826 --> 00:46:57.017
The function between t and
s is not a constant one.

00:46:57.017 --> 00:46:59.600
PROFESSOR TODA: It's going to
be a different parameterization,

00:46:59.600 --> 00:47:00.580
different speed.

00:47:00.580 --> 00:47:03.770
Sometimes-- OK, you
have to understand.

00:47:03.770 --> 00:47:06.740
Let's say I have a cone.

00:47:06.740 --> 00:47:10.230
And I'm going slow
at first, and I

00:47:10.230 --> 00:47:11.980
go faster and faster
and faster and faster

00:47:11.980 --> 00:47:13.900
to the end of the cone.

00:47:13.900 --> 00:47:18.390
But then I have the
same physical curve,

00:47:18.390 --> 00:47:21.040
and I parameterized
[INAUDIBLE] the length.

00:47:21.040 --> 00:47:24.310
And I say, no, I'm a mechanic.

00:47:24.310 --> 00:47:26.840
Or I'm the engineer
of the strain.

00:47:26.840 --> 00:47:29.420
I can make the motion
have a constant speed.

00:47:29.420 --> 00:47:33.130
So even if the helix
is no longer circular,

00:47:33.130 --> 00:47:36.560
and it's sort of a crazy helix
going on top of the mountain,

00:47:36.560 --> 00:47:39.330
as an engineer I
can just say, oh no,

00:47:39.330 --> 00:47:42.150
I want cruise control
for my little train.

00:47:42.150 --> 00:47:45.500
And I will go at the same speed.

00:47:45.500 --> 00:47:48.940
See, the problem is
the slope a constant.

00:47:48.940 --> 00:47:51.270
And thinking of
what they did that

00:47:51.270 --> 00:47:53.067
stand for, because
it didn't stand

00:47:53.067 --> 00:47:54.880
for [INAUDIBLE] in honors.

00:47:54.880 --> 00:47:57.280
We can do it in honors as well.

00:47:57.280 --> 00:47:58.700
We'll do it in a second.

00:47:58.700 --> 00:48:04.950
Now, k obviously is what?

00:48:04.950 --> 00:48:08.460
Some of you have
very good memory,

00:48:08.460 --> 00:48:13.250
and like the memory of a
medical doctor, which is great.

00:48:13.250 --> 00:48:14.560
Some of you don't.

00:48:14.560 --> 00:48:18.772
But if you don't you just go
back and look at the notes.

00:48:18.772 --> 00:48:20.756
What I'm trying to
do, but I don't know,

00:48:20.756 --> 00:48:22.960
it's also a matter
of money-- I don't

00:48:22.960 --> 00:48:26.040
want to use the math
department copier-- I'd

00:48:26.040 --> 00:48:29.850
like to make a stack of notes.

00:48:29.850 --> 00:48:33.090
So that's why I'm collecting
these notes, to bring them back

00:48:33.090 --> 00:48:33.971
to you.

00:48:33.971 --> 00:48:34.470
For free!

00:48:34.470 --> 00:48:36.470
I'm not going to
sell them to you.

00:48:36.470 --> 00:48:38.060
I'm [INAUDIBLE].

00:48:38.060 --> 00:48:41.515
So that you can have those
with you whenever you want,

00:48:41.515 --> 00:48:45.475
or put them in a spiral,
punch holes in them,

00:48:45.475 --> 00:48:48.680
and have them for
review at any time.

00:48:48.680 --> 00:48:51.450
Reminds me of what that
was-- that was in the notes.

00:48:51.450 --> 00:48:54.694
a over a squared plus b squared.

00:48:54.694 --> 00:48:57.300
So who can tell me, a
and b really quickly,

00:48:57.300 --> 00:49:00.870
so we don't waste too
much time, Mr. a is--?

00:49:00.870 --> 00:49:05.729


00:49:05.729 --> 00:49:07.020
STUDENT: So this is another way

00:49:07.020 --> 00:49:07.601
STUDENT: 2.

00:49:07.601 --> 00:49:08.350
PROFESSOR TODA: 2.

00:49:08.350 --> 00:49:13.037
STUDENT: So is this another
way of defining k in k of s?

00:49:13.037 --> 00:49:14.120
PROFESSOR TODA: Actually--

00:49:14.120 --> 00:49:16.895
STUDENT: That's the general
curvature for [INAUDIBLE].

00:49:16.895 --> 00:49:21.040
PROFESSOR TODA: You know
what is the magic thing?

00:49:21.040 --> 00:49:23.095
Even if-- the curvature
is an invariant.

00:49:23.095 --> 00:49:26.526
It doesn't depend the
reparametrization.

00:49:26.526 --> 00:49:29.830
There is a way maybe I'm going
to teach you, although this

00:49:29.830 --> 00:49:32.160
is not in the book.

00:49:32.160 --> 00:49:35.870
What are the formulas
corresponding

00:49:35.870 --> 00:49:41.580
to the [INAUDIBLE] t and v that
depend on curvature and torsion

00:49:41.580 --> 00:49:44.000
and the speed along the curve.

00:49:44.000 --> 00:49:48.750
And if you analyze the notion
of curvature, [INAUDIBLE],

00:49:48.750 --> 00:49:52.230
no matter what your
parameter will be, t, s, tau,

00:49:52.230 --> 00:49:56.690
God knows what, k will
still be the same number.

00:49:56.690 --> 00:49:59.300
So k is viewed as an
invariant with respect

00:49:59.300 --> 00:50:01.425
to the parametrization.

00:50:01.425 --> 00:50:04.120
STUDENT: So then that a over
a squared plus b squared,

00:50:04.120 --> 00:50:05.912
that's another way of finding k?

00:50:05.912 --> 00:50:07.120
PROFESSOR TODA: Say it again?

00:50:07.120 --> 00:50:09.285
STUDENT: So using a over
a squared plus b squared

00:50:09.285 --> 00:50:10.829
is another way of finding k?

00:50:10.829 --> 00:50:11.620
PROFESSOR TODA: No.

00:50:11.620 --> 00:50:13.916
Somebody gave you k.

00:50:13.916 --> 00:50:17.440
And then you say, if it's
a standard parametrization,

00:50:17.440 --> 00:50:25.290
and then I get 2/5,
can I be sure a is 2?

00:50:25.290 --> 00:50:28.220
I'm sure a is 2 from nothing.

00:50:28.220 --> 00:50:32.860
This is what makes me aware
that a is 2 the first place.

00:50:32.860 --> 00:50:36.640
Because its the radius
of the cylinder.

00:50:36.640 --> 00:50:39.290
This is x squared, x and y.

00:50:39.290 --> 00:50:41.860
You see, x squared plus
y squared is a squared.

00:50:41.860 --> 00:50:43.650
This is where I get a from.

00:50:43.650 --> 00:50:44.620
a is 2.

00:50:44.620 --> 00:50:47.140
I replace it in here
and I say, all righty,

00:50:47.140 --> 00:50:51.777
so I only have one
choice. a is 2 and b is?

00:50:51.777 --> 00:50:52.610
STUDENT: [INAUDIBLE]

00:50:52.610 --> 00:50:57.260


00:50:57.260 --> 00:51:00.470
PROFESSOR TODA: But can b
plus-- So what I'm saying,

00:51:00.470 --> 00:51:01.380
a is 2, right?

00:51:01.380 --> 00:51:04.390
We know that from this.

00:51:04.390 --> 00:51:08.440
If I block in here I have 4
and somebody says plus minus 1.

00:51:08.440 --> 00:51:09.520
No.

00:51:09.520 --> 00:51:11.000
b is always positive.

00:51:11.000 --> 00:51:13.380
So you remember the
last time we discussed

00:51:13.380 --> 00:51:16.640
about the standard
parametrization.

00:51:16.640 --> 00:51:20.280
But somebody will say,
but what if I put a minus?

00:51:20.280 --> 00:51:22.840
What if I'm going
to put a minus?

00:51:22.840 --> 00:51:24.150
That's an excellent question.

00:51:24.150 --> 00:51:27.072
What's going to happen
if you put minus t?

00:51:27.072 --> 00:51:28.010
[INTERPOSING VOICES]

00:51:28.010 --> 00:51:29.010
PROFESSOR TODA: Exactly.

00:51:29.010 --> 00:51:31.260
In the opposite direction.

00:51:31.260 --> 00:51:35.550
Instead of going
up, you go down.

00:51:35.550 --> 00:51:37.430
All right.

00:51:37.430 --> 00:51:41.095
Now, I'm gonna-- what else?

00:51:41.095 --> 00:51:43.270
Ah, I said, let's do this.

00:51:43.270 --> 00:51:47.986
Let's prove that the
angle is a constant,

00:51:47.986 --> 00:51:51.080
the angle that's
made by the velocity

00:51:51.080 --> 00:51:56.220
vector of the train with the
horizontal plane is a constant.

00:51:56.220 --> 00:51:57.840
Is this hard?

00:51:57.840 --> 00:51:58.340
Nah.

00:51:58.340 --> 00:51:58.840
Yes, sir?

00:51:58.840 --> 00:52:03.930
STUDENT: Are we still going
to find R of t given only k?

00:52:03.930 --> 00:52:05.550
PROFESSOR TODA: But didn't we?

00:52:05.550 --> 00:52:07.300
We did.

00:52:07.300 --> 00:52:13.750
R of t was 2 cosine
t, 2 sine t, and t.

00:52:13.750 --> 00:52:16.280
All right?

00:52:16.280 --> 00:52:17.470
OK, so we are done.

00:52:17.470 --> 00:52:18.640
What did I say?

00:52:18.640 --> 00:52:22.410
I said that let's
prove-- it's a proof.

00:52:22.410 --> 00:52:27.305
Let's prove that the angle made
by the velocity to the train--

00:52:27.305 --> 00:52:30.635
to the train?-- to the direction
of motion, which is the helix.

00:52:30.635 --> 00:52:37.438
And the horizontal
plane is a constant.

00:52:37.438 --> 00:52:38.426
Is this hard?

00:52:38.426 --> 00:52:39.908
How are we going to do that?

00:52:39.908 --> 00:52:42.872
Now I start waking up,
because I was very tired.

00:52:42.872 --> 00:52:44.259
STUDENT: [INAUDIBLE]

00:52:44.259 --> 00:52:45.342
PROFESSOR TODA: Excuse me.

00:52:45.342 --> 00:52:46.840
STUDENT: [INAUDIBLE]

00:52:46.840 --> 00:53:01.242
PROFESSOR TODA: So you see,
the helix contains this point.

00:53:01.242 --> 00:53:03.920
And I'm looking at
the velocity vector

00:53:03.920 --> 00:53:06.310
that is standard to the helix.

00:53:06.310 --> 00:53:09.320
And I'll call that R prime.

00:53:09.320 --> 00:53:10.980
And then you say,
yea, but how am I

00:53:10.980 --> 00:53:13.990
going to compute that angle?

00:53:13.990 --> 00:53:15.632
What is that angle?

00:53:15.632 --> 00:53:17.987
STUDENT: It's a function of b.

00:53:17.987 --> 00:53:20.820


00:53:20.820 --> 00:53:21.980
PROFESSOR TODA: It will be.

00:53:21.980 --> 00:53:24.820
But we have to do it rigorously.

00:53:24.820 --> 00:53:27.925
So what's going to happen
for me to draw that angle?

00:53:27.925 --> 00:53:30.094
First of all, I should
take-- from the tip

00:53:30.094 --> 00:53:33.240
of the vector I should
draw perpendicular

00:53:33.240 --> 00:53:36.235
to the horizontal plane
passing through the point.

00:53:36.235 --> 00:53:37.110
And I'll get P prime.

00:53:37.110 --> 00:53:37.693
God knows why.

00:53:37.693 --> 00:53:41.490
I don't know why, I don't know
why. [? Q. ?] And this is PR,

00:53:41.490 --> 00:53:42.910
and P-- not PR.

00:53:42.910 --> 00:53:46.990
PR is too much
[INAUDIBLE] radius, M.

00:53:46.990 --> 00:53:50.835
OK, so then you would
take PQ and then

00:53:50.835 --> 00:53:52.604
you would measure this angle.

00:53:52.604 --> 00:53:54.770
Well, you have to be a
little bit smarter than that,

00:53:54.770 --> 00:53:58.390
because you can
prove something else.

00:53:58.390 --> 00:54:02.930
This is the complement of
another angle that you love.

00:54:02.930 --> 00:54:07.095
And using chapter 9 you can
do that angle in no time.

00:54:07.095 --> 00:54:15.840


00:54:15.840 --> 00:54:20.800
So this is the
complement of the angle

00:54:20.800 --> 00:54:23.500
formed by the velocity vector
of prime with the normal.

00:54:23.500 --> 00:54:26.680


00:54:26.680 --> 00:54:29.720
But not the normal principle
normal to the curve,

00:54:29.720 --> 00:54:32.340
but the normal to the plane.

00:54:32.340 --> 00:54:34.510
And what is the
normal to the plane?

00:54:34.510 --> 00:54:38.960
Let's call the principal normal
n to the curve big N bar.

00:54:38.960 --> 00:54:42.110
So in order to avoid confusion,
I'll write this little n.

00:54:42.110 --> 00:54:42.945
How about that?

00:54:42.945 --> 00:54:45.240
Do you guys know-- like
they do in mechanics.

00:54:45.240 --> 00:54:48.360
If you have two normals,
they call that 1n.

00:54:48.360 --> 00:54:51.200
1 is little n, and
stuff like that.

00:54:51.200 --> 00:54:52.980
So this is the complement.

00:54:52.980 --> 00:54:55.430
If I were able to prove
that that complement

00:54:55.430 --> 00:54:59.600
is a constant-- this is the
Stanford [? property-- ?] then

00:54:59.600 --> 00:55:00.990
I will be happy.

00:55:00.990 --> 00:55:03.100
Is it hard?

00:55:03.100 --> 00:55:04.286
No, for god's sake.

00:55:04.286 --> 00:55:07.034
Who is little n?

00:55:07.034 --> 00:55:11.350
Little n would be-- is
that the normal to a plane

00:55:11.350 --> 00:55:12.330
that you love?

00:55:12.330 --> 00:55:13.340
What is your plane?

00:55:13.340 --> 00:55:14.090
STUDENT: xy plane.

00:55:14.090 --> 00:55:16.520
PROFESSOR TODA: Your
plane is horizontal plane.

00:55:16.520 --> 00:55:17.320
STUDENT: xy.

00:55:17.320 --> 00:55:18.570
PROFESSOR TODA: Yes, xy plane.

00:55:18.570 --> 00:55:22.120
Or xy plane shifted,
shifted, shifted, shifted.

00:55:22.120 --> 00:55:23.180
That's the normal change?

00:55:23.180 --> 00:55:23.679
No.

00:55:23.679 --> 00:55:24.886
Who is the normal?

00:55:24.886 --> 00:55:26.174
STUDENT: [INAUDIBLE]

00:55:26.174 --> 00:55:27.340
PROFESSOR TODA: [INAUDIBLE].

00:55:27.340 --> 00:55:28.308
STUDENT: 0, 0, 1.

00:55:28.308 --> 00:55:29.308
PROFESSOR TODA: 0, 0, 1.

00:55:29.308 --> 00:55:29.808
OK.

00:55:29.808 --> 00:55:32.110
When I put 0 I was [INAUDIBLE].

00:55:32.110 --> 00:55:33.720
So this is k.

00:55:33.720 --> 00:55:36.420


00:55:36.420 --> 00:55:37.730
All right.

00:55:37.730 --> 00:55:39.620
And what is our prime?

00:55:39.620 --> 00:55:42.300
I was-- that was
a piece of cake.

00:55:42.300 --> 00:55:47.360
We did it last time minus a
sine t, a equals sine t and b.

00:55:47.360 --> 00:55:50.720


00:55:50.720 --> 00:55:53.620
Let's find that angle.

00:55:53.620 --> 00:55:54.690
Well, I don't know.

00:55:54.690 --> 00:55:58.320
You have to teach me, because
you have chapter 9 fresher

00:55:58.320 --> 00:56:01.590
in your memory than I have it.

00:56:01.590 --> 00:56:03.920
Are you taking attendance also?

00:56:03.920 --> 00:56:07.177
Are you writing your name down?

00:56:07.177 --> 00:56:08.260
Oh, no problem whatsoever.

00:56:08.260 --> 00:56:09.391
STUDENT: We didn't get it.

00:56:09.391 --> 00:56:10.807
PROFESSOR TODA:
You didn't get it.

00:56:10.807 --> 00:56:11.750
Circulate it.

00:56:11.750 --> 00:56:16.660
All right, so who is going
to help me with the angle?

00:56:16.660 --> 00:56:19.870
What is the angle between
two vectors, guys?

00:56:19.870 --> 00:56:24.070
That should be review from what
we just covered in chapter 9.

00:56:24.070 --> 00:56:27.980
Let me call them
u and v. And who's

00:56:27.980 --> 00:56:29.916
going to tell me how
I get that angle?

00:56:29.916 --> 00:56:31.960
STUDENT: [INAUDIBLE] is equal
to the inverse cosine of the dot

00:56:31.960 --> 00:56:33.290
product of [? the magnitude. ?]

00:56:33.290 --> 00:56:35.081
PROFESSOR TODA: Do you
like me to write arc

00:56:35.081 --> 00:56:36.450
cosine or cosine [INAUDIBLE].

00:56:36.450 --> 00:56:37.760
Doesn't matter.

00:56:37.760 --> 00:56:39.850
Arc cosine of--

00:56:39.850 --> 00:56:40.960
STUDENT: The dot products.

00:56:40.960 --> 00:56:47.476
PROFESSOR TODA: The dot
product between u and v.

00:56:47.476 --> 00:56:48.940
STUDENT: Over magnitude.

00:56:48.940 --> 00:56:52.466
PROFESSOR TODA: Divided by the
product of their magnitudes.

00:56:52.466 --> 00:56:54.692
Look, I will change the
order, because you're not

00:56:54.692 --> 00:56:56.140
going to like it.

00:56:56.140 --> 00:56:56.910
Doesn't matter.

00:56:56.910 --> 00:56:57.680
OK?

00:56:57.680 --> 00:57:03.450
So the angle phi between
my favorite vectors

00:57:03.450 --> 00:57:08.460
here is going to be
simply the dot product.

00:57:08.460 --> 00:57:09.570
That's a blessing.

00:57:09.570 --> 00:57:10.242
It's a constant.

00:57:10.242 --> 00:57:11.908
STUDENT: So you're
doing the dot product

00:57:11.908 --> 00:57:13.416
between the normal [INAUDIBLE]?

00:57:13.416 --> 00:57:14.999
PROFESSOR TODA:
Between this and that.

00:57:14.999 --> 00:57:18.065
So this is u and this
is v. So the dot product

00:57:18.065 --> 00:57:22.340
would be 0 plus v.
So the dot product

00:57:22.340 --> 00:57:28.520
is arc cosine of v, which,
thank god, is a constant.

00:57:28.520 --> 00:57:30.310
I don't have to do
anything anymore.

00:57:30.310 --> 00:57:33.154
I'm done with the proof
bit, because arc cosine

00:57:33.154 --> 00:57:36.000
of a constant will
be a constant.

00:57:36.000 --> 00:57:36.720
OK?

00:57:36.720 --> 00:57:37.600
All right.

00:57:37.600 --> 00:57:40.850
So I have v over what?

00:57:40.850 --> 00:57:45.090
What is the length
of this vector?

00:57:45.090 --> 00:57:46.750
1. [INAUDIBLE].

00:57:46.750 --> 00:57:50.510
What's the length
of that vector?

00:57:50.510 --> 00:57:55.900
Square root of a
squared plus b squared.

00:57:55.900 --> 00:57:56.430
All right?

00:57:56.430 --> 00:58:01.831


00:58:01.831 --> 00:58:05.280
STUDENT: How did
you [INAUDIBLE].

00:58:05.280 --> 00:58:07.647
PROFESSOR TODA: So now
let me ask you one thing.

00:58:07.647 --> 00:58:11.362


00:58:11.362 --> 00:58:13.910
What kind of function
is arc cosine?

00:58:13.910 --> 00:58:16.430
Of course I said arc cosine
of a constant is a constant.

00:58:16.430 --> 00:58:18.390
What kind of a
function is arc cosine?

00:58:18.390 --> 00:58:21.740
I'm doing review with you
because I think it's useful.

00:58:21.740 --> 00:58:26.068
Arc cosine is defined on
what with values in what?

00:58:26.068 --> 00:58:30.289


00:58:30.289 --> 00:58:32.640
STUDENT: Repeat the question?

00:58:32.640 --> 00:58:33.890
PROFESSOR TODA: Arc cosine.

00:58:33.890 --> 00:58:36.100
Or cosine inverse,
like Ryan prefers.

00:58:36.100 --> 00:58:38.130
Cosine inverse is
the same thing.

00:58:38.130 --> 00:58:40.440
It's a function defined
by where to where?

00:58:40.440 --> 00:58:43.190
Cosine is defined
from where to where?

00:58:43.190 --> 00:58:46.136
From R to minus 1.

00:58:46.136 --> 00:58:47.740
It's a cosine of t.

00:58:47.740 --> 00:58:49.690
t could be any real number.

00:58:49.690 --> 00:58:51.800
The range is minus 1, 1.

00:58:51.800 --> 00:58:53.242
Close the interval.

00:58:53.242 --> 00:58:54.950
STUDENT: So it's-- so
I just wonder why--

00:58:54.950 --> 00:58:57.014
PROFESSOR TODA: Minus
1 to 1, close interval.

00:58:57.014 --> 00:58:58.315
But pay attention, please.

00:58:58.315 --> 00:59:03.040
Because it cannot go back to R.
It has to be a 1 to 1 function.

00:59:03.040 --> 00:59:05.960
You cannot have an inverse
function if you don't take

00:59:05.960 --> 00:59:09.242
a restriction of a
function to be 1 to 1.

00:59:09.242 --> 00:59:11.597
And we took that
restriction of a function.

00:59:11.597 --> 00:59:14.894
And do you remember what it was?

00:59:14.894 --> 00:59:15.840
[INTERPOSING VOICES]

00:59:15.840 --> 00:59:17.640
PROFESSOR TODA: 0 to pi.

00:59:17.640 --> 00:59:19.710
Now, on this one
I'm really happy.

00:59:19.710 --> 00:59:23.160
Because I asked
several people-- people

00:59:23.160 --> 00:59:27.133
come to my office to get
all sorts of transcripts,

00:59:27.133 --> 00:59:27.633
[INAUDIBLE].

00:59:27.633 --> 00:59:30.600
And in trigonometry
I asked one student,

00:59:30.600 --> 00:59:31.910
so you took trigonometry.

00:59:31.910 --> 00:59:32.910
So do you remember that?

00:59:32.910 --> 00:59:34.360
He didn't remember that.

00:59:34.360 --> 00:59:35.300
So I'm glad you do.

00:59:35.300 --> 00:59:40.130
How about when I had
the sine inverse?

00:59:40.130 --> 00:59:44.760
How was my restriction so that
would be a 1 to 1 function?

00:59:44.760 --> 00:59:46.900
It's got to go
from minus 1 to 1.

00:59:46.900 --> 00:59:48.180
What is the range?

00:59:48.180 --> 00:59:49.016
[INTERPOSING VOICES]

00:59:49.016 --> 00:59:51.300
PROFESSOR TODA: Minus pi over 2.

00:59:51.300 --> 00:59:53.253
You guys know your trig.

00:59:53.253 --> 00:59:53.752
Good.

00:59:53.752 --> 00:59:55.630
That's a very good thing.

00:59:55.630 --> 00:59:59.440
You were in high school
when you learned that?

00:59:59.440 --> 01:00:00.430
Here at Lubbock High?

01:00:00.430 --> 01:00:01.150
STUDENT: Yes.

01:00:01.150 --> 01:00:02.066
PROFESSOR TODA: Great.

01:00:02.066 --> 01:00:03.560
Good job, Lubbock High.

01:00:03.560 --> 01:00:06.230
But many students, I caught
them, who wanted credit

01:00:06.230 --> 01:00:08.350
for trig who didn't know that.

01:00:08.350 --> 01:00:09.680
Good.

01:00:09.680 --> 01:00:19.870
So since arc cosine is a
function that is of 0, pi,

01:00:19.870 --> 01:00:25.230
for example, what if my--
let me give you an example.

01:00:25.230 --> 01:00:26.910
What was last time, guys?

01:00:26.910 --> 01:00:30.800
a was 1. b was 1.

01:00:30.800 --> 01:00:32.010
For one example.

01:00:32.010 --> 01:00:33.810
In that case, 1 with 5b.

01:00:33.810 --> 01:00:36.327
STUDENT: [INAUDIBLE] ask you
for the example you just did?

01:00:36.327 --> 01:00:37.535
PROFESSOR TODA: No last time.

01:00:37.535 --> 01:00:39.560
STUDENT: A was 3 and b was--

01:00:39.560 --> 01:00:44.255
PROFESSOR TODA: So what would
that be, in this case 5?

01:00:44.255 --> 01:00:46.680
STUDENT: That would be
b over the square root--

01:00:46.680 --> 01:00:47.560
STUDENT: 3 over pi.

01:00:47.560 --> 01:00:49.985


01:00:49.985 --> 01:00:52.360
PROFESSOR TODA: a is 1 and b
is 1, like we did last time.

01:00:52.360 --> 01:00:55.050
STUDENT: [INAUDIBLE]
2, which is--

01:00:55.050 --> 01:00:57.059
PROFESSOR TODA: Plug
in 1 is a, b is 1.

01:00:57.059 --> 01:00:57.600
What is this?

01:00:57.600 --> 01:00:59.417
STUDENT: It's just pi over 4.

01:00:59.417 --> 01:01:00.500
PROFESSOR TODA: Pi over 4.

01:01:00.500 --> 01:01:06.800
So pi will be our cosine, of
1 over square root 2, which

01:01:06.800 --> 01:01:12.090
is 45 degree angle, which is--
you said pi over 4, right?

01:01:12.090 --> 01:01:14.540
[INAUDIBLE].

01:01:14.540 --> 01:01:19.800
So exactly, you would
have that over here.

01:01:19.800 --> 01:01:22.580
This is where the
cosine [INAUDIBLE].

01:01:22.580 --> 01:01:28.220
Now you see, guys, the way we
have, the way I assume a and b,

01:01:28.220 --> 01:01:30.980
the way anybody-- the
book also introduces

01:01:30.980 --> 01:01:33.350
a and b to be positive numbers.

01:01:33.350 --> 01:01:37.230
Can you tell me what kind
of angle phi will be,

01:01:37.230 --> 01:01:39.900
not only restricted to 0 pi?

01:01:39.900 --> 01:01:41.360
Well, a is positive.

01:01:41.360 --> 01:01:42.480
b is positive.

01:01:42.480 --> 01:01:44.360
a doesn't matter.

01:01:44.360 --> 01:01:46.670
The whole thing
will be positive.

01:01:46.670 --> 01:01:50.510
Arc cosine of a
positive number--

01:01:50.510 --> 01:01:52.010
STUDENT: Between
0 and pi over 2.

01:01:52.010 --> 01:01:53.010
PROFESSOR TODA: That is.

01:01:53.010 --> 01:01:56.326
Yeah, so it has to be
between 0 and pi over 2.

01:01:56.326 --> 01:01:57.950
So it's going to be
only this quadrant.

01:01:57.950 --> 01:01:59.640
Does that make sense?

01:01:59.640 --> 01:02:03.388
Yes, think with the
imagination of your eyes,

01:02:03.388 --> 01:02:05.220
or the eyes of your imagination.

01:02:05.220 --> 01:02:06.430
OK.

01:02:06.430 --> 01:02:08.360
You have a cylinder.

01:02:08.360 --> 01:02:10.270
And you are moving
along that cylinder.

01:02:10.270 --> 01:02:12.160
And this is how you turn.

01:02:12.160 --> 01:02:14.400
You turn with that little train.

01:02:14.400 --> 01:02:16.580
Du-du-du-du-du, you go up.

01:02:16.580 --> 01:02:19.910
When you turn the
velocity vector and you

01:02:19.910 --> 01:02:23.123
look at the-- mm.

01:02:23.123 --> 01:02:23.956
STUDENT: The normal.

01:02:23.956 --> 01:02:24.860
PROFESSOR TODA: The normal!

01:02:24.860 --> 01:02:25.360
Thank you.

01:02:25.360 --> 01:02:30.245
The z axis, you always have an
angle between 0 and pi over 2.

01:02:30.245 --> 01:02:31.712
So it makes sense.

01:02:31.712 --> 01:02:34.157
I'm going to go ahead and
erase the whole thing.

01:02:34.157 --> 01:02:41.020


01:02:41.020 --> 01:02:47.720
So we reviewed, more or less, s
of t, integration, derivation,

01:02:47.720 --> 01:02:52.020
moving from position vector
to velocity to acceleration

01:02:52.020 --> 01:02:56.260
and back, acceleration to
velocity to position vector,

01:02:56.260 --> 01:02:58.500
the meaning of arclength.

01:02:58.500 --> 01:03:00.890
There are some things I
would like to tell you,

01:03:00.890 --> 01:03:07.829
because Ryan asked me a few more
questions about the curvature.

01:03:07.829 --> 01:03:11.560
The curvature
formula depends very

01:03:11.560 --> 01:03:16.830
much on the type of formula
you used for the curve.

01:03:16.830 --> 01:03:18.800
So you say, wait,
wait, wait, Magdelena,

01:03:18.800 --> 01:03:21.290
you told us-- you
are confusing us.

01:03:21.290 --> 01:03:23.710
You told us that the
curvature is uniquely

01:03:23.710 --> 01:03:33.740
defined as the magnitude
of the acceleration vector

01:03:33.740 --> 01:03:36.800
when the law of motion
is an arclength.

01:03:36.800 --> 01:03:38.850
And that is correct.

01:03:38.850 --> 01:03:43.190
So suppose my original law of
motion was R of t [INAUDIBLE]

01:03:43.190 --> 01:03:47.750
time, any time, t,
any time parameter.

01:03:47.750 --> 01:03:49.370
I'm making a face.

01:03:49.370 --> 01:03:53.290
But then from that we switch
to something beautiful,

01:03:53.290 --> 01:03:56.445
which is called the
arclength parametrization.

01:03:56.445 --> 01:03:58.280
Why am I so happy?

01:03:58.280 --> 01:04:04.970
Because in this parametrization
the magnitude of the speed

01:04:04.970 --> 01:04:07.120
is 1.

01:04:07.120 --> 01:04:17.700
And I define k to
be the magnitude

01:04:17.700 --> 01:04:19.870
of R double prime of s, right?

01:04:19.870 --> 01:04:22.430
The acceleration only in
the arclength [? time ?]

01:04:22.430 --> 01:04:23.426
parameterization.

01:04:23.426 --> 01:04:24.920
And then this was
the definition.

01:04:24.920 --> 01:04:30.410


01:04:30.410 --> 01:04:36.550
A. Can you prove-- what?

01:04:36.550 --> 01:04:40.190
Can you prove the
following formula?

01:04:40.190 --> 01:04:52.200


01:04:52.200 --> 01:04:58.514
T prime of s equals
k times N of s.

01:04:58.514 --> 01:05:02.550
This is famous for people
who do-- not for everybody.

01:05:02.550 --> 01:05:05.530
But imagine you have
an engineer who does

01:05:05.530 --> 01:05:08.430
research of the laws of motion.

01:05:08.430 --> 01:05:13.130
Maybe he works for
the railways and he's

01:05:13.130 --> 01:05:17.170
looking at skew
curves, or he is one

01:05:17.170 --> 01:05:20.480
of those people who
project the ski slopes,

01:05:20.480 --> 01:05:25.360
or all sorts of winter sports
slope or something, that

01:05:25.360 --> 01:05:29.150
involve a lot of
curvatures and torsions.

01:05:29.150 --> 01:05:31.240
That guy has to know
the Frenet formula.

01:05:31.240 --> 01:05:34.260
So this is the famous
first Frenet formula.

01:05:34.260 --> 01:05:40.140


01:05:40.140 --> 01:05:46.690
Frenet was a mathematician
who gave the name to the TNB

01:05:46.690 --> 01:05:48.476
vectors, the trihedron.

01:05:48.476 --> 01:05:49.970
You have the T was what?

01:05:49.970 --> 01:05:52.958
The T was the tangent
[INAUDIBLE] vector.

01:05:52.958 --> 01:05:58.180
The N was the
principal unit normal.

01:05:58.180 --> 01:06:00.930
In those videos that I'm
watching that I also sent you--

01:06:00.930 --> 01:06:02.400
I like most of them.

01:06:02.400 --> 01:06:05.660
I like the Khan Academy
more than everything.

01:06:05.660 --> 01:06:09.100
Also I like the one that
was made by Dr. [? Gock ?]

01:06:09.100 --> 01:06:12.540
But Dr. [? Gock ?] made a
little bit of a mistake.

01:06:12.540 --> 01:06:13.920
A conceptual mistake.

01:06:13.920 --> 01:06:17.013
We all make mistakes by
misprinting or misreading

01:06:17.013 --> 01:06:18.366
or goofy mistake.

01:06:18.366 --> 01:06:20.690
But he said this is
the normal vector.

01:06:20.690 --> 01:06:22.930
This is not-- it's the
principle normal vectors.

01:06:22.930 --> 01:06:24.726
There are many normals.

01:06:24.726 --> 01:06:26.630
There is only one
tangent direction,

01:06:26.630 --> 01:06:29.010
but in terms of normals
there are many that

01:06:29.010 --> 01:06:30.914
are-- all of these are normals.

01:06:30.914 --> 01:06:34.940
All the perpendicular in
the plane-- [INAUDIBLE]

01:06:34.940 --> 01:06:39.780
so this is my law of motion,
T. All this plane is normal.

01:06:39.780 --> 01:06:41.960
So any of these
vectors is a normal.

01:06:41.960 --> 01:06:44.990
The one we choose and
defined as T prime

01:06:44.990 --> 01:06:47.220
over T prime [INAUDIBLE]
absolute values

01:06:47.220 --> 01:06:48.990
called the principal normal.

01:06:48.990 --> 01:06:51.350
It's like the principal
of a high school.

01:06:51.350 --> 01:06:53.230
He is important.

01:06:53.230 --> 01:06:58.352
So T and B-- B goes
down, or goes-- down.

01:06:58.352 --> 01:07:04.540
Well, yeah, because B is T cross
N. So when you find the Frenet

01:07:04.540 --> 01:07:10.440
Trihedron, TNB, it's like that.

01:07:10.440 --> 01:07:15.685
T, N, and B. What's special,
why do we call it the frame,

01:07:15.685 --> 01:07:18.460
is that every
[? payer ?] of vectors

01:07:18.460 --> 01:07:20.090
are mutually orthogonal.

01:07:20.090 --> 01:07:22.270
And they are all unit vectors.

01:07:22.270 --> 01:07:25.910
This is the famous Frenet frame.

01:07:25.910 --> 01:07:27.580
Now, Mr. Frenet was a smart guy.

01:07:27.580 --> 01:07:32.330
He found-- I don't know whether
he was adopting mathematics

01:07:32.330 --> 01:07:33.050
or not.

01:07:33.050 --> 01:07:34.290
Doesn't matter.

01:07:34.290 --> 01:07:37.970
He found a bunch of formulas,
of which this is the first one.

01:07:37.970 --> 01:07:42.265
And it's called a
first Frenet formula.

01:07:42.265 --> 01:07:44.230
That's one thing
I want to ask you.

01:07:44.230 --> 01:07:47.170
And then I'm going to give you
more formulas for curvatures,

01:07:47.170 --> 01:07:50.460
depending on how you
define your curve.

01:07:50.460 --> 01:08:08.832
So for example, base B
based on the definition one

01:08:08.832 --> 01:08:18.870
can prove that for a curve
that is not parametrizing

01:08:18.870 --> 01:08:22.870
arclength-- you say, ugh,
forget about parametrization

01:08:22.870 --> 01:08:23.590
in arclength.

01:08:23.590 --> 01:08:26.840
This time you're
assuming, I want to know!

01:08:26.840 --> 01:08:29.410
I'm coming to this
because Ryan asked.

01:08:29.410 --> 01:08:32.381
I want to know, what is
the formula directly?

01:08:32.381 --> 01:08:34.439
Is there a direct
formula that comes

01:08:34.439 --> 01:08:38.529
from here for the curvature?

01:08:38.529 --> 01:08:41.310
Yeah, but it's a lot
more complicated.

01:08:41.310 --> 01:08:45.265
When I was a freshman, maybe
a freshman or a sophomore,

01:08:45.265 --> 01:08:48.090
I don't remember, when
I was asked to memorize

01:08:48.090 --> 01:08:52.689
that, I did not memorize it.

01:08:52.689 --> 01:08:56.649
Then when I started working
as a faculty member,

01:08:56.649 --> 01:09:01.810
I saw that I am supposed
to ask it from my students.

01:09:01.810 --> 01:09:05.578
So this is going to be
R prime plus product

01:09:05.578 --> 01:09:12.057
R double prime in magnitude
over R prime cubed.

01:09:12.057 --> 01:09:14.550
So how am I supposed
to remember that?

01:09:14.550 --> 01:09:15.658
It's not so easy.

01:09:15.658 --> 01:09:17.800
Are you cold there?

01:09:17.800 --> 01:09:18.649
It's cold there.

01:09:18.649 --> 01:09:22.590
I don't know how
these roofs are made.

01:09:22.590 --> 01:09:24.670
Velocity times acceleration.

01:09:24.670 --> 01:09:26.620
This is what I try
to teach myself.

01:09:26.620 --> 01:09:29.810
I was old already, 26 or 27.

01:09:29.810 --> 01:09:32.979
Velocity times
acceleration, cross product,

01:09:32.979 --> 01:09:35.760
take the magnitude,
divide by the speed, cube.

01:09:35.760 --> 01:09:36.810
Oh my god.

01:09:36.810 --> 01:09:41.340
So I was supposed to know
that when I was 18 or 19.

01:09:41.340 --> 01:09:44.510
Now, I was teaching majors
of mechanical engineering.

01:09:44.510 --> 01:09:45.840
They knew that by heart.

01:09:45.840 --> 01:09:48.210
I didn't, so I had to learn it.

01:09:48.210 --> 01:09:51.475
So if one is too
lazy or it's simply

01:09:51.475 --> 01:09:54.915
inconvenient to try to
reparametrize from R of T

01:09:54.915 --> 01:10:00.790
being arclength parametrization
R of s and do that thing here,

01:10:00.790 --> 01:10:05.300
one can just plug in and
find the curvature like that.

01:10:05.300 --> 01:10:08.450
For example, guys,
as Ryan asked,

01:10:08.450 --> 01:10:13.290
if I have A cosine, [INAUDIBLE],
and I do this with respect

01:10:13.290 --> 01:10:16.740
to T, can I get k
without-- k will not

01:10:16.740 --> 01:10:18.950
depend on T or s or tau.

01:10:18.950 --> 01:10:20.860
It will always be the same.

01:10:20.860 --> 01:10:23.400
I will still get A
over A squared plus B

01:10:23.400 --> 01:10:25.260
squared, no matter what.

01:10:25.260 --> 01:10:28.930
So even if I use this
formula for my helix,

01:10:28.930 --> 01:10:30.950
I'm going to get the same thing.

01:10:30.950 --> 01:10:33.060
I'll get A over A
squared plus B squared,

01:10:33.060 --> 01:10:35.390
because curvature
is an invariant.

01:10:35.390 --> 01:10:38.510
There is another invariant
that's-- the other invariant,

01:10:38.510 --> 01:10:40.550
of course, in space
is called torsion.

01:10:40.550 --> 01:10:43.680
We want to talk a little
bit about that later.

01:10:43.680 --> 01:10:48.780
So is this hard?

01:10:48.780 --> 01:10:49.280
No.

01:10:49.280 --> 01:10:50.450
It shouldn't be hard.

01:10:50.450 --> 01:10:54.930
And you guys should be able
to help me on that, hopefully.

01:10:54.930 --> 01:10:56.600
How do we prove that?

01:10:56.600 --> 01:10:58.480
STUDENT: N is G
prime [INAUDIBLE].

01:10:58.480 --> 01:11:01.950


01:11:01.950 --> 01:11:03.450
PROFESSOR TODA:
That's right, proof.

01:11:03.450 --> 01:11:06.080
And that's a very good
start, wouldn't you say?

01:11:06.080 --> 01:11:09.090
So what were the definitions?

01:11:09.090 --> 01:11:14.350
Let me start from
the definition of T.

01:11:14.350 --> 01:11:17.180
That's going to be-- I
am in hard planes, right?

01:11:17.180 --> 01:11:21.050
So you say, wait, why do
you write it as a quotient?

01:11:21.050 --> 01:11:22.430
You're being silly.

01:11:22.430 --> 01:11:24.530
You are in arclength, Magdalena.

01:11:24.530 --> 01:11:25.470
I am.

01:11:25.470 --> 01:11:26.340
I am.

01:11:26.340 --> 01:11:29.860
I just pretend that
I cannot see that.

01:11:29.860 --> 01:11:32.160
So if I'm in
arclength, that means

01:11:32.160 --> 01:11:35.870
that the denominator is 1.

01:11:35.870 --> 01:11:37.320
So I'm being silly.

01:11:37.320 --> 01:11:44.380
So R prime of s is
T. Say it again.

01:11:44.380 --> 01:11:49.280
R prime of s is T. OK.

01:11:49.280 --> 01:11:53.720
Now, did we know that
T and N are orthogonal?

01:11:53.720 --> 01:12:00.530


01:12:00.530 --> 01:12:04.350
How did we know that T
and N were orthogonal?

01:12:04.350 --> 01:12:07.524
We proved that last
time, actually.

01:12:07.524 --> 01:12:11.003
T and N are orthogonal.

01:12:11.003 --> 01:12:12.991
How do I write
that? [INAUDIBLE].

01:12:12.991 --> 01:12:15.973


01:12:15.973 --> 01:12:21.990
Meaning that T is
perpendicular to N, right?

01:12:21.990 --> 01:12:24.110
From the definition.

01:12:24.110 --> 01:12:26.000
You said it right, Sandra.

01:12:26.000 --> 01:12:28.000
But why is it from
the definition

01:12:28.000 --> 01:12:30.900
that I can jump to
conclusions and say, oh,

01:12:30.900 --> 01:12:36.120
since I have T prime here, then
this is perpendicular to T?

01:12:36.120 --> 01:12:37.435
Well, we did that last time.

01:12:37.435 --> 01:12:39.236
STUDENT: Two parallel vectors.

01:12:39.236 --> 01:12:41.110
PROFESSOR TODA: We did
it-- how did we do it?

01:12:41.110 --> 01:12:42.250
We did this last.

01:12:42.250 --> 01:12:45.800
We said T dot T equals 1.

01:12:45.800 --> 01:12:47.960
Prime the whole thing.

01:12:47.960 --> 01:12:54.270
T prime times T plus T times T
prime, T dot T prime will be 0.

01:12:54.270 --> 01:12:57.030
So T and T prime are
perpendicular always.

01:12:57.030 --> 01:12:58.150
Right?

01:12:58.150 --> 01:13:03.030
OK, so the whole thing is a
colinear vector to T prime.

01:13:03.030 --> 01:13:05.025
It's just T prime
times the scalar.

01:13:05.025 --> 01:13:08.320
So he must be
perpendicular to T.

01:13:08.320 --> 01:13:10.720
So T and N are perpendicular.

01:13:10.720 --> 01:13:14.680
So I do have the
direction of motion.

01:13:14.680 --> 01:13:19.482
I know that I must
have some scalar here.

01:13:19.482 --> 01:13:22.796


01:13:22.796 --> 01:13:26.630
How do I prove that this
scalar is the curvature?

01:13:26.630 --> 01:13:30.510


01:13:30.510 --> 01:13:35.995
So if I have-- if they
are colinear-- why are

01:13:35.995 --> 01:13:36.740
they colinear?

01:13:36.740 --> 01:13:42.200
T perpendicular to T
prime implies that T prime

01:13:42.200 --> 01:13:46.020
is colinear to N. Say it again.

01:13:46.020 --> 01:13:49.860
If T and T prime are
perpendicular to one another,

01:13:49.860 --> 01:13:53.260
that means T prime is
calling it to the normal.

01:13:53.260 --> 01:13:58.471
So here I may have
alph-- no alpha.

01:13:58.471 --> 01:14:00.260
I don't know!

01:14:00.260 --> 01:14:03.700
Alpha over [INAUDIBLE]
sounds like a curve.

01:14:03.700 --> 01:14:04.616
Give me some function.

01:14:04.616 --> 01:14:08.864


01:14:08.864 --> 01:14:09.530
STUDENT: u of s?

01:14:09.530 --> 01:14:11.050
PROFESSOR TODA: Gamma of s.

01:14:11.050 --> 01:14:15.360
u of s, I don't know.

01:14:15.360 --> 01:14:17.410
So how did I conclude that?

01:14:17.410 --> 01:14:19.500
From T perpendicular to T prime.

01:14:19.500 --> 01:14:22.210
Now from here on, you
have to tell me why

01:14:22.210 --> 01:14:29.430
gamma must be exactly kappa.

01:14:29.430 --> 01:14:33.712
Well, let's take
T prime from here.

01:14:33.712 --> 01:14:38.090
T prime from here
will give me what?

01:14:38.090 --> 01:14:40.730
T prime is our prime prime.

01:14:40.730 --> 01:14:42.300
Say what?

01:14:42.300 --> 01:14:43.200
Our prime prime.

01:14:43.200 --> 01:14:44.657
What is our prime prime?

01:14:44.657 --> 01:14:46.565
Our [? problem ?] prime of s.

01:14:46.565 --> 01:14:48.582
STUDENT: You have one
too many primes inside.

01:14:48.582 --> 01:14:49.665
PROFESSOR TODA: Oh my god.

01:14:49.665 --> 01:14:50.165
Yeah.

01:14:50.165 --> 01:14:52.770


01:14:52.770 --> 01:14:54.120
So R prime prime.

01:14:54.120 --> 01:14:58.430
So T prime in
absolute value will

01:14:58.430 --> 01:15:02.650
be exactly R double prime of s.

01:15:02.650 --> 01:15:04.990
Oh, OK.

01:15:04.990 --> 01:15:10.130
Note that from here also T
prime of s in absolute value,

01:15:10.130 --> 01:15:13.840
in magnitude, I'm sorry,
has to be gamma of s.

01:15:13.840 --> 01:15:14.820
Why is that?

01:15:14.820 --> 01:15:17.470
Because the magnitude of N is 1.

01:15:17.470 --> 01:15:20.930
N is unique vector
by definition.

01:15:20.930 --> 01:15:24.870
So these two guys
have to coincide.

01:15:24.870 --> 01:15:27.132
So R double prime,
the best thing

01:15:27.132 --> 01:15:28.590
that I need to do,
it must coincide

01:15:28.590 --> 01:15:30.500
with the scalar gamma of s.

01:15:30.500 --> 01:15:32.840
So who is the
mysterious gamma of s?

01:15:32.840 --> 01:15:36.340
He has no chance
but being this guy.

01:15:36.340 --> 01:15:38.660
But this guy has a name.

01:15:38.660 --> 01:15:41.920
This guy, he's the curvature
[? cap ?] of s by definition.

01:15:41.920 --> 01:15:45.738


01:15:45.738 --> 01:15:49.126
Remember, Ryan, this
is the definition.

01:15:49.126 --> 01:15:51.546
So by definition the
curvature was the magnitude

01:15:51.546 --> 01:15:55.010
of the acceleration
in arclength.

01:15:55.010 --> 01:15:55.910
OK.

01:15:55.910 --> 01:15:58.460
Both of these guys are
T prime in magnitude.

01:15:58.460 --> 01:16:01.770
So they must be equal
from here and here.

01:16:01.770 --> 01:16:04.696
It implies that my
gamma must be kappa.

01:16:04.696 --> 01:16:07.780
And I prove the formula.

01:16:07.780 --> 01:16:09.130
OK.

01:16:09.130 --> 01:16:10.911
How do you say
something is proved?

01:16:10.911 --> 01:16:12.294
Because this is what we wanted.

01:16:12.294 --> 01:16:16.235
We wanted to replace this
generic scalar function

01:16:16.235 --> 01:16:20.080
to prove that this is
just the curvature.

01:16:20.080 --> 01:16:20.580
QED.

01:16:20.580 --> 01:16:24.420


01:16:24.420 --> 01:16:26.960
That's exactly what
we wanted to prove.

01:16:26.960 --> 01:16:29.046
Now, whatever scalar
function you have here,

01:16:29.046 --> 01:16:30.170
that must be the curvature.

01:16:30.170 --> 01:16:34.460


01:16:34.460 --> 01:16:36.415
Very smart guy, this Mr. Frenet.

01:16:36.415 --> 01:16:39.720


01:16:39.720 --> 01:16:40.990
I'm now going to take a break.

01:16:40.990 --> 01:16:43.830
If you want to go use the
bathroom really quickly,

01:16:43.830 --> 01:16:45.025
feel free to do it.

01:16:45.025 --> 01:16:47.590


01:16:47.590 --> 01:16:49.470
I'm just going to
clean the board,

01:16:49.470 --> 01:16:51.694
and I'll keep going
in a few minutes.

01:16:51.694 --> 01:17:50.501


01:17:50.501 --> 01:17:51.334
STUDENT: [INAUDIBLE]

01:17:51.334 --> 01:17:55.799


01:17:55.799 --> 01:17:57.298
PROFESSOR TODA: I
will do it-- well,

01:17:57.298 --> 01:18:01.274
actually I want to do a
different example, simple one,

01:18:01.274 --> 01:18:06.244
which is a plain curve, and show
that the curvature has a very

01:18:06.244 --> 01:18:11.036
pretty formula that you
could [INAUDIBLE] memorize,

01:18:11.036 --> 01:18:13.516
that in essence is the same.

01:18:13.516 --> 01:18:17.484
But it depends on
y equals f of x.

01:18:17.484 --> 01:18:19.468
[INAUDIBLE] So if
somebody gives you

01:18:19.468 --> 01:18:22.444
a plane called y
equals f of x, can you

01:18:22.444 --> 01:18:25.420
write that curvature
[INAUDIBLE] function of f?

01:18:25.420 --> 01:18:26.908
And you can.

01:18:26.908 --> 01:18:30.545
And again, I was deep in
that when I was 18 or 19

01:18:30.545 --> 01:18:31.868
as a freshman.

01:18:31.868 --> 01:18:35.836
But unfortunately for me I
didn't learn it at that time.

01:18:35.836 --> 01:18:41.340
And several years later when
I started teaching engineers,

01:18:41.340 --> 01:18:43.776
well, they are
mostly mechanical.

01:18:43.776 --> 01:18:46.770
And mechanical
engineering [INAUDIBLE].

01:18:46.770 --> 01:18:50.263
They knew those, and they needed
those in every research paper.

01:18:50.263 --> 01:18:54.255
So I had to learn it
together with them.

01:18:54.255 --> 01:18:57.748
I'll worry about [INAUDIBLE].

01:18:57.748 --> 01:19:01.241
STUDENT: Can you do a really
ugly one, like [INAUDIBLE]?

01:19:01.241 --> 01:19:04.734
PROFESSOR TODA: I can
do some ugly ones.

01:19:04.734 --> 01:20:37.240


01:20:37.240 --> 01:20:47.670
And once you know the
general parametrization,

01:20:47.670 --> 01:20:51.908
it will give you a curvature.

01:20:51.908 --> 01:20:53.327
Now I'm testing your memory.

01:20:53.327 --> 01:20:55.230
Let's see what you remember.

01:20:55.230 --> 01:20:59.854
Um-- don't look at the notes.

01:20:59.854 --> 01:21:03.270
A positive function,
absolute-- actually,

01:21:03.270 --> 01:21:06.680
magnitude of what vector?

01:21:06.680 --> 01:21:07.580
STUDENT: R prime.

01:21:07.580 --> 01:21:17.060
PROFESSOR TODA: R prime velocity
plus acceleration speed cubed.

01:21:17.060 --> 01:21:18.540
Right?

01:21:18.540 --> 01:21:19.040
OK.

01:21:19.040 --> 01:21:24.070
Now, can we take advantage
of what we just learned

01:21:24.070 --> 01:21:30.055
and find-- you find
with me, of course, not

01:21:30.055 --> 01:21:33.654
as professor and student,
but like a group of students

01:21:33.654 --> 01:21:35.070
together.

01:21:35.070 --> 01:21:46.286
Let's find a simple
formula corresponding

01:21:46.286 --> 01:21:52.202
to the curvature
of a plane curve.

01:21:52.202 --> 01:21:59.597


01:21:59.597 --> 01:22:05.280
And the plane curve
could be [INAUDIBLE]

01:22:05.280 --> 01:22:09.020
in two different ways,
just because I want

01:22:09.020 --> 01:22:14.510
you to practice more on that.

01:22:14.510 --> 01:22:18.360
Either given as a general
parametrization-- guys,

01:22:18.360 --> 01:22:20.090
what is the general
parametrization

01:22:20.090 --> 01:22:24.510
I'm talking about
for a plane curve?

01:22:24.510 --> 01:22:26.400
x of t, y of t, right?

01:22:26.400 --> 01:22:28.660
x equals x of t.

01:22:28.660 --> 01:22:29.870
y equals y of t.

01:22:29.870 --> 01:22:34.400
So one should not have
to do that all the time,

01:22:34.400 --> 01:22:37.310
not have to do that for a
simplification like a playing

01:22:37.310 --> 01:22:38.190
card.

01:22:38.190 --> 01:22:41.710
We have to find another
formula that's pretty, right?

01:22:41.710 --> 01:22:43.210
Well, maybe it's not as pretty.

01:22:43.210 --> 01:22:45.250
But when is it really pretty?

01:22:45.250 --> 01:22:49.090
I bet it's going to be really
pretty if you have a plane

01:22:49.090 --> 01:22:54.610
curve even as you're used
to in an explicit form--

01:22:54.610 --> 01:22:56.600
I keep going.

01:22:56.600 --> 01:22:59.910
No stop. [INAUDIBLE].

01:22:59.910 --> 01:23:01.110
I think it's better.

01:23:01.110 --> 01:23:03.450
We make better use
of time this way.

01:23:03.450 --> 01:23:06.890
Or y equals f of x.

01:23:06.890 --> 01:23:12.600


01:23:12.600 --> 01:23:17.953
This is an explicit way to
write the equation of a curve.

01:23:17.953 --> 01:23:20.660


01:23:20.660 --> 01:23:23.330
OK, so what do we need to do?

01:23:23.330 --> 01:23:26.430
That should be really easy.

01:23:26.430 --> 01:23:32.958
R of t being the first case of
our general parametrization,

01:23:32.958 --> 01:23:40.720
x equals x of t, y equals y of
t will be-- who tells me, guys,

01:23:40.720 --> 01:23:43.470
that-- this is in your hands.

01:23:43.470 --> 01:23:47.700
Now you convinced me
that, for whatever reason,

01:23:47.700 --> 01:23:49.880
you [INAUDIBLE].

01:23:49.880 --> 01:23:51.800
You became friends
with these curves.

01:23:51.800 --> 01:23:52.760
I don't know when.

01:23:52.760 --> 01:23:54.680
I guess in the process
of doing homework.

01:23:54.680 --> 01:23:55.650
Am I right?

01:23:55.650 --> 01:24:00.480
I think you did not quite like
them before or the last week.

01:24:00.480 --> 01:24:02.470
But I think you're
friends with them now.

01:24:02.470 --> 01:24:06.556
x of t, y of t.

01:24:06.556 --> 01:24:07.990
Let people talk.

01:24:07.990 --> 01:24:12.770


01:24:12.770 --> 01:24:13.740
STUDENT: 0.

01:24:13.740 --> 01:24:15.670
PROFESSOR TODA: So.

01:24:15.670 --> 01:24:16.170
Great.

01:24:16.170 --> 01:24:21.110
And then R prime of t will be
x prime of t, y prime of t,

01:24:21.110 --> 01:24:21.760
and 0.

01:24:21.760 --> 01:24:24.487
I assume this to
be always non-zero.

01:24:24.487 --> 01:24:26.235
I have a regular curve.

01:24:26.235 --> 01:24:30.680
R double prime will be--
x double prime where

01:24:30.680 --> 01:24:34.146
double prime-- we
did the review today

01:24:34.146 --> 01:24:36.450
of the lasting acceleration.

01:24:36.450 --> 01:24:39.630
Now, your friends over
here, are they nice or mean?

01:24:39.630 --> 01:24:42.600
I hope they are not so mean.

01:24:42.600 --> 01:24:45.810
The cross product is
a friendly fellow.

01:24:45.810 --> 01:24:48.970
You have i, j, k, and
then the second row

01:24:48.970 --> 01:24:50.640
would be x prime, y prime, 0.

01:24:50.640 --> 01:24:54.846
The last row would be x double
prime, y double prime, 0.

01:24:54.846 --> 01:24:58.550
And it's a piece of cake.

01:24:58.550 --> 01:25:02.146


01:25:02.146 --> 01:25:03.520
OK, piece of cake,
piece of cake.

01:25:03.520 --> 01:25:08.960
But I want to know
what the answer is.

01:25:08.960 --> 01:25:15.630
So you have exactly 15 seconds
to answer this question.

01:25:15.630 --> 01:25:23.298
Who is R prime plus R double
prime as a [? coordinate. ?]

01:25:23.298 --> 01:25:25.266
[INTERPOSING VOICES]

01:25:25.266 --> 01:25:28.710


01:25:28.710 --> 01:25:29.700
PROFESSOR TODA: Good.

01:25:29.700 --> 01:25:35.372
x prime, y double prime minus x
double prime, y prime times k.

01:25:35.372 --> 01:25:37.720
And it doesn't matter
when I take the magnitude,

01:25:37.720 --> 01:25:40.600
because magnitude of k is 1.

01:25:40.600 --> 01:25:42.380
So I discovered some.

01:25:42.380 --> 01:25:46.700
This is how mathematicians like
to discover new formulas based

01:25:46.700 --> 01:25:48.730
on the formulas they
[? knew. ?] They

01:25:48.730 --> 01:25:50.090
have a lot of satisfaction.

01:25:50.090 --> 01:25:51.020
Look what I got.

01:25:51.020 --> 01:25:56.630
Of course, they in general have
more complicated things to do,

01:25:56.630 --> 01:25:58.952
and they have to
check and recheck.

01:25:58.952 --> 01:26:06.305
But every piece of a
computation is a challenge.

01:26:06.305 --> 01:26:10.310
And that gives
people satisfaction.

01:26:10.310 --> 01:26:14.900
And when they make a mistake, it
brings a lot of tears as well.

01:26:14.900 --> 01:26:21.190
So what-- could be written
on the bottom, what's

01:26:21.190 --> 01:26:24.680
the speed cubed?

01:26:24.680 --> 01:26:26.740
Speed is coming from this guy.

01:26:26.740 --> 01:26:32.055
So the speed of the velocity,
the magnitude of the velocity

01:26:32.055 --> 01:26:32.990
is the speed.

01:26:32.990 --> 01:26:35.100
And that-- going
to give you square.

01:26:35.100 --> 01:26:37.010
I'm not going to write
down [INAUDIBLE].

01:26:37.010 --> 01:26:39.144
Square root of x squared,
x prime squared times

01:26:39.144 --> 01:26:42.740
y prime squared,
and I cube that.

01:26:42.740 --> 01:26:46.370
Many people, and I saw
that in engineering, they

01:26:46.370 --> 01:26:49.760
don't like to put that
square root anymore.

01:26:49.760 --> 01:26:53.950
And they just write x prime
squared plus y prime squared

01:26:53.950 --> 01:26:55.110
to the what power?

01:26:55.110 --> 01:26:55.690
STUDENT: 3/2.

01:26:55.690 --> 01:26:56.540
PROFESSOR TODA: 3/2.

01:26:56.540 --> 01:27:01.650
So this is very useful
for engineering styles,

01:27:01.650 --> 01:27:05.110
when you have to deal
with plane curves, motions

01:27:05.110 --> 01:27:08.560
in plane curves.

01:27:08.560 --> 01:27:13.770
But now what do you
have in the case,

01:27:13.770 --> 01:27:19.220
in the happy case, when
you have y equals f of x?

01:27:19.220 --> 01:27:21.335
I'm going to do
that in a second.

01:27:21.335 --> 01:27:26.460


01:27:26.460 --> 01:27:29.144
I want to keep this
formula on the board.

01:27:29.144 --> 01:27:38.108


01:27:38.108 --> 01:27:40.598
What's the simplest
parametrization?

01:27:40.598 --> 01:27:43.088
Because that's why we
need it, to look over

01:27:43.088 --> 01:27:46.580
parametrizations
again and again.

01:27:46.580 --> 01:27:52.190
R of t for this plane
curve will be-- what is t?

01:27:52.190 --> 01:27:53.750
x is t, right?

01:27:53.750 --> 01:27:56.000
x is t, y is f of t.

01:27:56.000 --> 01:27:57.050
Piece of cake.

01:27:57.050 --> 01:28:00.310
So you have t and f of t.

01:28:00.310 --> 01:28:03.355
And how many of you watched
the videos that I sent you?

01:28:03.355 --> 01:28:06.394


01:28:06.394 --> 01:28:08.890
Do you prefer Khan
Academy, or do you

01:28:08.890 --> 01:28:12.880
prefer the guys, [INAUDIBLE]
guys who are lecturing?

01:28:12.880 --> 01:28:15.585
The professors who are lecturing
in front of a board or in front

01:28:15.585 --> 01:28:17.955
of a-- what is that?

01:28:17.955 --> 01:28:21.350
A projector screen?

01:28:21.350 --> 01:28:22.620
I like all of them.

01:28:22.620 --> 01:28:25.140
I think they're very good.

01:28:25.140 --> 01:28:27.572
I think you can learn
a lot from three

01:28:27.572 --> 01:28:29.530
or four different
instructors at the same time.

01:28:29.530 --> 01:28:32.010
That's ideal.

01:28:32.010 --> 01:28:35.590
I guess that you have
this chance only now

01:28:35.590 --> 01:28:36.980
in the past few years.

01:28:36.980 --> 01:28:41.300
Because 20 years ago, if you're
didn't like your instructor

01:28:41.300 --> 01:28:45.790
or just you couldn't stand
them, you had no other chance.

01:28:45.790 --> 01:28:48.030
There was no
YouTube, no internet,

01:28:48.030 --> 01:28:50.972
no way to learn from others.

01:28:50.972 --> 01:29:00.220
R prime of t would
be 1 f prime of t.

01:29:00.220 --> 01:29:02.840
But instead of t I'll
out x, because x is t.

01:29:02.840 --> 01:29:03.930
I don't care.

01:29:03.930 --> 01:29:07.470
R double prime of t would
be 0, f double prime of x.

01:29:07.470 --> 01:29:12.430
So I feel that, hey, I know
what's going to come up.

01:29:12.430 --> 01:29:15.390
And I'm ready.

01:29:15.390 --> 01:29:17.680
Well, we are ready
to write it down.

01:29:17.680 --> 01:29:20.190
This is going to be Mr. x prime.

01:29:20.190 --> 01:29:22.790
This is going to be
replacing Mr. y prime.

01:29:22.790 --> 01:29:25.780
This is going to replace
Mr. a double prime.

01:29:25.780 --> 01:29:29.350
This is going to be replacing
Mr. y double prime of x.

01:29:29.350 --> 01:29:31.120
Oh, OK, all right.

01:29:31.120 --> 01:29:38.840
So k, our old friend from
here will become what?

01:29:38.840 --> 01:29:42.490
And I'd better shut up,
because I'm talking too much.

01:29:42.490 --> 01:29:45.020
STUDENT: [INAUDIBLE]
double prime [INAUDIBLE].

01:29:45.020 --> 01:29:48.092
PROFESSOR TODA: That is
the absolute value, mm-hmm.

01:29:48.092 --> 01:29:54.041
[? n ?] double prime
of x, and nothing else.

01:29:54.041 --> 01:29:54.540
Right, guys?

01:29:54.540 --> 01:29:55.628
Are you with me?

01:29:55.628 --> 01:29:56.960
Divided by--

01:29:56.960 --> 01:29:57.850
STUDENT: [INAUDIBLE]

01:29:57.850 --> 01:29:59.558
PROFESSOR TODA: Should
I add square root?

01:29:59.558 --> 01:30:00.890
I love square roots.

01:30:00.890 --> 01:30:01.880
I'm crazy about them.

01:30:01.880 --> 01:30:11.950
So you go 1 plus f
prime squared cubed.

01:30:11.950 --> 01:30:16.410
So that's going to
be-- any questions?

01:30:16.410 --> 01:30:18.090
Are you guys with me?

01:30:18.090 --> 01:30:21.296
That's going to be the
formula that I'm going

01:30:21.296 --> 01:30:22.420
to use in the next example.

01:30:22.420 --> 01:30:25.540


01:30:25.540 --> 01:30:29.820
In case somebody
wants to know-- I got

01:30:29.820 --> 01:30:32.010
this question from one of you.

01:30:32.010 --> 01:30:35.060
Suppose we get a
parametrization of a circle

01:30:35.060 --> 01:30:37.410
in the midterm or the final.

01:30:37.410 --> 01:30:44.335
Somebody says, I have x
of t, just like we did it

01:30:44.335 --> 01:30:47.555
today, a cosine t plus 0.

01:30:47.555 --> 01:30:52.380
And y of t equals
a sine t plus y 0.

01:30:52.380 --> 01:30:53.990
What is this, guys?

01:30:53.990 --> 01:31:04.495
This is a circle, a center
at 0, y, 0, and radius a.

01:31:04.495 --> 01:31:08.630


01:31:08.630 --> 01:31:14.916
Can use a better formula-- that
anticipated my action today--

01:31:14.916 --> 01:31:18.535
to actually prove that k
is going to be [? 1/a? ?]

01:31:18.535 --> 01:31:19.260
Precisely.

01:31:19.260 --> 01:31:20.550
Can we do that in the exam?

01:31:20.550 --> 01:31:23.120
Yes.

01:31:23.120 --> 01:31:25.490
So while I told
you a long time ago

01:31:25.490 --> 01:31:28.590
that engineers and
mathematicians observed

01:31:28.590 --> 01:31:31.000
hundreds of years
ago-- actually,

01:31:31.000 --> 01:31:32.770
somebody said, no,
you're not right.

01:31:32.770 --> 01:31:35.010
The Egyptians already saw that.

01:31:35.010 --> 01:31:38.243
They had the notion of
inverse proportionality

01:31:38.243 --> 01:31:42.390
in Egypt, which makes sense
if you look at the pyramids.

01:31:42.390 --> 01:31:47.750
So one look at the radius,
it says if the radius is 2,

01:31:47.750 --> 01:31:50.640
then the curvature
is not very bent.

01:31:50.640 --> 01:31:52.760
So the curvature's inverse
proportion [INAUDIBLE]

01:31:52.760 --> 01:31:53.480
the radius.

01:31:53.480 --> 01:31:57.480
So if this is 2, we said
the curvature's 1/2.

01:31:57.480 --> 01:32:01.680
If you take a big
circle, the bigger

01:32:01.680 --> 01:32:04.060
the radius, the
smaller the bending

01:32:04.060 --> 01:32:07.640
of the arc of the circle,
the smaller of the curvature.

01:32:07.640 --> 01:32:10.670
Apparently the ancient
world knew that already.

01:32:10.670 --> 01:32:12.195
They Egyptians knew that.

01:32:12.195 --> 01:32:13.140
The Greeks knew that.

01:32:13.140 --> 01:32:15.432
But I think they
never formalized it--

01:32:15.432 --> 01:32:16.624
not that I know.

01:32:16.624 --> 01:32:19.290


01:32:19.290 --> 01:32:24.530
So if you are asked to
do this in any exam,

01:32:24.530 --> 01:32:26.942
do you think that
would be a problem?

01:32:26.942 --> 01:32:28.150
Of course we would do review.

01:32:28.150 --> 01:32:31.730
Because people are going to
forget this formula, or even

01:32:31.730 --> 01:32:33.370
the definition.

01:32:33.370 --> 01:32:36.010
You can compute k
for this formula.

01:32:36.010 --> 01:32:39.330
And we are going to
get k to the 1/a.

01:32:39.330 --> 01:32:41.340
This is a piece
of cake, actually.

01:32:41.340 --> 01:32:44.110
You may not believe me, but
once you plug in the equations

01:32:44.110 --> 01:32:46.290
it's very easy.

01:32:46.290 --> 01:32:48.425
Or you can do it
from the definition

01:32:48.425 --> 01:32:50.640
that gives you k of s.

01:32:50.640 --> 01:32:52.560
You'll reparametrize
this in arclength.

01:32:52.560 --> 01:32:54.710
You can do that as well.

01:32:54.710 --> 01:32:57.600
And you still get 1/a.

01:32:57.600 --> 01:32:59.880
The question that
I got by email,

01:32:59.880 --> 01:33:01.440
and I get a lot of email.

01:33:01.440 --> 01:33:04.660
I told you, that
keeps me busy a lot,

01:33:04.660 --> 01:33:08.490
about 200 emails every day.

01:33:08.490 --> 01:33:10.680
I really like the emails
I get from students,

01:33:10.680 --> 01:33:13.620
because I get emails from
all sorts of sources--

01:33:13.620 --> 01:33:15.340
Got some spam also.

01:33:15.340 --> 01:33:21.780
Anyway, what I'm trying to say,
I got this question last time

01:33:21.780 --> 01:33:24.680
saying, if on the midterm
we get such a question,

01:33:24.680 --> 01:33:28.073
can we say simply, curvature's
1/a, a is the radius.

01:33:28.073 --> 01:33:30.820
Is that enough?

01:33:30.820 --> 01:33:34.250
Depends on how the
problem was formulated.

01:33:34.250 --> 01:33:39.360
Most likely I'm going to make
it through that or show that.

01:33:39.360 --> 01:33:43.450
Even if you state something,
like, yes, it's 1/a,

01:33:43.450 --> 01:33:46.270
with a little argument,
it's inverse proportional

01:33:46.270 --> 01:33:50.380
to the radius, I will
still give partial credit.

01:33:50.380 --> 01:33:53.650
For any argument that
is valid, especially

01:33:53.650 --> 01:33:56.390
if it's based on
empirical observation,

01:33:56.390 --> 01:33:58.920
I do give some extra
credit, even if you didn't

01:33:58.920 --> 01:34:02.600
use the specific formula.

01:34:02.600 --> 01:34:04.910
Let's see one example.

01:34:04.910 --> 01:34:07.615
Let's take y equals e to the x.

01:34:07.615 --> 01:34:11.870


01:34:11.870 --> 01:34:15.726
No, let's take e
to the negative x.

01:34:15.726 --> 01:34:16.690
Doesn't matter.

01:34:16.690 --> 01:34:20.560


01:34:20.560 --> 01:34:26.375
y equals e to the negative x.

01:34:26.375 --> 01:34:30.830
And let's make x
between 0 and 1.

01:34:30.830 --> 01:34:35.290


01:34:35.290 --> 01:34:36.970
I'll say, write the curvature.

01:34:36.970 --> 01:34:40.610


01:34:40.610 --> 01:34:45.246
Write the equation or the
formula of the curvature.

01:34:45.246 --> 01:34:50.230


01:34:50.230 --> 01:34:54.700
And I know it's 2 o'clock
and I am answering questions.

01:34:54.700 --> 01:34:58.070
This was a question that one of
you had during the short break

01:34:58.070 --> 01:34:59.300
we took.

01:34:59.300 --> 01:35:00.480
Can we do such a problem?

01:35:00.480 --> 01:35:01.544
Like she said.

01:35:01.544 --> 01:35:02.960
Yes, I [INAUDIBLE]
to the negative

01:35:02.960 --> 01:35:05.520
x because I want
to catch somebody

01:35:05.520 --> 01:35:06.790
not knowing the derivative.

01:35:06.790 --> 01:35:08.990
I don't know why I'm doing this.

01:35:08.990 --> 01:35:10.510
Right?

01:35:10.510 --> 01:35:15.480
So if I were to draw that, OK,
try and draw that, but not now.

01:35:15.480 --> 01:35:18.620
Now, what formula
are you going to use?

01:35:18.620 --> 01:35:21.860
Of course, you could
do this in many ways.

01:35:21.860 --> 01:35:24.510
All those formulas are
equivalent for the curvature.

01:35:24.510 --> 01:35:27.360
What's the simplest
way to do it?

01:35:27.360 --> 01:35:30.020
Do y prime.

01:35:30.020 --> 01:35:33.770
Minus it to the minus x.

01:35:33.770 --> 01:35:36.790
Note here in this problem that
even if you mess up and forget

01:35:36.790 --> 01:35:39.950
the minus sign, you still
get the final answer correct.

01:35:39.950 --> 01:35:46.050
But I may subtract a few points
if I see something nonsensical.

01:35:46.050 --> 01:35:47.300
y double prime equals--

01:35:47.300 --> 01:35:48.870
[INTERPOSING VOICES]

01:35:48.870 --> 01:35:51.660
--plus e to the minus x.

01:35:51.660 --> 01:35:56.738
And what is the
curvature k of t?

01:35:56.738 --> 01:35:59.200
STUDENT: y prime over--

01:35:59.200 --> 01:36:01.340
PROFESSOR TODA: Oh, I
didn't say one more thing.

01:36:01.340 --> 01:36:04.620
I want the curvature, but
I also want the curvature

01:36:04.620 --> 01:36:08.486
in three separate moments,
in the beginning, in the end,

01:36:08.486 --> 01:36:09.360
and in the middle.

01:36:09.360 --> 01:36:11.430
STUDENT: Don't we
need to parametrize it

01:36:11.430 --> 01:36:15.250
so we can [INAUDIBLE]
x prime [INAUDIBLE]?

01:36:15.250 --> 01:36:16.740
PROFESSOR TODA: No.

01:36:16.740 --> 01:36:18.340
Did I erase it?

01:36:18.340 --> 01:36:19.330
STUDENT: Yeah, you did.

01:36:19.330 --> 01:36:20.710
PROFESSOR TODA: [INAUDIBLE].

01:36:20.710 --> 01:36:24.356
And one of my colleagues
said, Magda, you are smart,

01:36:24.356 --> 01:36:28.000
but you are like one
of those people who,

01:36:28.000 --> 01:36:29.729
in the anecdotes
about math professors,

01:36:29.729 --> 01:36:31.520
gets out of their office
and starts walking

01:36:31.520 --> 01:36:32.930
and stops a student.

01:36:32.930 --> 01:36:34.845
Was I going this
way or that way?

01:36:34.845 --> 01:36:36.070
And that's me.

01:36:36.070 --> 01:36:37.650
And I'm sorry about that.

01:36:37.650 --> 01:36:41.933
I should not have erased that.

01:36:41.933 --> 01:36:44.000
I'm going to go
ahead and rewrite it,

01:36:44.000 --> 01:36:48.101
because I'm a goofball.

01:36:48.101 --> 01:36:55.496
So the one that I wanted to use
k of x will be f double prime.

01:36:55.496 --> 01:36:56.975
STUDENT: And cubed.

01:36:56.975 --> 01:36:58.454
PROFESSOR TODA: Cubed!

01:36:58.454 --> 01:36:59.440
Thank you.

01:36:59.440 --> 01:37:02.920


01:37:02.920 --> 01:37:09.636
So that 3/2, remember it,
[INAUDIBLE] 3/2 [INAUDIBLE]

01:37:09.636 --> 01:37:10.990
square root cubed.

01:37:10.990 --> 01:37:13.750
Now, for this one, is it hard?

01:37:13.750 --> 01:37:15.080
No.

01:37:15.080 --> 01:37:16.190
That's a piece of cake.

01:37:16.190 --> 01:37:18.610
I said I like it in
general, but I also

01:37:18.610 --> 01:37:22.910
like it-- find the curvature
of this curve in the beginning.

01:37:22.910 --> 01:37:24.100
You travel on me.

01:37:24.100 --> 01:37:27.590
From time 0 to 1
o'clock, whatever.

01:37:27.590 --> 01:37:28.430
One second.

01:37:28.430 --> 01:37:32.770
That's saying this is in seconds
to make it more physical.

01:37:32.770 --> 01:37:39.950
I want the k at 0, I want k
at 1/2, and I want k at 1.

01:37:39.950 --> 01:37:42.430
And I'd like you to
compare those values.

01:37:42.430 --> 01:37:46.270


01:37:46.270 --> 01:37:49.120
And I'll give you one
more task after that.

01:37:49.120 --> 01:37:50.580
But let me start working.

01:37:50.580 --> 01:37:53.090
So you say you help me on that.

01:37:53.090 --> 01:37:55.100
[INAUDIBLE]

01:37:55.100 --> 01:38:02.516
Minus x over square
root of 1 plus--

01:38:02.516 --> 01:38:04.380
STUDENT: [INAUDIBLE]

01:38:04.380 --> 01:38:05.312
PROFESSOR TODA: Right.

01:38:05.312 --> 01:38:08.190
So can I write this differently,
a little bit differently?

01:38:08.190 --> 01:38:09.940
Like what?

01:38:09.940 --> 01:38:12.020
I don't want to square
each of the minus 2x.

01:38:12.020 --> 01:38:14.350
Can I do that?

01:38:14.350 --> 01:38:19.540
And then the whole thing
I can say to the 3/2

01:38:19.540 --> 01:38:24.530
or I can use the square root,
whichever is your favorite.

01:38:24.530 --> 01:38:28.990
Now, what is k of 0?

01:38:28.990 --> 01:38:29.490
STUDENT: 0.

01:38:29.490 --> 01:38:32.990
Or 1.

01:38:32.990 --> 01:38:34.462
PROFESSOR TODA: Really?

01:38:34.462 --> 01:38:36.226
STUDENT: 1/2.

01:38:36.226 --> 01:38:36.961
3/2.

01:38:36.961 --> 01:38:38.710
PROFESSOR TODA: So
let's take this slowly.

01:38:38.710 --> 01:38:43.584
Because we can all make
mistakes, goofy mistakes.

01:38:43.584 --> 01:38:45.000
That doesn't mean
we're not smart.

01:38:45.000 --> 01:38:46.770
We're very smart, right?

01:38:46.770 --> 01:38:51.210
But it's just a matter of
book-keeping and paying

01:38:51.210 --> 01:38:53.010
attention, being attentive.

01:38:53.010 --> 01:38:55.420
OK.

01:38:55.420 --> 01:39:00.240
When I take 0 and replace--
this is drying fast.

01:39:00.240 --> 01:39:02.858
I'm trying to draw it.

01:39:02.858 --> 01:39:10.000
I have 1 over 1
plus 1 to the 3/2.

01:39:10.000 --> 01:39:15.160
I have a student in one exam
who was just-- I don't know.

01:39:15.160 --> 01:39:16.580
He was rushing.

01:39:16.580 --> 01:39:20.950
He didn't realize that
he had to take it slowly.

01:39:20.950 --> 01:39:22.790
He was extremely smart, though.

01:39:22.790 --> 01:39:29.760
1 over-- you have
that 1 plus 1 is 2.

01:39:29.760 --> 01:39:33.710
2 to the 1/2 would be
square root of 2 cubed.

01:39:33.710 --> 01:39:35.890
It would be exactly
2 square root of 2.

01:39:35.890 --> 01:39:39.750
And more you can write
this as rationalized.

01:39:39.750 --> 01:39:42.050
Now, I have a question for you.

01:39:42.050 --> 01:39:43.010
[INAUDIBLE]

01:39:43.010 --> 01:39:47.099
I'm When we were kids, if you
remember-- you are too young.

01:39:47.099 --> 01:39:48.140
Maybe you don't remember.

01:39:48.140 --> 01:39:52.710
But I remember when I was a kid,
my teacher would always ask me,

01:39:52.710 --> 01:39:53.980
rationalize your answer.

01:39:53.980 --> 01:39:56.920
Rationalize your answer.

01:39:56.920 --> 01:40:00.362
Put the rational number
in the denominator.

01:40:00.362 --> 01:40:02.740
Why do you think that was?

01:40:02.740 --> 01:40:05.156
For hundreds of years
people did that.

01:40:05.156 --> 01:40:07.430
STUDENT: [INAUDIBLE]

01:40:07.430 --> 01:40:11.900
PROFESSOR TODA: Because they
didn't have a calculator.

01:40:11.900 --> 01:40:16.303
So we used to, even I used to be
able to get the square root out

01:40:16.303 --> 01:40:17.518
by hand.

01:40:17.518 --> 01:40:20.920
Has anybody taught you how to
compute square root by hand?

01:40:20.920 --> 01:40:21.649
You know that.

01:40:21.649 --> 01:40:22.378
Who taught you?

01:40:22.378 --> 01:40:23.350
STUDENT: I don't remember it.

01:40:23.350 --> 01:40:24.940
My seventh grade
teacher taught us.

01:40:24.940 --> 01:40:26.440
PROFESSOR TODA:
There is a technique

01:40:26.440 --> 01:40:29.450
of taking groups of twos
and then fitting the--

01:40:29.450 --> 01:40:31.020
and they still teach that.

01:40:31.020 --> 01:40:33.250
I was amazed, they
still teach that

01:40:33.250 --> 01:40:35.460
in half of the Asian countries.

01:40:35.460 --> 01:40:39.360
And it's hard, but kids
in fifth and sixth grade

01:40:39.360 --> 01:40:45.190
have that practice, which some
of us learned and forgot about.

01:40:45.190 --> 01:40:50.029
So imagine that how people would
have done this, and of course,

01:40:50.029 --> 01:40:51.070
square root of 2 is easy.

01:40:51.070 --> 01:40:53.550
1.4142, blah blah blah.

01:40:53.550 --> 01:40:54.250
Divide by 2.

01:40:54.250 --> 01:40:56.350
You can do it by hand.

01:40:56.350 --> 01:40:57.860
At least a good approximation.

01:40:57.860 --> 01:41:01.970
But imagine having a nasty
square root there to compute,

01:41:01.970 --> 01:41:05.850
and then you would divide
by that natural number.

01:41:05.850 --> 01:41:09.140
You have to rely on your
own computation to do it.

01:41:09.140 --> 01:41:11.190
There were no calculators.

01:41:11.190 --> 01:41:14.098
How about k of 1?

01:41:14.098 --> 01:41:14.598
How is that?

01:41:14.598 --> 01:41:15.562
What is that?

01:41:15.562 --> 01:41:20.382


01:41:20.382 --> 01:41:23.756
e to the minus 1.

01:41:23.756 --> 01:41:26.540
That's a little bit
harder to compute, right?

01:41:26.540 --> 01:41:28.611
1 plus [INAUDIBLE].

01:41:28.611 --> 01:41:31.760
What is that going to be?

01:41:31.760 --> 01:41:34.040
Minus 2.

01:41:34.040 --> 01:41:37.143
Replace it by 1 to the 3/2.

01:41:37.143 --> 01:41:41.930
I would like you to go
home and do the following.

01:41:41.930 --> 01:41:45.810
[INAUDIBLE]-- Not now, not now.

01:41:45.810 --> 01:41:48.310
We stay a little
bit longer together.

01:41:48.310 --> 01:41:51.900
k of 0, k of 1/2, and k of 1.

01:41:51.900 --> 01:41:53.034
Which one is bigger?

01:41:53.034 --> 01:41:59.930


01:41:59.930 --> 01:42:03.395
And one last question about
that, how much extra credit

01:42:03.395 --> 01:42:04.220
should I give you?

01:42:04.220 --> 01:42:05.870
One point?

01:42:05.870 --> 01:42:08.420
One point if you turn this in.

01:42:08.420 --> 01:42:11.200
Um, yeah.

01:42:11.200 --> 01:42:13.380
Four, [? maybe ?] two points.

01:42:13.380 --> 01:42:19.430
Compare all these
three values, and find

01:42:19.430 --> 01:42:32.792
the maximum and the
minimum of kappa of t,

01:42:32.792 --> 01:42:37.530
kappa of x, for
the interval where

01:42:37.530 --> 01:42:46.690
x is in the interval 0, 1.

01:42:46.690 --> 01:42:48.870
0, closed 1.

01:42:48.870 --> 01:42:49.940
Close it.

01:42:49.940 --> 01:42:53.930
Now, don't ask me,
because it's extra credit.

01:42:53.930 --> 01:42:58.740
One question was, by email,
can I ask my tutor to help me?

01:42:58.740 --> 01:43:02.170
As long as your tutor doesn't
write down your solution,

01:43:02.170 --> 01:43:03.720
you are in good shape.

01:43:03.720 --> 01:43:07.020
Your tutor should help you
understand some constants,

01:43:07.020 --> 01:43:08.290
spend time with you.

01:43:08.290 --> 01:43:12.773
But they should not write
your assignment themselves.

01:43:12.773 --> 01:43:13.272
OK?

01:43:13.272 --> 01:43:16.540
So it's not a big deal.

01:43:16.540 --> 01:43:22.030
Not I want to tell you one
secret that I normally don't

01:43:22.030 --> 01:43:26.730
tell my Calculus 3 students.

01:43:26.730 --> 01:43:29.370
But the more I get
to know you, the more

01:43:29.370 --> 01:43:34.020
I realize that you are worth
me telling you about that.

01:43:34.020 --> 01:43:35.420
STUDENT: [INAUDIBLE]

01:43:35.420 --> 01:43:38.080
PROFESSOR TODA: No.

01:43:38.080 --> 01:43:41.900
There is a beautiful
theory that engineers

01:43:41.900 --> 01:43:48.142
use when they start the motions
of curves and parametrizations

01:43:48.142 --> 01:43:51.058
in space.

01:43:51.058 --> 01:43:53.420
And that includes
the Frenet formulas.

01:43:53.420 --> 01:43:56.054


01:43:56.054 --> 01:43:58.510
And you already
know the first one.

01:43:58.510 --> 01:44:04.520
And I was debating, I was just
reviewing what I taught you,

01:44:04.520 --> 01:44:07.321
and I was happy with
what I taught you.

01:44:07.321 --> 01:44:10.580
And I said, they know
about position vector.

01:44:10.580 --> 01:44:13.100
They know about
velocity, acceleration.

01:44:13.100 --> 01:44:15.927
They know how to get back
and forth from one another.

01:44:15.927 --> 01:44:16.760
They know our claim.

01:44:16.760 --> 01:44:18.468
They know how to
[? reparameterize our ?]

01:44:18.468 --> 01:44:19.940
claims.

01:44:19.940 --> 01:44:25.040
They know the [INAUDIBLE]
and B. They know already

01:44:25.040 --> 01:44:27.010
the first Frenet formula.

01:44:27.010 --> 01:44:28.010
They know the curvature.

01:44:28.010 --> 01:44:29.992
What else can I teach them?

01:44:29.992 --> 01:44:34.070
I want to show you--
one of you asked me,

01:44:34.070 --> 01:44:38.280
is this all that we should know?

01:44:38.280 --> 01:44:41.875
This is all that a regular
student should know in Calculus

01:44:41.875 --> 01:44:43.746
3, but there is more.

01:44:43.746 --> 01:44:44.870
And you are honor students.

01:44:44.870 --> 01:44:49.930
And I want to show you some
beautiful equations here.

01:44:49.930 --> 01:44:54.930
So do you remember that
if I introduce r of s

01:44:54.930 --> 01:45:03.900
as a curving arclength,
that is a regular curve.

01:45:03.900 --> 01:45:11.050
I said there is a certain famous
formula that is T prime of s

01:45:11.050 --> 01:45:13.900
called-- leave space.

01:45:13.900 --> 01:45:15.320
Leave a little bit of space.

01:45:15.320 --> 01:45:15.945
You'll see why.

01:45:15.945 --> 01:45:17.850
It's a surprise.

01:45:17.850 --> 01:45:22.950
k times-- why
don't I say k of s?

01:45:22.950 --> 01:45:26.220
Because I want to point
out that k is an invariant.

01:45:26.220 --> 01:45:28.580
Even if you have
another parameter,

01:45:28.580 --> 01:45:29.905
would be the same function.

01:45:29.905 --> 01:45:38.565
But yes, as a function of s,
would be k times N bar, bar.

01:45:38.565 --> 01:45:40.906
More bars because
they are free vectors.

01:45:40.906 --> 01:45:42.920
They are not bound
to a certain point.

01:45:42.920 --> 01:45:44.550
They're not married
to a certain point.

01:45:44.550 --> 01:45:49.180
They are free to shift
by parallelism in space.

01:45:49.180 --> 01:45:54.250
However, I'm going to review
them as bound at the point

01:45:54.250 --> 01:45:55.200
where they are.

01:45:55.200 --> 01:45:58.110
So they-- no way they
are married to the point

01:45:58.110 --> 01:46:03.640
that they belong to.

01:46:03.640 --> 01:46:07.200
Maybe the [? bend ?]
will change.

01:46:07.200 --> 01:46:09.394
I don't know how it's
going to change like crazy.

01:46:09.394 --> 01:46:18.170


01:46:18.170 --> 01:46:19.250
Something like that.

01:46:19.250 --> 01:46:26.819
At every point you have a T, an
N, and it's a 90 degree angle.

01:46:26.819 --> 01:46:30.900
Then you have the binormal,
which makes a 90 degree

01:46:30.900 --> 01:46:33.280
angle-- [INAUDIBLE].

01:46:33.280 --> 01:46:36.860
So the way you should
imagine these corners

01:46:36.860 --> 01:46:39.360
would be something
like that, right?

01:46:39.360 --> 01:46:40.860
90-90-90.

01:46:40.860 --> 01:46:43.360
It's just hard to draw them.

01:46:43.360 --> 01:46:51.730
Between the vectors you have--
If you draw T and N, am I

01:46:51.730 --> 01:46:53.430
right, that is coming out?

01:46:53.430 --> 01:46:54.110
No.

01:46:54.110 --> 01:46:56.050
I have to switch them.

01:46:56.050 --> 01:46:57.650
T and N. Now, am I right?

01:46:57.650 --> 01:46:59.340
Now I'm thinking of
the [? faucet. ?]

01:46:59.340 --> 01:47:02.440
If I move T-- yeah,
now it's coming out.

01:47:02.440 --> 01:47:08.120
So this is not getting
into the formula.

01:47:08.120 --> 01:47:09.490
So this is the first formula.

01:47:09.490 --> 01:47:10.350
You say, so what?

01:47:10.350 --> 01:47:11.185
You've taught that.

01:47:11.185 --> 01:47:12.450
We proved it together.

01:47:12.450 --> 01:47:14.070
What do you want from us?

01:47:14.070 --> 01:47:17.560
I want to teach you
two more formulas.

01:47:17.560 --> 01:47:18.540
N prime.

01:47:18.540 --> 01:47:21.970


01:47:21.970 --> 01:47:24.420
And I'd like you to
leave more space here.

01:47:24.420 --> 01:47:27.360


01:47:27.360 --> 01:47:30.746
So you have like an empty field
here and an empty field here

01:47:30.746 --> 01:47:32.024
[INAUDIBLE].

01:47:32.024 --> 01:47:35.815
If you were to compute
T prime, the magic thing

01:47:35.815 --> 01:47:40.452
is that T prime is a vector.

01:47:40.452 --> 01:47:41.378
N prime is a vector.

01:47:41.378 --> 01:47:42.770
B prime is a vector.

01:47:42.770 --> 01:47:44.430
They're all vectors.

01:47:44.430 --> 01:47:48.630
They are the derivatives
of the vectors T and NB.

01:47:48.630 --> 01:47:50.970
And you say, why would I
care about the derivatives

01:47:50.970 --> 01:47:52.210
of the vectors T and NB?

01:47:52.210 --> 01:47:54.310
I'll tell you in a second.

01:47:54.310 --> 01:47:58.070
So if you were to
compute in prime,

01:47:58.070 --> 01:48:00.050
you're going to get here.

01:48:00.050 --> 01:48:04.357
Minus k of s times T of s.

01:48:04.357 --> 01:48:07.219
Leave room.

01:48:07.219 --> 01:48:09.570
Leave room, because there
is no component that

01:48:09.570 --> 01:48:13.760
depends on N. No such component
that that depends on N.

01:48:13.760 --> 01:48:14.680
This is [INAUDIBLE].

01:48:14.680 --> 01:48:17.260
There is nothing in
N. And then in the end

01:48:17.260 --> 01:48:28.580
you'll say, plus tau of s
times B. There is missing--

01:48:28.580 --> 01:48:30.350
something is.

01:48:30.350 --> 01:48:32.990
And finally, if
you take B prime,

01:48:32.990 --> 01:48:34.950
there is nothing
here, nothing here.

01:48:34.950 --> 01:48:42.865
In the middle you have
minus tau of s times N of s.

01:48:42.865 --> 01:48:45.700


01:48:45.700 --> 01:48:49.730
And now you know that nobody
else but you knows that.

01:48:49.730 --> 01:48:54.060
The other regular sections
don't know these formulas.

01:48:54.060 --> 01:48:57.420


01:48:57.420 --> 01:49:02.220
What do you observe about
this bunch of equations?

01:49:02.220 --> 01:49:04.160
Say, oh, wait a minute.

01:49:04.160 --> 01:49:06.520
First of all, why did
you put it like that?

01:49:06.520 --> 01:49:07.840
Looks like a cross.

01:49:07.840 --> 01:49:09.160
It is a cross.

01:49:09.160 --> 01:49:12.830
It is like one is shaped in the
name of the Father, of the Son,

01:49:12.830 --> 01:49:13.730
and so on.

01:49:13.730 --> 01:49:17.040
So does it have anything
to do with religion?

01:49:17.040 --> 01:49:17.540
No.

01:49:17.540 --> 01:49:23.260
But it's going to help you
memorize better the equations.

01:49:23.260 --> 01:49:27.004
These are the famous
Frenet equations.

01:49:27.004 --> 01:49:30.310


01:49:30.310 --> 01:49:33.980
You only saw the first one.

01:49:33.980 --> 01:49:35.062
What do they represent?

01:49:35.062 --> 01:49:38.230


01:49:38.230 --> 01:49:40.090
If somebody asks you, what is k?

01:49:40.090 --> 01:49:42.910
What it is k of s?

01:49:42.910 --> 01:49:44.130
What's the curvature?

01:49:44.130 --> 01:49:44.880
You go to a party.

01:49:44.880 --> 01:49:46.820
There are only nerds.

01:49:46.820 --> 01:49:47.440
It's you.

01:49:47.440 --> 01:49:50.370
Some people taking advanced
calculus or some people

01:49:50.370 --> 01:49:54.790
from Physics, and they say, OK,
have you heard of the Frenet

01:49:54.790 --> 01:49:56.920
motion, Frenet
formulas, and you say,

01:49:56.920 --> 01:49:58.760
I know everything about it.

01:49:58.760 --> 01:50:02.310
What if they ask you, what
is the curvature of k?

01:50:02.310 --> 01:50:07.640
You say, curvature measures
how a curve is bent.

01:50:07.640 --> 01:50:11.820
And they say, yeah, but the
Frenet formula tells you

01:50:11.820 --> 01:50:13.610
more about that.

01:50:13.610 --> 01:50:17.720
Not only k shows you
how bent the curve is.

01:50:17.720 --> 01:50:27.080
But k is a measure of
how fast T changes.

01:50:27.080 --> 01:50:28.240
And he sees why.

01:50:28.240 --> 01:50:31.030
Practically, if you take
the [INAUDIBLE] to the bat,

01:50:31.030 --> 01:50:37.310
this is the speed of T. So how
fast the teaching will change.

01:50:37.310 --> 01:50:39.890
That will be magnitude,
will be just k.

01:50:39.890 --> 01:50:42.440
Because magnitude of N is 1.

01:50:42.440 --> 01:50:48.820
So note that k of s is
the length of T prime.

01:50:48.820 --> 01:51:04.387
This measures the change
in T. So how fast T varies.

01:51:04.387 --> 01:51:08.610


01:51:08.610 --> 01:51:11.320
What does the torsion represent?

01:51:11.320 --> 01:51:16.690
Well, how fast the
binormal varies.

01:51:16.690 --> 01:51:20.596
But if you want to
think of a helix,

01:51:20.596 --> 01:51:25.640
and it's a little
bit hard to imagine,

01:51:25.640 --> 01:51:30.160
the curvature measures how
bent a certain curve is.

01:51:30.160 --> 01:51:33.800
And it measures how
bent a plane curve is.

01:51:33.800 --> 01:51:38.720
For example, for the circle you
have radius a, 1/a, and so on.

01:51:38.720 --> 01:51:40.870
But there must be
also a function that

01:51:40.870 --> 01:51:46.090
shows you how a curve twists.

01:51:46.090 --> 01:51:50.060
Because you have not
just a plane curve where

01:51:50.060 --> 01:51:52.370
you care about curvature only.

01:51:52.370 --> 01:51:58.570
But in the space curve you
care how the curves twist.

01:51:58.570 --> 01:52:03.190
How fast do they move
away from a certain plane?

01:52:03.190 --> 01:52:10.956
Now, if I were to draw-- is
it hard to memorize these?

01:52:10.956 --> 01:52:11.456
No.

01:52:11.456 --> 01:52:14.060
I memorized them easily
based on the fact

01:52:14.060 --> 01:52:19.850
that everything looks
like a decomposition

01:52:19.850 --> 01:52:23.920
of a vector in terms of
T, N, and B. So in my mind

01:52:23.920 --> 01:52:28.470
it was like, I take any vector
I want, B. And this is T,

01:52:28.470 --> 01:52:32.700
this is N, and this is B.
Just the weight was IJK.

01:52:32.700 --> 01:52:36.680
Instead if I, I have T. Instead
of J, I have N. Instead of K,

01:52:36.680 --> 01:52:40.040
I have B. They are
still unit vectors.

01:52:40.040 --> 01:52:42.610
So locally at the
point I have this frame

01:52:42.610 --> 01:52:44.230
and I have any vector.

01:52:44.230 --> 01:52:46.950
This vector-- I'm a physicist.

01:52:46.950 --> 01:52:50.940
So let's say I'm going to
represent that as v1 times

01:52:50.940 --> 01:52:54.050
the T plus v2
times-- instead of J,

01:52:54.050 --> 01:52:57.790
we'll use that N plus
B3 times-- that's

01:52:57.790 --> 01:52:59.980
the last element of the bases.

01:52:59.980 --> 01:53:03.605
Instead of k I have v.
So it's the same here.

01:53:03.605 --> 01:53:06.360
You try to pick a
vector and decompose

01:53:06.360 --> 01:53:09.950
that in terms of T, N, and B.
Will I put that on the final?

01:53:09.950 --> 01:53:10.720
No.

01:53:10.720 --> 01:53:12.925
But I would like you to
remember it, especially

01:53:12.925 --> 01:53:17.070
if you are an engineering
major or physics major,

01:53:17.070 --> 01:53:19.692
that there is this
kind of Frenet frame.

01:53:19.692 --> 01:53:26.010
For those of you who are taking
a-- for differential equations,

01:53:26.010 --> 01:53:28.760
you already do some matrices
and built-in systems

01:53:28.760 --> 01:53:31.590
of equations, systems of
differential equations.

01:53:31.590 --> 01:53:33.215
I'm not going to get there.

01:53:33.215 --> 01:53:38.072
But suppose you don't know
differential equations,

01:53:38.072 --> 01:53:41.730
but you know a little
bit of linear algebra.

01:53:41.730 --> 01:53:44.950
And I know you know how
to multiply matrices.

01:53:44.950 --> 01:53:47.120
You know how I know
you multiply matrices,

01:53:47.120 --> 01:53:49.540
no matter how much
mathematics you learn.

01:53:49.540 --> 01:53:52.670
And most of you, you are not in
general algebra this semester.

01:53:52.670 --> 01:53:55.070
Only two of you are
in general algebra.

01:53:55.070 --> 01:54:03.250
When I took a C++ course,
the first homework I got was

01:54:03.250 --> 01:54:06.530
to program a matrix
multiplication.

01:54:06.530 --> 01:54:07.730
I have to give in matrices.

01:54:07.730 --> 01:54:10.900
I have to program that in C++.

01:54:10.900 --> 01:54:14.600
And freshmen knew that.

01:54:14.600 --> 01:54:20.440
So that means you know how
to write this as a matrix

01:54:20.440 --> 01:54:21.410
multiplication.

01:54:21.410 --> 01:54:23.050
Can anybody help me?

01:54:23.050 --> 01:54:25.880
So T, N, B is the magic triple.

01:54:25.880 --> 01:54:28.980
T, N, B's the magic corner.

01:54:28.980 --> 01:54:32.000
T, N, and B are the Three
Musketeers who are all

01:54:32.000 --> 01:54:34.326
orthogonal to one another.

01:54:34.326 --> 01:54:37.985
And then I do derivative
with respect to s.

01:54:37.985 --> 01:54:42.290
If I want to be
elegant, I'll put d/ds.

01:54:42.290 --> 01:54:44.280
OK.

01:54:44.280 --> 01:54:47.330
How am I going to
fill in this matrix?

01:54:47.330 --> 01:54:50.650
So somebody who wants to know
about differential equations,

01:54:50.650 --> 01:54:51.610
this would be a--

01:54:51.610 --> 01:54:52.790
STUDENT: 0, k, 0.

01:54:52.790 --> 01:54:53.873
PROFESSOR TODA: Very good.

01:54:53.873 --> 01:55:04.770
0, k, 0, minus k 0
tau, 0 minus tau 0.

01:55:04.770 --> 01:55:07.362
This is called the
skew symmetric matrix.

01:55:07.362 --> 01:55:11.810


01:55:11.810 --> 01:55:14.740
Such matrices are very
important in robotics.

01:55:14.740 --> 01:55:17.430
If you've ever been
to a robotics team,

01:55:17.430 --> 01:55:20.040
like one of those
projects, you should

01:55:20.040 --> 01:55:22.990
know that when we study
motions of-- let's say

01:55:22.990 --> 01:55:26.620
that my arm performs
two rotations in a row.

01:55:26.620 --> 01:55:30.500
All these motions
are described based

01:55:30.500 --> 01:55:35.320
on some groups of rotations.

01:55:35.320 --> 01:55:39.950
And if I go into details,
it's going to be really hard.

01:55:39.950 --> 01:55:45.580
But practically
in such a setting

01:55:45.580 --> 01:55:49.800
we have to deal with matrices
that either have determined

01:55:49.800 --> 01:55:53.520
one, like all rotations
actually have,

01:55:53.520 --> 01:55:58.300
or have some other
properties, like this guy.

01:55:58.300 --> 01:56:00.410
What's the determinant
of this guy?

01:56:00.410 --> 01:56:02.010
What do you guys think?

01:56:02.010 --> 01:56:02.750
Just look at it.

01:56:02.750 --> 01:56:03.250
STUDENT: 0?

01:56:03.250 --> 01:56:04.000
PROFESSOR TODA: 0.

01:56:04.000 --> 01:56:05.660
It has determinant 0.

01:56:05.660 --> 01:56:08.470
And moreover, it
looks in the mirror.

01:56:08.470 --> 01:56:11.195
So this comes from
a group of motion,

01:56:11.195 --> 01:56:14.690
which is little s over 3,
the linear algebra, actually.

01:56:14.690 --> 01:56:17.190
So when k is looking
in the mirror,

01:56:17.190 --> 01:56:20.820
it becomes minus k tau,
is becoming minus tau.

01:56:20.820 --> 01:56:24.190
It is antisymmetric
or skew symmetric.

01:56:24.190 --> 01:56:27.010
Skew symmetric or
antisymmetric is the same.

01:56:27.010 --> 01:56:29.960
STUDENT: Antisymmetric,
skew symmetric matrix.

01:56:29.960 --> 01:56:31.935
PROFESSOR TODA: Skew
symmetric or antisymmetric

01:56:31.935 --> 01:56:33.370
is exactly the same thing.

01:56:33.370 --> 01:56:34.292
They are synonyms.

01:56:34.292 --> 01:56:37.060


01:56:37.060 --> 01:56:40.130
So it looks in the mirror
and picks up the minus sign,

01:56:40.130 --> 01:56:41.820
has 0 in the bag.

01:56:41.820 --> 01:56:43.135
What am I going to put here?

01:56:43.135 --> 01:56:44.460
You already got the idea.

01:56:44.460 --> 01:56:47.060
So when Ryan gave
me this, he meant

01:56:47.060 --> 01:56:50.460
that he knew what I'm going
to put here, as a vector,

01:56:50.460 --> 01:56:54.024
as a column vector.

01:56:54.024 --> 01:56:54.936
STUDENT: [INAUDIBLE]

01:56:54.936 --> 01:56:56.019
PROFESSOR TODA: No, no no.

01:56:56.019 --> 01:56:57.040
How do I multiply?

01:56:57.040 --> 01:56:58.510
TNB, right?

01:56:58.510 --> 01:57:01.300
So guys, how do you
multiply matrices?

01:57:01.300 --> 01:57:05.420
You go first row
and first column.

01:57:05.420 --> 01:57:06.400
So you go like this.

01:57:06.400 --> 01:57:13.629
0 times T plus k times 10
plus 0 times B. Here it is.

01:57:13.629 --> 01:57:15.170
So I'm teaching you
a little bit more

01:57:15.170 --> 01:57:18.620
than-- if you are going to
stick with linear algebra

01:57:18.620 --> 01:57:21.250
and stick with
differential equations,

01:57:21.250 --> 01:57:25.440
this is a good introduction
to more of those mathematics.

01:57:25.440 --> 01:57:26.111
Yes, sir?

01:57:26.111 --> 01:57:28.059
STUDENT: Why don't
you use Cramer's rule?

01:57:28.059 --> 01:57:28.850
PROFESSOR TODA: Uh?

01:57:28.850 --> 01:57:31.270
STUDENT: Why don't you
use the Cramer's rule?

01:57:31.270 --> 01:57:32.722
PROFESSOR TODA:
The Cramer's rule?

01:57:32.722 --> 01:57:34.835
STUDENT: Yeah. [INAUDIBLE].

01:57:34.835 --> 01:57:35.626
PROFESSOR TODA: No.

01:57:35.626 --> 01:57:44.070
First of all, Crarmer's rule is
to solve systems of equations

01:57:44.070 --> 01:57:47.810
that don't involve derivatives,
like a linear system

01:57:47.810 --> 01:57:51.960
like Ax equals B.
I'm going to have,

01:57:51.960 --> 01:57:56.760
for example, 3x1
plus 2x3 equals 1.

01:57:56.760 --> 01:58:01.000
5x1 plus x2 plus x3
equals something else.

01:58:01.000 --> 01:58:03.400
So for that I can
use Cramer's rule.

01:58:03.400 --> 01:58:04.690
But look at that!

01:58:04.690 --> 01:58:06.060
This is really complicated.

01:58:06.060 --> 01:58:07.840
It's a dynamical system.

01:58:07.840 --> 01:58:11.580
At every moment of time
the vectors are changing.

01:58:11.580 --> 01:58:13.420
So it's a crazy [INAUDIBLE].

01:58:13.420 --> 01:58:19.095
Like A of t times
something, so some vector

01:58:19.095 --> 01:58:22.858
that is also depending on
time equals the derivative

01:58:22.858 --> 01:58:25.000
of that vector that [INAUDIBLE].

01:58:25.000 --> 01:58:31.560
So that's a OD system that
you should learn in 3351.

01:58:31.560 --> 01:58:33.360
So I don't know what
your degree plan is,

01:58:33.360 --> 01:58:35.162
but most of you in
engineering will

01:58:35.162 --> 01:58:43.802
take my class, 2316 in algebra,
OD1 3350 where they teach you

01:58:43.802 --> 01:58:45.010
about differential equations.

01:58:45.010 --> 01:58:48.070
These are all differential
equations, all three of them.

01:58:48.070 --> 01:58:51.326
In 3351 you learn
about this system

01:58:51.326 --> 01:58:54.460
which is a system of
differential equation.

01:58:54.460 --> 01:58:57.210
And then you
practically say, now I

01:58:57.210 --> 01:58:59.840
know everything I need to
know in math, and you say,

01:58:59.840 --> 01:59:01.100
goodbye math.

01:59:01.100 --> 01:59:02.740
If you guys wanted
to learn more,

01:59:02.740 --> 01:59:06.223
of course I would be very
happy to learn that, hey, I

01:59:06.223 --> 01:59:08.810
like math, I'd like
to be a double major.

01:59:08.810 --> 01:59:12.320
I'd like to be not just an
engineering, but also math

01:59:12.320 --> 01:59:14.550
major if you really like it.

01:59:14.550 --> 01:59:18.170
Many people already
have a minor.

01:59:18.170 --> 01:59:20.240
Many of you have a
minor in your plan.

01:59:20.240 --> 01:59:22.835
Like for that minor
you only need--

01:59:22.835 --> 01:59:24.170
STUDENT: One extra math course.

01:59:24.170 --> 01:59:25.753
PROFESSOR TODA: One
extra math course.

01:59:25.753 --> 01:59:30.720
For example, with 3350 you
don't need 3351 for a minor.

01:59:30.720 --> 01:59:31.380
Why?

01:59:31.380 --> 01:59:34.380
Because you are taking the
probability in stats anyway.

01:59:34.380 --> 01:59:35.260
You have to.

01:59:35.260 --> 01:59:38.600
They force you to do that, 3342.

01:59:38.600 --> 01:59:44.970
So if you take 3351 it's on top
of the minor that we give you.

01:59:44.970 --> 01:59:46.400
I know because that's what I do.

01:59:46.400 --> 01:59:47.980
I look at the degree plans.

01:59:47.980 --> 01:59:51.960
And I work closely to the
math adviser, with Patty.

01:59:51.960 --> 01:59:54.265
She has all the [INAUDIBLE].

01:59:54.265 --> 01:59:55.390
STUDENT: So is [INAUDIBLE]?

01:59:55.390 --> 01:59:59.527


01:59:59.527 --> 02:00:00.860
PROFESSOR TODA: You mean double?

02:00:00.860 --> 02:00:01.840
Double degree?

02:00:01.840 --> 02:00:04.214
We have this already in place.

02:00:04.214 --> 02:00:05.380
We've had it for many years.

02:00:05.380 --> 02:00:07.400
It's an excellent plan.

02:00:07.400 --> 02:00:09.870
162 hours it is now.

02:00:09.870 --> 02:00:12.870
It used to be 159.

02:00:12.870 --> 02:00:17.700
Double major, computer
science and mathematics.

02:00:17.700 --> 02:00:22.840
And I could say they were
some of the most successful

02:00:22.840 --> 02:00:26.520
in terms of finding jobs.

02:00:26.520 --> 02:00:28.660
What would you take
on top of that?

02:00:28.660 --> 02:00:30.945
Well, as a math major you
have a few more courses

02:00:30.945 --> 02:00:32.860
to take one top of that.

02:00:32.860 --> 02:00:36.720
You can link your computer
science with the mathematics,

02:00:36.720 --> 02:00:39.630
for example, by taking
numerical analysis.

02:00:39.630 --> 02:00:42.300
If you love computers
and you like calculus

02:00:42.300 --> 02:00:46.700
and you want to put
together all the information

02:00:46.700 --> 02:00:49.140
you have in both, then
numerical analysis

02:00:49.140 --> 02:00:50.430
would be your best bet.

02:00:50.430 --> 02:00:55.215
And they require that in
both computer science degree

02:00:55.215 --> 02:00:58.230
if you are a double major,
and your math degree.

02:00:58.230 --> 02:01:03.050
So the good thing is that some
things count for both degrees.

02:01:03.050 --> 02:01:06.746
And so with those 160
hours you are very happy.

02:01:06.746 --> 02:01:10.060
Oh, I'm done, I got
a few more hours.

02:01:10.060 --> 02:01:12.420
Many math majors
already have around 130.

02:01:12.420 --> 02:01:13.830
They're not supposed to.

02:01:13.830 --> 02:01:16.080
They are supposed
to stop at 120.

02:01:16.080 --> 02:01:19.690
So why not go the extra 20 hours
and get two degrees in one?

02:01:19.690 --> 02:01:21.133
STUDENT: It's a semester.

02:01:21.133 --> 02:01:22.095
PROFESSOR TODA: Yeah.

02:01:22.095 --> 02:01:23.428
Of course, it's a lot more work.

02:01:23.428 --> 02:01:26.430
But we have people
who like-- really they

02:01:26.430 --> 02:01:30.170
are nerdy people who loved
computer science from when

02:01:30.170 --> 02:01:31.990
they were three or four.

02:01:31.990 --> 02:01:33.410
And they also like math.

02:01:33.410 --> 02:01:37.160
And they say, OK,
I want to do both.

02:01:37.160 --> 02:01:41.640
OK, a little bit more
and I'll let you go.

02:01:41.640 --> 02:01:44.822
Now I want you to ask
me other questions

02:01:44.822 --> 02:01:48.486
you may have had about the
homework, anything that

02:01:48.486 --> 02:01:59.224
gave you headache, anything that
you feel you need a little bit

02:01:59.224 --> 02:02:00.712
more of an explanation about.

02:02:00.712 --> 02:02:12.471


02:02:12.471 --> 02:02:12.970
Yes?

02:02:12.970 --> 02:02:14.011
STUDENT: I just have one.

02:02:14.011 --> 02:02:15.795
In WeBWork, what
is the easiest way

02:02:15.795 --> 02:02:17.650
to take the square
root of something?

02:02:17.650 --> 02:02:18.610
STUDENT: sqrt.

02:02:18.610 --> 02:02:21.966
PROFESSOR TODA: sqrt
is what you type.

02:02:21.966 --> 02:02:24.870
But of course you can
also go to the caret 1/2.

02:02:24.870 --> 02:02:27.930


02:02:27.930 --> 02:02:29.490
Something non-technical?

02:02:29.490 --> 02:02:34.354
Any question, yes sir,
from the homework?

02:02:34.354 --> 02:02:38.690
Or in relation to [INAUDIBLE]?

02:02:38.690 --> 02:02:41.506
STUDENT: I don't understand
why is the tangent unit vector,

02:02:41.506 --> 02:02:44.162
it's just the slope off
of that line, right?

02:02:44.162 --> 02:02:45.144
The drunk bug?

02:02:45.144 --> 02:02:47.100
Whatever line the
drunk bug is on?

02:02:47.100 --> 02:02:49.120
PROFESSOR TODA: So it
would be the tangent

02:02:49.120 --> 02:02:52.040
to the directional
motion, which is a curve.

02:02:52.040 --> 02:02:54.620


02:02:54.620 --> 02:02:58.140
And normalized to
have length one.

02:02:58.140 --> 02:03:01.700
Because otherwise our
prime is-- you may say,

02:03:01.700 --> 02:03:04.210
why do you need T to be unitary?

02:03:04.210 --> 02:03:07.150


02:03:07.150 --> 02:03:11.355
OK, computations become
horrible unless your speed

02:03:11.355 --> 02:03:13.815
is 1 or 5 or 9.

02:03:13.815 --> 02:03:18.140
If the speed is a constant,
everything else becomes easier.

02:03:18.140 --> 02:03:20.182
So that's one reason.

02:03:20.182 --> 02:03:22.086
STUDENT: And why
is the derivative

02:03:22.086 --> 02:03:24.466
of T then perpendicular?

02:03:24.466 --> 02:03:26.502
Why does it always turn into--

02:03:26.502 --> 02:03:27.960
PROFESSOR TODA:
Perpendicular to T?

02:03:27.960 --> 02:03:30.940
We've done that last time,
but I'm glad to do it again.

02:03:30.940 --> 02:03:34.430
And I forgot what we
wrote in the book,

02:03:34.430 --> 02:03:36.720
and I also saw in
the book this thing

02:03:36.720 --> 02:03:42.800
that if you have R, in
absolute value, constant--

02:03:42.800 --> 02:03:44.810
and I've done that
with you guys--

02:03:44.810 --> 02:03:52.020
prove that R and R prime had
every point perpendicular.

02:03:52.020 --> 02:03:54.850
So if you have-- we've
done that before.

02:03:54.850 --> 02:03:57.090
Now, what do you do then?

02:03:57.090 --> 02:04:00.564
T [INAUDIBLE] T is 1.

02:04:00.564 --> 02:04:04.500
The scalar [INAUDIBLE]
the product.

02:04:04.500 --> 02:04:09.540
T prime times T plus
T prime T prime.

02:04:09.540 --> 02:04:12.090
So 0.

02:04:12.090 --> 02:04:16.540
And T is perpendicular
to T prime,

02:04:16.540 --> 02:04:20.610
because that means T
or T prime equals 0.

02:04:20.610 --> 02:04:27.860


02:04:27.860 --> 02:04:30.360
When you run in a
circle, you say--

02:04:30.360 --> 02:04:33.790
OK, let's run in a circle.

02:04:33.790 --> 02:04:40.650
I say, this is my T. I can feel
that there is something that's

02:04:40.650 --> 02:04:42.530
trying to bend me this way.

02:04:42.530 --> 02:04:44.366
That is my acceleration.

02:04:44.366 --> 02:04:49.040
And I have to-- but I don't
know-- how familiar are you

02:04:49.040 --> 02:04:51.385
with the winter sports?

02:04:51.385 --> 02:04:54.150


02:04:54.150 --> 02:04:58.070
In many winter sports, the
Frenet Trihedron is crucial.

02:04:58.070 --> 02:05:01.090
Imagine that you have
one of those slopes,

02:05:01.090 --> 02:05:04.890
and all of the sudden the
torsion becomes too weak.

02:05:04.890 --> 02:05:06.680
That means it becomes dangerous.

02:05:06.680 --> 02:05:09.870
That means that the
vehicle you're in,

02:05:09.870 --> 02:05:15.010
the snow vehicle or any kind
of-- your skis, [INAUDIBLE],

02:05:15.010 --> 02:05:20.850
if the torsion of your body
moving can become too big,

02:05:20.850 --> 02:05:21.880
that will be a problem.

02:05:21.880 --> 02:05:24.670
So you have to redesign
that some more.

02:05:24.670 --> 02:05:26.660
And this is what they do.

02:05:26.660 --> 02:05:28.570
You know there have
been many accidents.

02:05:28.570 --> 02:05:32.360
And many times they say,
even in Formula One,

02:05:32.360 --> 02:05:38.170
the people who project
a certain racetrack,

02:05:38.170 --> 02:05:41.620
like a track in
Indianapolis or Montecarlo

02:05:41.620 --> 02:05:44.460
or whatever, they
have to have in mind

02:05:44.460 --> 02:05:47.660
that Frenet frame every second.

02:05:47.660 --> 02:05:50.690
So there are
simulators showing how

02:05:50.690 --> 02:05:52.874
the Frenet frame is changing.

02:05:52.874 --> 02:05:55.718
There are programs that
measure the curvature

02:05:55.718 --> 02:05:59.950
in a torsion for those
simulators at every point.

02:05:59.950 --> 02:06:02.690
Neither the curvature
nor the torsion

02:06:02.690 --> 02:06:04.360
can exceed a certain value.

02:06:04.360 --> 02:06:06.900
Otherwise it becomes dangerous.

02:06:06.900 --> 02:06:09.545
You say, oh, I thought
only the speed is a danger.

02:06:09.545 --> 02:06:10.910
Nope.

02:06:10.910 --> 02:06:14.846
It's also the way that the
motion, if it's a skew curve,

02:06:14.846 --> 02:06:16.710
it's really complicated.

02:06:16.710 --> 02:06:20.030
Because you twist and turn
and bend in many ways.

02:06:20.030 --> 02:06:22.239
And it can become
really dangerous.

02:06:22.239 --> 02:06:23.280
Speed is not [INAUDIBLE].

02:06:23.280 --> 02:06:26.262


02:06:26.262 --> 02:06:30.252
STUDENT: So the torsion was
the twists in the track?

02:06:30.252 --> 02:06:31.960
PROFESSOR TODA: The
torsion is the twist.

02:06:31.960 --> 02:06:34.580
And by the way, keep your idea.

02:06:34.580 --> 02:06:37.290
You wanted to ask
something more?

02:06:37.290 --> 02:06:43.090
When you twist-- suppose you
have something like a race car.

02:06:43.090 --> 02:06:47.190
And the race car is at
the walls of the track.

02:06:47.190 --> 02:06:57.980
And here's-- when you have
a very abrupt curvature

02:06:57.980 --> 02:07:03.760
and torsion, and you can have
that in Formula One as well,

02:07:03.760 --> 02:07:09.922
why do they build one wall
a lot higher than the other?

02:07:09.922 --> 02:07:13.980
Because the poor car-- I
don't know how passionate you

02:07:13.980 --> 02:07:19.552
are about Formula
One or car races--

02:07:19.552 --> 02:07:24.690
the poor car is going
to be close to the wall.

02:07:24.690 --> 02:07:28.382
It's going to bend like that,
that wall would be round.

02:07:28.382 --> 02:07:32.850
And as a builder, you have to
build the wall really high.

02:07:32.850 --> 02:07:35.818
Because that kind of high
speed, high velocity,

02:07:35.818 --> 02:07:39.350
high curvature, the poor
car's going szhhhhh-- then

02:07:39.350 --> 02:07:42.050
again on a normal track.

02:07:42.050 --> 02:07:45.140
Imagine what happens if the
wall is not high enough.

02:07:45.140 --> 02:07:48.490
The wheels of the car
will go up and get over.

02:07:48.490 --> 02:07:50.041
And it's going to be a disaster.

02:07:50.041 --> 02:07:52.740


02:07:52.740 --> 02:07:57.490
So that engineer ha to study
all the parametric equations

02:07:57.490 --> 02:08:01.240
and the Frenet frame and
deep down make a simulator,

02:08:01.240 --> 02:08:04.450
compute how tall the walls
should be in order for the car

02:08:04.450 --> 02:08:10.220
not to get over on the other
side or get off the track.

02:08:10.220 --> 02:08:12.350
It's really complicated stuff.

02:08:12.350 --> 02:08:14.890
It's all mathematics
and physics,

02:08:14.890 --> 02:08:18.680
but all the applications are
run by engineers and-- yes, sir?

02:08:18.680 --> 02:08:22.033
STUDENT: What's the difference
[INAUDIBLE] centrifugal force?

02:08:22.033 --> 02:08:23.950
PROFESSOR TODA: The
centrifugal force

02:08:23.950 --> 02:08:26.380
is related to our double prime.

02:08:26.380 --> 02:08:32.130
Our double prime is related
to N and T at the same time.

02:08:32.130 --> 02:08:36.160
So at some point, let me ask you
one last question and I'm done.

02:08:36.160 --> 02:08:39.210


02:08:39.210 --> 02:08:43.236
What's the relationship between
acceleration or double prime?

02:08:43.236 --> 02:08:45.982
And are they the same thing?

02:08:45.982 --> 02:08:50.297
And when are they
not the same thing?

02:08:50.297 --> 02:08:52.380
Because you say, OK,
practically the centrifugal--

02:08:52.380 --> 02:08:54.180
STUDENT: They're
the same on a curve.

02:08:54.180 --> 02:08:55.740
PROFESSOR TODA:
They are the same--

02:08:55.740 --> 02:08:56.823
STUDENT: Like on a circle.

02:08:56.823 --> 02:08:58.520
PROFESSOR TODA: On a circle!

02:08:58.520 --> 02:09:00.090
And you are getting so close.

02:09:00.090 --> 02:09:01.370
It's hot, hot, hot.

02:09:01.370 --> 02:09:08.100
On a circle and on a helix they
are the same up to a constant.

02:09:08.100 --> 02:09:11.310
So what do you think the
magic answer will be?

02:09:11.310 --> 02:09:12.480
N was what, guys?

02:09:12.480 --> 02:09:15.220
N was-- remind me again.

02:09:15.220 --> 02:09:18.475
That was T prime over
absolute value of T prime.

02:09:18.475 --> 02:09:22.290
But that doesn't mean,
does not equal, in general,

02:09:22.290 --> 02:09:26.350
does not equal to
R double prime.

02:09:26.350 --> 02:09:28.096
When is it equal?

02:09:28.096 --> 02:09:29.930
In general it's not equal.

02:09:29.930 --> 02:09:31.070
When is it equal?

02:09:31.070 --> 02:09:35.440
If you are in aclength, you
see the advantage of aclength.

02:09:35.440 --> 02:09:36.965
It's wonderful.

02:09:36.965 --> 02:09:40.067
In arclength, T is R prime of s.

02:09:40.067 --> 02:09:45.940
And in arclength that means T
prime is R double prime of s.

02:09:45.940 --> 02:09:48.590
And in arclength
I just told you,

02:09:48.590 --> 02:09:50.330
T prime is the first
Frenet formula.

02:09:50.330 --> 02:09:55.510
It'll be curvature times the N.

02:09:55.510 --> 02:10:02.180
So the acceleration
practically and the N

02:10:02.180 --> 02:10:06.560
will be the same in arclength,
up to a scalar multiplication.

02:10:06.560 --> 02:10:11.678
But what if your speed
is not even constant?

02:10:11.678 --> 02:10:12.530
Then God help you.

02:10:12.530 --> 02:10:16.850
Because the acceleration
R double prime and N

02:10:16.850 --> 02:10:19.780
are not colinear.

02:10:19.780 --> 02:10:24.370
So if I were to draw-- and
that's my last picture--

02:10:24.370 --> 02:10:26.950
let me give you a
wild motion here.

02:10:26.950 --> 02:10:32.466
You start slow and then you go
crazy and fast and slow down.

02:10:32.466 --> 02:10:35.657
Just like most of the
physical models from the bugs

02:10:35.657 --> 02:10:38.900
and the flies and so on.

02:10:38.900 --> 02:10:44.950
In that kind of crazy motion you
have a T and N at every point.

02:10:44.950 --> 02:10:45.450
[INAUDIBLE]

02:10:45.450 --> 02:10:48.430


02:10:48.430 --> 02:10:50.860
[? v ?] will be down.

02:10:50.860 --> 02:10:53.360
And T is here.

02:10:53.360 --> 02:10:56.940
So can you draw arc
double prime for me?

02:10:56.940 --> 02:10:59.433
It will still be
towards the inside.

02:10:59.433 --> 02:11:04.200
But it's still going to
coincide with N. Maybe this one.

02:11:04.200 --> 02:11:12.640
What's the magic thing is
that T, N, and R double prime

02:11:12.640 --> 02:11:15.745
are in the same plane always.

02:11:15.745 --> 02:11:18.170
That's another
secret other students

02:11:18.170 --> 02:11:19.630
don't know in Calculus 3.

02:11:19.630 --> 02:11:22.456
That same thing is
called osculating plane.

02:11:22.456 --> 02:11:25.630


02:11:25.630 --> 02:11:31.270
We have a few magic
names for these things.

02:11:31.270 --> 02:11:36.509
So T and N, the plane that
is-- how shall I say that?

02:11:36.509 --> 02:11:37.050
I don't know.

02:11:37.050 --> 02:11:43.638
The plane given by T and N
is called osculating plane.

02:11:43.638 --> 02:11:46.710


02:11:46.710 --> 02:11:49.080
The acceleration is
always on that plane.

02:11:49.080 --> 02:11:52.460
So imagine T and N are
in the same shaded plane.

02:11:52.460 --> 02:11:55.500
R double prime is
in the same plane.

02:11:55.500 --> 02:11:56.550
OK?

02:11:56.550 --> 02:11:58.510
Now, can you guess
the other two names?

02:11:58.510 --> 02:12:03.639
So this is T, this
is N. And B is up.

02:12:03.639 --> 02:12:04.805
This is my body's direction.

02:12:04.805 --> 02:12:06.200
T and N, look at me.

02:12:06.200 --> 02:12:10.160
T, N, and B. I'm the
Frenet Trihedron.

02:12:10.160 --> 02:12:13.360
Which one is the
osculating plane?

02:12:13.360 --> 02:12:16.740
It's the horizontal xy plane.

02:12:16.740 --> 02:12:20.940
OK, do you know-- maybe you're
a mechanical engineering major,

02:12:20.940 --> 02:12:23.650
and after that I
will let you go.

02:12:23.650 --> 02:12:25.920
No extra credit,
though for this task.

02:12:25.920 --> 02:12:29.130
Maybe I'm going to start asking
questions and give you $1.

02:12:29.130 --> 02:12:31.330
I used to do that a lot
in differential equations,

02:12:31.330 --> 02:12:34.580
like ask a hard question,
whoever gets it first,

02:12:34.580 --> 02:12:36.420
give her a dollar.

02:12:36.420 --> 02:12:41.595
Until a point when they asked
me to teach Honors 3350 when

02:12:41.595 --> 02:12:44.210
I started having three or four
people answering the question

02:12:44.210 --> 02:12:44.925
at the same time.

02:12:44.925 --> 02:12:49.020
And that was a
significant expense,

02:12:49.020 --> 02:12:52.215
because I had to give $4
away at the same time.

02:12:52.215 --> 02:12:53.840
STUDENT: I feel like
you should've just

02:12:53.840 --> 02:12:54.590
split it between--

02:12:54.590 --> 02:12:57.950
PROFESSOR TODA: So that's
normal and binormal.

02:12:57.950 --> 02:13:00.800
This is me, the binormal,
and this is the normal.

02:13:00.800 --> 02:13:03.080
Does anybody know the
name of this plane,

02:13:03.080 --> 02:13:05.850
between normal and bionormal?

02:13:05.850 --> 02:13:08.170
This would be this plane.

02:13:08.170 --> 02:13:10.750
STUDENT: The skew [INAUDIBLE].

02:13:10.750 --> 02:13:12.250
PROFESSOR TODA:
Normal and binormal.

02:13:12.250 --> 02:13:13.884
They call that normal plane.

02:13:13.884 --> 02:13:16.540


02:13:16.540 --> 02:13:22.510
So it's tricky if you are not
a mechanical engineering major.

02:13:22.510 --> 02:13:28.460
But some of you are maybe
and will learn that later.

02:13:28.460 --> 02:13:29.940
Any other questions for me?

02:13:29.940 --> 02:13:33.890
Now, in my office I'm
going to do review.

02:13:33.890 --> 02:13:37.836
I was wondering
if you have time,

02:13:37.836 --> 02:13:39.960
I don't know if you have
time to come to my office,

02:13:39.960 --> 02:13:43.340
but should you have any kind
of homework related question,

02:13:43.340 --> 02:13:46.290
I'll be very happy
to answer it now.

02:13:46.290 --> 02:13:49.000
3:00 to 5:00.

02:13:49.000 --> 02:13:51.020
Now, one time I
had a student who

02:13:51.020 --> 02:13:53.175
only had seven questions left.

02:13:53.175 --> 02:13:55.690
He came to my office and
he left with no homework.

02:13:55.690 --> 02:13:57.593
We finished all of them.

02:13:57.593 --> 02:13:58.380
And I felt guilty.

02:13:58.380 --> 02:14:00.870
But at the same, he
said, well, no, it's

02:14:00.870 --> 02:14:03.170
better I came to you instead
of going to my tutor.

02:14:03.170 --> 02:14:05.180
It was fine.

02:14:05.180 --> 02:14:08.670
So we can try some
problems together today

02:14:08.670 --> 02:14:11.850
if you want between 3:00 and
5:00, if you have the time.

02:14:11.850 --> 02:14:13.838
Some of you don't have the time.

02:14:13.838 --> 02:14:14.832
All right?

02:14:14.832 --> 02:14:16.323
If you don't have
the time today,

02:14:16.323 --> 02:14:19.305
and you would like to
be helped [INAUDIBLE],

02:14:19.305 --> 02:14:21.293
click Email Instructor.

02:14:21.293 --> 02:14:24.275
I'm going to get the
questions [INAUDIBLE].

02:14:24.275 --> 02:14:26.014
You're welcome to
ask me anything

02:14:26.014 --> 02:14:27.257
at any time over there.

02:14:27.257 --> 02:14:38.191


02:14:38.191 --> 02:14:41.173
[CLASSROOM CHATTER]

02:14:41.173 --> 02:15:12.348


02:15:12.348 --> 02:15:14.472
PROFESSOR TODA: I have
somebody who's taking notes.

02:15:14.472 --> 02:15:15.167
STUDENT: Yeah, I know.

02:15:15.167 --> 02:15:15.963
And that's why I was like--

02:15:15.963 --> 02:15:16.957
PROFESSOR TODA: He's
going to make a copy

02:15:16.957 --> 02:15:18.448
and I'll give you a copy.

02:15:18.448 --> 02:15:19.442
STUDENT: Yeah.

02:15:19.442 --> 02:15:23.621
My Cal 1 teacher,
Dr. [INAUDIBLE].

02:15:23.621 --> 02:15:24.412
STUDENT: Thank you.

02:15:24.412 --> 02:15:25.495
PROFESSOR TODA: Yes, yeah.

02:15:25.495 --> 02:15:26.897
Have a nice day.

02:15:26.897 --> 02:15:29.030
STUDENT: --got really mad
when I don't take notes.

02:15:29.030 --> 02:15:34.680
Because he felt like
I was not, I guess--

02:15:34.680 --> 02:15:36.503