## TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 3

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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
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[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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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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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
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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
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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
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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.
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I mean, this problem
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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,
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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
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So you have R10 of
t, 1/2 sine of 2t
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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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So I got the long motion back.
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Similarly, you could find,
if somebody gives you
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the acceleration of a
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this is the acceleration.
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And I give you some
initial values.
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And you have to find
first the velocity,
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going backwards one step.
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And from the velocity,
backwards a second step,
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get the position vector.
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And that sounds a little
bit more elaborate.
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But it doesn't have to
be a long computation.
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In general, we do not
focus on giving you
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an awfully long computation.
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We just want to test your
understanding of the concepts.
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And having this in mind,
I picked another example.
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I would like to
see what that is.
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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.
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So somebody gives you the
acceleration in the velocity
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vector at 0 and is asking you
to find the velocity vector So
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let me give it to you
for t between 0 and 2 pi.
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I give you the
acceleration vector,
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it will be nice and sassy.
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Let's see, that's going to be
cosine of t, sine of t and 0.
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And you'll say, oh, I
know how to do those.
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Of course you know.
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But I want you to pay
attention to the constraints
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of integration.
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This is why I do this
kind of exercise again.
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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,
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and let's say this would be--
I have no idea what I want.
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Let's say i, j, and that's it.
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How do you do that?
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V of t.
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Let's integrate together.
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You don't like this?
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I hope that by now,
you've got used to it.
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A bracket, I'm doing a
bracket, like in the book.
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So sine t plus a constant.
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What's the integral
of sine, class?
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V equals sine t plus a constant.
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And C3 is a constant.
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And there I go.
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You say, oh my god,
what am I having?
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V of 0-- is as a
vector, I presented it
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in the canonical standard
basis as 1, 1, and 0.
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So from that one, you
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and say, yes, I'm going to
plug in 0, see what I get.
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In the general formula,
when you plug in 0,
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you get C1-- what
is cosine of 0?
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Minus 1, I have here, plus C2.
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And C3, that is always there.
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And then V of 0 is
what I got here.
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V of 0 has to be compared to
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So C1 is 1, C2 is 2, and C3 is--
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So let me replace it.
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I say the answer will be--
cosine t plus 1, sine t plus 2,
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and the constants.
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But then somebody, who is
really an experimental guy,
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says well--
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STUDENT: You have it backwards.
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It's sine of t plus
1, and then you
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have the cosine of t plus 2.
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PROFESSOR TODA: Oh, yeah.
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Wait a minute.
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This is-- I
miscopied looking up.
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So I have sine t, I was
supposed to-- minus cosine t
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and I'm done.
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So thank you for telling me.
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So sum t plus 1 minus
cosine t plus 2 and 0
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are the functions that I put
here by replacing C1, C2, C3.
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And then, somebody
says, wait a minute,
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now let me give you V of 0.
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Let me give you R of 0.
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And you were supposed
to get R from here.
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So what is R of t, the
position vector, find it.
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V of t is given.
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Actually, it's given by
you, because you found it
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at the previous step.
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And R of 0 is given as well.
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And let's say that would
be-- let's say 1, 1, and 1.
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So what do you need to do next?
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You have R prime given.
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That leaves you to
integrate to get R t.
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And R of t is going to be what?
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Who is going to tell me
what I have to write down?
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Minus cosine t plus t plus--
let's use the constant K1
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integration.
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And then what?
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STUDENT: Sine of t.
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PROFESSOR TODA: I think
it's minus sine, right?
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Minus sine of t plus 2t
plus K2 and K3, right?
• 17:56 - 18:04
So R of 0 is going to be what?
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First of all, we use this
piece of information.
• 18:08 - 18:12
Second of all, we identify
from the formula we got.
• 18:12 - 18:16
So from the formula I
got, just plugging in 0,
• 18:16 - 18:23
it should come out straight
as minus 1 plus K1.
• 18:23 - 18:28
0 for this guy, 0 for the
second term, K2 and K3.
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So who is helping me solve
the system really quickly?
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K1 is 2.
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K2 is--
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STUDENT: 1.
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PROFESSOR TODA: K3 is 1.
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And I'm going back
to R and replace it.
• 18:51 - 18:55
for this two-step problem.
• 18:55 - 18:58
So I have a two-step integration
from the acceleration
• 18:58 - 19:00
to the velocity,
from the velocity
• 19:00 - 19:05
to the position vector.
• 19:05 - 19:08
Minus cosine t plus t plus 2.
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Remind me, because I have
a tendency to miscopy,
• 19:12 - 19:13
an I looking in the right place?
• 19:13 - 19:14
Yes.
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So I have minus sine t plus
2t plus 1 and K3 is one.
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So this is the process you
are supposed to remember
• 19:30 - 19:32
for the rest of the semester.
• 19:32 - 19:33
It's not a hard one.
• 19:33 - 19:37
It's something that
everybody should master.
• 19:37 - 19:38
Is it hard?
• 19:38 - 19:40
How many of you understood this?
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• 19:42 - 19:45
Oh, no problem, good.
• 19:45 - 19:52
Now would you tell me--
I'm not going to ask you
• 19:52 - 19:54
what kind of motion this is.
• 19:54 - 19:57
It's a little bit close to
a circular motion but not
• 19:57 - 19:58
a circular motion.
• 19:58 - 20:01
However, can you tell
me anything interesting
• 20:01 - 20:05
that I have, in terms
• 20:05 - 20:06
of the acceleration vector?
• 20:06 - 20:11
The acceleration
vector is beautiful,
• 20:11 - 20:14
just like in the
case of the washer.
• 20:14 - 20:19
That was a vector
that-- like this
• 20:19 - 20:21
would be the circular motion.
• 20:21 - 20:23
The acceleration would
be this unique vector
• 20:23 - 20:25
that comes inside.
• 20:25 - 20:27
Is this going outside
or coming inside?
• 20:27 - 20:30
Is it a unit vector?
• 20:30 - 20:33
Yes, it is a unit vector.
• 20:33 - 20:37
So suppose that I'm
looking at the trajectory,
• 20:37 - 20:40
if it were more or
less a motion that has
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to do with mixing into a bowl.
• 20:45 - 20:49
Would this go inside or outside?
• 20:49 - 20:52
Towards the outside
or towards the inside?
• 20:52 - 20:58
I plugged j-- depends on
what I'm looking at, in terms
• 20:58 - 21:00
of surface that I'm on, right?
• 21:00 - 21:02
Do you remember
from last time we
• 21:02 - 21:04
was on a cylinder.
• 21:04 - 21:08
is that [INAUDIBLE] pointing?
• 21:08 - 21:12
And it was pointing
outside of the cylinder,
• 21:12 - 21:16
in the direction
towards the outside.
• 21:16 - 21:27
Coming back to the
review, there are
• 21:27 - 21:31
several things I'd like to
review but not all of them.
• 21:31 - 21:34
Because some of the
examples we have there,
• 21:34 - 21:38
you understood them really well.
• 21:38 - 21:40
I was very proud
of you, and I saw
• 21:40 - 21:44
that you finished--
almost all of you
• 21:44 - 21:46
finished the
homework number one.
• 21:46 - 21:49
So I was looking outside
at homework number
• 21:49 - 21:53
two that is over
these three sections.
• 21:53 - 21:58
So I was hoping you would ask
me today, between two and three,
• 21:58 - 22:01
if you have any difficulties
with homework two.
• 22:01 - 22:04
That's due February 11.
• 22:04 - 22:13
And then the latest homework
that I posted yesterday, I
• 22:13 - 22:15
don't know how many
of you logged in.
• 22:15 - 22:19
But last night I
posted a homework
• 22:19 - 22:22
that is getting a huge
• 22:22 - 22:23
is the 28th of February.
• 22:23 - 22:29
Because somebody's
birthday is February 29.
• 22:29 - 22:35
I was just thinking why would
somebody need be a whole month?
• 22:35 - 22:37
You would need the whole
month to have a good view
• 22:37 - 22:39
of the whole chapter 11.
• 22:39 - 22:41
I sent you the videos
for chapter 11.
• 22:41 - 22:44
And for chapter 11, you
have this huge homework
• 22:44 - 22:47
which is 49 problems.
• 22:47 - 22:50
do not leave it
• 22:50 - 22:52
to the last five
days or six days,
• 22:52 - 22:56
because it's going to kill you.
• 22:56 - 22:57
There are people who
say, I can finish
• 22:57 - 22:59
this in the next five days.
• 22:59 - 23:00
I know you can.
• 23:00 - 23:02
I know you can,
I don't doubt it.
• 23:02 - 23:04
That's why I left
you so much freedom.
• 23:04 - 23:08
But you have-- today is
the second or the third?
• 23:08 - 23:11
So practically you have
25 days to work on this.
• 23:11 - 23:15
On the 28th at 11 PM
it's going to close.
• 23:15 - 23:19
I would work a few
problems every other day.
• 23:19 - 23:22
Because I need a break,
so I would alternate.
• 23:22 - 23:25
But don't leave it--
even if you have help,
• 23:25 - 23:28
especially if you have help,
like a tutor or tutoring
• 23:28 - 23:30
services here that are
free in the department.
• 23:30 - 23:32
Do not leave it
to the last days.
• 23:32 - 23:35
Because you're putting pressure
• 23:35 - 23:37
• 23:37 - 23:38
Yes sir.
• 23:38 - 23:39
STUDENT: So that's
homework three?
• 23:39 - 23:40
PROFESSOR TODA:
That's homework three,
• 23:40 - 23:43
and it's a huge homework
over chapter 11.
• 23:43 - 23:46
STUDENT: You said
there are 49 problems?
• 23:46 - 23:49
PROFESSOR TODA: I don't
remember exactly but 47, 49.
• 23:49 - 23:50
I don't remember how many.
• 23:50 - 23:53
STUDENT: Between 45 and 50.
• 23:53 - 23:56
PROFESSOR TODA:
Between 45 and 50, yes.
• 23:56 - 23:59
If you encounter any bug--
although there shouldn't
• 23:59 - 24:02
be bugs, maybe 1 in 1,000.
• 24:02 - 24:05
If you encounter any
bug that the programmer
• 24:05 - 24:10
of those problems may
have accidentally put in,
• 24:10 - 24:11
you let me know.
• 24:11 - 24:14
So I can contact them.
• 24:14 - 24:17
If there is a problem that I
consider shouldn't be there,
• 24:17 - 24:20
I will eliminate that later on.
• 24:20 - 24:23
But hopefully, everything
will be doable,
• 24:23 - 24:28
everything will be fair and
you will be able to solve it.
• 24:28 - 24:32
• 24:32 - 24:35
Any questions?
• 24:35 - 24:37
Particular questions
from the homework?
• 24:37 - 24:40
• 24:40 - 24:44
STUDENT: [INAUDIBLE] is it to
parametrize a circle of a set,
• 24:44 - 24:48
like of a certain
• 24:48 - 24:49
PROFESSOR TODA:
Shall we do that?
• 24:49 - 24:53
Do you want me to do that
in general, in xy-plane, OK.
• 24:53 - 24:55
STUDENT: [INAUDIBLE]
in the xy-plane.
• 24:55 - 24:59
• 24:59 - 25:05
PROFESSOR TODA: xy-plane and
then what was the equation?
• 25:05 - 25:10
Was it like a equals sine
of t or a equals sine of bt?
• 25:10 - 25:12
Because it's a
little bit different,
• 25:12 - 25:16
depending on how the
parametrization was given.
• 25:16 - 25:17
again, I forgot.
• 25:17 - 25:19
I don't know what to refer you.
• 25:19 - 25:20
STUDENT: Ryder.
• 25:20 - 25:22
• 25:22 - 25:25
PROFESSOR TODA: Was that part
of what's due on the 11th?
• 25:25 - 25:28
STUDENT: It doesn't-- yes, it
doesn't give a revision set.
• 25:28 - 25:29
It says--
• 25:29 - 25:33
PROFESSOR TODA: Let me quickly
• 25:33 - 25:38
of the circle of radius 7 in
the xy-plane, centered at 3, 1,
• 25:38 - 25:41
oriented counterclockwise.
• 25:41 - 25:43
The point 10, 1
should be connected--
• 25:43 - 25:45
STUDENT: Just one more second.
• 25:45 - 25:46
PROFESSOR TODA: Do
you mind if I put it.
• 25:46 - 25:47
I'll take good care of it.
• 25:47 - 25:48
I won't drop it.
• 25:48 - 25:52
• 25:52 - 25:58
So the point-- parametrization
• 25:58 - 26:02
7 in the xy-plane,
centered at 3, 1.
• 26:02 - 26:12
So circle centered at-- and
I'll say it x0, 1 0, being 3, 1.
• 26:12 - 26:16
• 26:16 - 26:19
No, because then I'm
• 26:19 - 26:21
But I'm solving
• 26:21 - 26:23
even if I change
change the numbers.
• 26:23 - 26:26
• 26:26 - 26:28
Why don't I change
the numbers, and then
• 26:28 - 26:31
you do it for the given numbers.
• 26:31 - 26:34
Let's say 1, 0.
• 26:34 - 26:40
And it's the same type
of problem, right?
• 26:40 - 26:43
Oriented counterclockwise.
• 26:43 - 26:43
That's important.
• 26:43 - 26:52
• 26:52 - 26:54
So you have circle radius 7.
• 26:54 - 26:57
I think people could
have any other,
• 26:57 - 27:01
because problems are-- sometimes
you get a random assignment.
• 27:01 - 27:05
So you have R
equals 2, let's say.
• 27:05 - 27:08
• 27:08 - 27:14
And you have the point,
how to make up something.
• 27:14 - 27:21
The point corresponding
to t equals
• 27:21 - 27:30
0 will be given as you have
[INAUDIBLE], 1, 0, whatever.
• 27:30 - 27:32
OK?
• 27:32 - 27:37
Use the t as the parameter
• 27:37 - 27:39
So use t as a parameter
• 27:39 - 27:43
and the answers are written in
the interactive field as x of t
• 27:43 - 27:45
equals what and y
of t equals what,
• 27:45 - 27:47
and it's waiting for
you to fill them in.
• 27:47 - 27:49
You know.
• 27:49 - 27:54
OK, now I was talking
to [INAUDIBLE].
• 27:54 - 27:57
I'm going to give
this back to you.
• 27:57 - 27:58
Thank you, Ryan.
• 27:58 - 28:03
So when you said it's a
little bit frustrating,
• 28:03 - 28:08
and I agree wit you, that
in this variant of webwork
• 28:08 - 28:11
problems you have to enter
both of them correctly
• 28:11 - 28:15
in order to say yes, correct.
• 28:15 - 28:18
I was used to another library--
the library was outdated
• 28:18 - 28:22
[INAUDIBLE]-- where if I
enter this correctly I get 50%
• 28:22 - 28:26
credit, and if I enter this
incorrectly it's not going
• 28:26 - 28:27
to penalize me.
• 28:27 - 28:30
So I a little bit
• 28:30 - 28:32
and I was shown the
old library where
• 28:32 - 28:36
I can go ahead and go
back and assign problems
• 28:36 - 28:39
correct for this one
• 28:39 - 28:42
and incorrect for this one,
and you get partial credit.
• 28:42 - 28:47
So I'm probably going
to switch to that.
• 28:47 - 28:47
Let's do that.
• 28:47 - 28:49
This is a very good problem.
• 28:49 - 28:52
I'm glad you brought it up.
• 28:52 - 28:57
conics in high school?
• 28:57 - 29:00
well, it depends.
• 29:00 - 29:01
• 29:01 - 29:03
• 29:03 - 29:04
• 29:04 - 29:07
Some of you put them down
for me for extra credit.
• 29:07 - 29:09
I was very happy you did that.
• 29:09 - 29:10
It's a good exercise.
• 29:10 - 29:12
If you have-- Alex, yes?
• 29:12 - 29:14
STUDENT: I was just
thinking, does that say 1, 0?
• 29:14 - 29:18
• 29:18 - 29:19
The point corresponding
to t0 [INAUDIBLE]?
• 29:19 - 29:20
PROFESSOR TODA: I think
that's what I meant.
• 29:20 - 29:22
I don't know, I just
came up with it.
• 29:22 - 29:23
• 29:23 - 29:23
1, 0.
• 29:23 - 29:24
I make up all my problems.
• 29:24 - 29:26
STUDENT: But the center
of the circle isn't 1, 0.
• 29:26 - 29:27
PROFESSOR TODA: Oh, oops.
• 29:27 - 29:30
Yes.
• 29:30 - 29:32
Sorry.
• 29:32 - 29:33
So 2, 0.
• 29:33 - 29:34
No--
• 29:34 - 29:35
[INTERPOSING VOICES]
• 29:35 - 29:38
PROFESSOR TODA:
• 29:38 - 29:41
This is the problem when you
don't think very [INAUDIBLE].
• 29:41 - 29:44
I always like to make
up my own problems.
• 29:44 - 29:48
When an author, when we came up
with the problems in the book,
• 29:48 - 29:52
of course we had to think, draw,
and make sure they made sense.
• 29:52 - 29:55
But when you just come up with
a problem out of the middle
• 29:55 - 29:57
of nowhere-- thank you so much.
• 29:57 - 29:59
Of course, we
would have realized
• 29:59 - 30:01
that was nonsense
in just a minute.
• 30:01 - 30:04
But it's good that you told me.
• 30:04 - 30:07
So x of t, y of t.
• 30:07 - 30:11
• 30:11 - 30:12
Let's find it.
• 30:12 - 30:13
Based on what?
• 30:13 - 30:16
What is the general
equation of a circle?
• 30:16 - 30:22
x minus x0 squared plus y minus
y0 squared equals R squared.
• 30:22 - 30:25
And you have learned
that in high school.
• 30:25 - 30:27
Am I right or not?
• 30:27 - 30:27
You have.
• 30:27 - 30:28
OK.
• 30:28 - 30:29
Good.
• 30:29 - 30:36
Now, in our case what
is x0 and what is y0?
• 30:36 - 30:40
x0 is 1 and y0 is 0.
• 30:40 - 30:43
Because that's
why-- I don't know.
• 30:43 - 30:44
• 30:44 - 30:47
And I said that's the center.
• 30:47 - 30:49
I'll draw.
• 30:49 - 30:51
I should have drawn
it in the beginning,
• 30:51 - 30:54
and that would have
helped me not come up
• 30:54 - 31:00
with some nonsensical data.
• 31:00 - 31:02
c is 1, 0.
• 31:02 - 31:03
• 31:03 - 31:05
So I'm going this way.
• 31:05 - 31:07
What point is this way, guys?
• 31:07 - 31:09
Just by the way.
• 31:09 - 31:10
Because [INAUDIBLE]
is 1, 0, right?
• 31:10 - 31:16
And this way the other
extreme, the antipode is 3, 0.
• 31:16 - 31:20
So that's exactly what
Alexander was saying.
• 31:20 - 31:22
And now it makes sense.
• 31:22 - 31:25
• 31:25 - 31:26
Well, I cannot draw today.
• 31:26 - 31:27
STUDENT: [INAUDIBLE]
• 31:27 - 31:30
• 31:30 - 31:32
PROFESSOR TODA:
It looks horrible.
• 31:32 - 31:37
It looks like an egg that
is shaped-- disabled egg.
• 31:37 - 31:41
• 31:41 - 31:42
OK.
• 31:42 - 31:43
All right.
• 31:43 - 31:50
So the motion of-- the
motion will come like that.
• 31:50 - 31:54
From t equals 0, when I'm
here, counterclockwise,
• 31:54 - 31:57
I have to draw-- any kind of
circle you have in the homework
• 31:57 - 32:01
should be drawn on the board.
• 32:01 - 32:06
If you have a general, you
don't know what the data is.
• 32:06 - 32:09
the general problem.
• 32:09 - 32:11
For the original problem,
which is a circle
• 32:11 - 32:15
of center x, 0, y, 0 and
• 32:15 - 32:20
what is the parametrization
without data?
• 32:20 - 32:20
Without specific data.
• 32:20 - 32:23
What is the parametrization?
• 32:23 - 32:26
And I want you to pay
attention very well.
• 32:26 - 32:27
You are paying attention.
• 32:27 - 32:30
You are very careful today.
• 32:30 - 32:31
[INAUDIBLE]
• 32:31 - 32:34
So what do you have?
• 32:34 - 32:36
STUDENT: Cosine.
• 32:36 - 32:38
PROFESSOR TODA:
Before that cosine
• 32:38 - 32:40
there is an R, excellent.
• 32:40 - 32:44
So [INAUDIBLE]
there R cosine of t.
• 32:44 - 32:47
I'm not done.
• 32:47 - 32:47
What do I put here?
• 32:47 - 32:48
STUDENT: Over d.
• 32:48 - 32:49
PROFESSOR TODA: No, no.
• 32:49 - 32:51
I'm continuing.
• 32:51 - 32:52
STUDENT: Plus x0.
• 32:52 - 32:54
PROFESSOR TODA: Plus x0.
• 32:54 - 32:57
And R sine t plus y0.
• 32:57 - 33:00
Who taught me that?
• 33:00 - 33:03
First of all, this
is not unique.
• 33:03 - 33:04
It's not unique.
• 33:04 - 33:06
I could put sine t
here and cosine t here
• 33:06 - 33:09
and it would be the same
type of parametrization.
• 33:09 - 33:11
But we usually put
the cosine first
• 33:11 - 33:14
because we look at the
x-axis corresponding
• 33:14 - 33:18
to the cosine and the y-axis
corresponding to the sine.
• 33:18 - 33:21
If I don't know that,
because I happen to know that
• 33:21 - 33:24
from when I was 16 in high
school, if I don't know that,
• 33:24 - 33:25
what do I know?
• 33:25 - 33:28
I cook up my own
parametrization.
• 33:28 - 33:29
And that's a very good thing.
• 33:29 - 33:31
• 33:31 - 33:33
How does one come up with this?
• 33:33 - 33:34
Do we have to memorize?
• 33:34 - 33:38
In mathematics, thank god,
we don't memorize much.
• 33:38 - 33:42
The way we cook up things
is just from, in this case,
• 33:42 - 33:45
from the Pythagorean
theorem of-- no.
• 33:45 - 33:47
Pythagorean theorem
of trigonometry?
• 33:47 - 33:49
The fundamental identity
of trigonometry,
• 33:49 - 33:53
which is the same thing as
the Pythagorean theorem.
• 33:53 - 33:55
What's the fundamental
identity of trigonometry?
• 33:55 - 33:58
Cosine squared plus
sin squared equals 1.
• 33:58 - 34:04
If I have a problem
like that, I must
• 34:04 - 34:09
have that this is R cosine
t and this is R sine t.
• 34:09 - 34:11
Because when I take
the red guys and I
• 34:11 - 34:14
square them and I
• 34:14 - 34:18
I'm going to have R squared.
• 34:18 - 34:19
All righty, good.
• 34:19 - 34:23
So no matter what
kind of data you have,
• 34:23 - 34:28
you should be able to come
up with this on your own.
• 34:28 - 34:34
And what else is
going to be happening?
• 34:34 - 34:38
When I solve for x of-- the
point corresponding to t
• 34:38 - 34:39
equals 0.
• 34:39 - 34:44
x of 0 and y of 0 will
therefore be what?
• 34:44 - 34:49
It will be R plus x0.
• 34:49 - 34:51
This is going to be what?
• 34:51 - 34:54
Just y0.
• 34:54 - 34:56
Does anybody give them to me?
• 34:56 - 34:59
STUDENT: 3, 0.
• 34:59 - 35:02
PROFESSOR TODA: Alexander
gave me the correct ones.
• 35:02 - 35:06
They will be 3 and 0.
• 35:06 - 35:07
Are you guys with me?
• 35:07 - 35:11
They could be anything,
anything that makes sense.
• 35:11 - 35:15
All right, for example somebody
would say, I'm starting here.
• 35:15 - 35:17
I give you other points.
• 35:17 - 35:20
Then you put them in, you
plug in that initial point,
• 35:20 - 35:23
meaning that you're
• 35:23 - 35:26
And you do go around
the circle one
• 35:26 - 35:32
because, you take [INAUDIBLE]
only between 0 and 2 pi.
• 35:32 - 35:33
Alexander.
• 35:33 - 35:34
STUDENT: I have [INAUDIBLE].
• 35:34 - 35:35
PROFESSOR TODA: OK.
• 35:35 - 35:36
STUDENT: [INAUDIBLE]
• 35:36 - 35:38
PROFESSOR TODA: No, I thought
that I misprinted something
• 35:38 - 35:39
again.
• 35:39 - 35:41
STUDENT: No, I was about to
say something really dumb.
• 35:41 - 35:42
PROFESSOR TODA: OK.
• 35:42 - 35:44
• 35:44 - 35:49
So how do we make sense
of what we have here?
• 35:49 - 35:52
Well, y0 corresponds
to what I said.
• 35:52 - 35:56
So this is a
superfluous equation.
• 35:56 - 35:58
I don't need that.
• 35:58 - 36:01
What do I know from that?
• 36:01 - 36:06
R will be 2.
• 36:06 - 36:08
x1 is 1.
• 36:08 - 36:10
I have a superfluous equation.
• 36:10 - 36:14
I have to get identities
in that case, right?
• 36:14 - 36:15
OK, now.
• 36:15 - 36:20
• 36:20 - 36:27
What is going to be my--
my bunch of equations
• 36:27 - 36:48
will be x of t equals 2
cosine t plus 1 and y of t
• 36:48 - 36:49
equals-- I don't
like this marker.
• 36:49 - 36:50
I hate it.
• 36:50 - 36:51
Where did I get it?
• 36:51 - 36:52
In the math department.
• 36:52 - 36:53
And it's a new one.
• 36:53 - 36:55
I got it as a new one.
• 36:55 - 36:57
It's not working.
• 36:57 - 36:58
OK, y of t.
• 36:58 - 37:01
• 37:01 - 37:04
The blue contrast is invisible.
• 37:04 - 37:08
I have 2 sine t.
• 37:08 - 37:08
Okey dokey.
• 37:08 - 37:12
When you finish a
problem, always quickly
• 37:12 - 37:16
verify if what you
got makes sense.
• 37:16 - 37:20
And obviously if I
look at everything,
• 37:20 - 37:21
it's matching the whole point.
• 37:21 - 37:22
Right?
• 37:22 - 37:23
OK.
• 37:23 - 37:30
Now going back to-- this is
reminding me of something in 3d
• 37:30 - 37:35
that I wanted to talk
• 37:35 - 37:37
This is [INAUDIBLE].
• 37:37 - 37:43
• 37:43 - 37:45
I'm going to give
you, in a similar way
• 37:45 - 37:48
with this simple
problem, I'm going
• 37:48 - 37:50
to give you something
more complicated
• 37:50 - 38:17
and say find the
parametrization of a helix.
• 38:17 - 38:20
And you say, well,
I'm happy that this
• 38:20 - 38:22
• 38:22 - 38:24
I have to be a little
bit more careful
• 38:24 - 38:27
so that they make sense.
• 38:27 - 38:44
Of a helix R of t such that
it is contained or it lies,
• 38:44 - 39:00
it lies on the circular
cylinder x squared
• 39:00 - 39:04
plus y squared equals 4.
• 39:04 - 39:05
Why is that a cylinder?
• 39:05 - 39:08
The z's missing, so it's
going to be a cylinder whose
• 39:08 - 39:09
main axis is the z axis.
• 39:09 - 39:10
Right?
• 39:10 - 39:11
Are you guys with me?
• 39:11 - 39:15
I think we are on the same page.
• 39:15 - 39:19
And you cannot solve the
problem just with this data.
• 39:19 - 39:22
Do you agree with me?
• 39:22 - 39:47
And knowing that, the
curvature of the helix is k
• 39:47 - 40:04
equals 2/5 at every point.
• 40:04 - 40:06
And of course it's an oxymoron.
• 40:06 - 40:08
Because what I
proved last time is
• 40:08 - 40:13
that the curvature of
a helix is a constant.
• 40:13 - 40:27
So remember, we got the
curvature of a helix
• 40:27 - 40:30
as being a constant.
• 40:30 - 40:34
• 40:34 - 40:36
STUDENT: What's that last
word of the sentence?
• 40:36 - 40:39
It's "the curvature
is at every" what?
• 40:39 - 40:40
PROFESSOR TODA: At every point.
• 40:40 - 40:45
I'm sorry I said, it very--
I abbreviated [INAUDIBLE].
• 40:45 - 40:48
So at every point you
have the same curvature.
• 40:48 - 40:51
When you draw a
helix you say, wait,
• 40:51 - 40:54
the helix is bent uniformly.
• 40:54 - 40:59
If you were to play with a
spring taken from am old bed,
• 40:59 - 41:02
you would go with your
hands along the spring.
• 41:02 - 41:05
And then you say, oh,
• 41:05 - 41:06
Yes, it does.
• 41:06 - 41:09
And that means the
curvature is the same.
• 41:09 - 41:12
How would you
solve this problem?
• 41:12 - 41:16
This problem is hard,
because you cannot integrate
• 41:16 - 41:17
the curvature.
• 41:17 - 41:19
Well, what is the curvature?
• 41:19 - 41:21
The curvature would be--
• 41:21 - 41:22
STUDENT: Absolute value.
• 41:22 - 41:24
PROFESSOR TODA: Just
absolute value of R
• 41:24 - 41:28
double prime if it were in s.
• 41:28 - 41:31
And you cannot integrate.
• 41:31 - 41:34
If somebody gave you
the vector equation
• 41:34 - 41:37
of double prime of
this, them you say,
• 41:37 - 41:39
yes, I can integrate
one step going back.
• 41:39 - 41:40
I get R prime of s.
• 41:40 - 41:42
Then I go back to R of s.
• 41:42 - 41:43
But this is a little
bit complicated.
• 41:43 - 41:45
I'm giving you a scalar.
• 41:45 - 41:51
You have to be a little bit
aware of what you did last time
• 41:51 - 41:55
and try to remember
what we did last time.
• 41:55 - 41:56
What did we do last time?
• 41:56 - 41:58
I would not give you
a problem like that
• 41:58 - 42:03
on the final, because it would
assume that you have solved
• 42:03 - 42:06
the problem we did last
time in terms of R of t
• 42:06 - 42:10
equals A equals sine t.
• 42:10 - 42:11
A sine t and [? vt. ?]
• 42:11 - 42:16
And we said, this is the
standard parametrized helix
• 42:16 - 42:22
that sits on a cylinder of
radius A and has the phb.
• 42:22 - 42:28
So the distance between
consecutive spirals
• 42:28 - 42:29
really matters.
• 42:29 - 42:30
That really makes
the difference.
• 42:30 - 42:31
STUDENT: I have a question.
• 42:31 - 42:33
PROFESSOR TODA: You wanted
• 42:33 - 42:34
STUDENT: Is s always
the reciprocal of t?
• 42:34 - 42:36
Are they always--
• 42:36 - 42:37
PROFESSOR TODA:
No, not reciprocal.
• 42:37 - 42:46
You mean s of t is a function
is from t0 to t of the speed.
• 42:46 - 42:50
R prime and t-- d tau, right?
• 42:50 - 42:52
Tau not t. [INAUDIBLE].
• 42:52 - 42:54
• 42:54 - 43:01
t and s are
different parameters.
• 43:01 - 43:02
Different times.
• 43:02 - 43:04
Different parameter times.
• 43:04 - 43:05
And you say--
• 43:05 - 43:07
STUDENT: Isn't s
the parameter time
• 43:07 - 43:09
when [INAUDIBLE] parametrized?
• 43:09 - 43:10
PROFESSOR TODA: Very good.
• 43:10 - 43:12
So what is the magic s?
• 43:12 - 43:14
I'm proud of you.
• 43:14 - 43:16
This is the important
thing to remember.
• 43:16 - 43:18
t could be any time.
• 43:18 - 43:20
I start measuring
wherever I want.
• 43:20 - 43:24
I can set my watch to start now.
• 43:24 - 43:25
It could be crazy.
• 43:25 - 43:27
Doesn't have to be uniform.
• 43:27 - 43:28
Motion, I don't care.
• 43:28 - 43:31
• 43:31 - 43:33
s is a friend of
yours that says,
• 43:33 - 43:38
I am that special time
so that according to me
• 43:38 - 43:41
the speed will become one.
• 43:41 - 43:46
So for a physicist to measure
the speed with respect to this,
• 43:46 - 43:49
parameter s time, the speed
will always become one.
• 43:49 - 43:52
That is the arclength
time and position.
• 43:52 - 43:54
How you get from one
another, I told you last time
• 43:54 - 43:57
that for both of them
you have-- this is R of t
• 43:57 - 43:59
and this is little r of s.
• 43:59 - 44:01
And there is a composition.
• 44:01 - 44:03
s can be viewed as
a function of t,
• 44:03 - 44:06
and t can be viewed
as a function of s.
• 44:06 - 44:10
As functions they are
inverse to one another.
• 44:10 - 44:13
So going back to who they
are, a very good question,
• 44:13 - 44:16
because this is a review
anyway, [? who wants ?]
• 44:16 - 44:19
s as a function of t for
this particular problem?
• 44:19 - 44:24
I hope you remember, we were
like-- have you seen this movie
• 44:24 - 44:28
with Mickey Mouse going
on a mountain that
• 44:28 - 44:32
was more like a cylinder.
• 44:32 - 44:35
And this is the train
going at a constant slope.
• 44:35 - 44:43
And one of my colleagues,
actually, he's at Stanford,
• 44:43 - 44:47
was telling me that he
gave his students in Calc 1
• 44:47 - 44:52
to prove, formally prove,
that the angle formed
• 44:52 - 44:57
by the law of motion
by the velocity vector,
• 44:57 - 45:02
with the horizontal plane
passing through the particle,
• 45:02 - 45:04
is always a constant.
• 45:04 - 45:07
in now, but of course we can.
• 45:07 - 45:09
We could do that.
• 45:09 - 45:11
So maybe the next
thing would be, like,
• 45:11 - 45:13
if you [INAUDIBLE]
an extra problem, can
• 45:13 - 45:17
we show that angle between the
velocity vector on the helix
• 45:17 - 45:21
and the horizontal plane through
that point is a constant.
• 45:21 - 45:23
STUDENT: Wouldn't it
just be, because B of t
• 45:23 - 45:24
is just a constant times t?
• 45:24 - 45:25
PROFESSOR TODA: Yeah.
• 45:25 - 45:26
We'll get to that.
• 45:26 - 45:27
We'll get to that in a second.
• 45:27 - 45:32
So he reminded me of an old
movie from like 70 years ago,
• 45:32 - 45:34
with Mickey Mouse and the train.
• 45:34 - 45:39
And the train going
up at the same speed.
• 45:39 - 45:41
You have to maintain
the same speed.
• 45:41 - 45:45
Because if you risk it
not, then you sort of
• 45:45 - 45:46
are getting trouble.
• 45:46 - 45:48
So you never stop.
• 45:48 - 45:49
If you stop you go back.
• 45:49 - 45:51
So it's a regular curve.
• 45:51 - 45:53
What I have here is
that such a curve.
• 45:53 - 45:55
Regular curve, never stop.
• 45:55 - 45:57
Get up with a constant speed.
• 45:57 - 45:59
Do you guys remember the
speed from last time?
• 45:59 - 46:01
We'll square root the a
squared plus b squared.
• 46:01 - 46:04
When we did the
velocity thingie.
• 46:04 - 46:11
And I get square root a
squared plus b squared times t.
• 46:11 - 46:19
Now, today I would like
• 46:19 - 46:22
What if-- Ryan brought this up.
• 46:22 - 46:22
It's very good.
• 46:22 - 46:24
b is a constant.
• 46:24 - 46:27
What if b would
not be a constant,
• 46:27 - 46:29
or maybe could be worse?
• 46:29 - 46:33
another linear function with t,
• 46:33 - 46:36
but something that contains
higher powers of t.
• 46:36 - 46:39
• 46:39 - 46:43
Then you don't go at the
constant speed anymore.
• 46:43 - 46:45
You can say goodbye
to the cartoon.
• 46:45 - 46:46
Yes, sir?
• 46:46 - 46:49
STUDENT: And then
it's [INAUDIBLE].
• 46:49 - 46:50
One that goes [INAUDIBLE].
• 46:50 - 46:52
PROFESSOR TODA: I
mean, it's still--
• 46:52 - 46:55
STUDENT: s is not
multiplied by a constant.
• 46:55 - 46:57
The function between t and
s is not a constant one.
• 46:57 - 47:00
PROFESSOR TODA: It's going to
be a different parameterization,
• 47:00 - 47:01
different speed.
• 47:01 - 47:04
Sometimes-- OK, you
have to understand.
• 47:04 - 47:07
Let's say I have a cone.
• 47:07 - 47:10
And I'm going slow
at first, and I
• 47:10 - 47:12
go faster and faster
and faster and faster
• 47:12 - 47:14
to the end of the cone.
• 47:14 - 47:18
But then I have the
same physical curve,
• 47:18 - 47:21
and I parameterized
[INAUDIBLE] the length.
• 47:21 - 47:24
And I say, no, I'm a mechanic.
• 47:24 - 47:27
Or I'm the engineer
of the strain.
• 47:27 - 47:29
I can make the motion
have a constant speed.
• 47:29 - 47:33
So even if the helix
is no longer circular,
• 47:33 - 47:37
and it's sort of a crazy helix
going on top of the mountain,
• 47:37 - 47:39
as an engineer I
can just say, oh no,
• 47:39 - 47:42
I want cruise control
for my little train.
• 47:42 - 47:46
And I will go at the same speed.
• 47:46 - 47:49
See, the problem is
the slope a constant.
• 47:49 - 47:51
And thinking of
what they did that
• 47:51 - 47:53
stand for, because
it didn't stand
• 47:53 - 47:55
for [INAUDIBLE] in honors.
• 47:55 - 47:57
We can do it in honors as well.
• 47:57 - 47:59
We'll do it in a second.
• 47:59 - 48:05
Now, k obviously is what?
• 48:05 - 48:08
Some of you have
very good memory,
• 48:08 - 48:13
and like the memory of a
medical doctor, which is great.
• 48:13 - 48:15
Some of you don't.
• 48:15 - 48:19
But if you don't you just go
back and look at the notes.
• 48:19 - 48:21
What I'm trying to
do, but I don't know,
• 48:21 - 48:23
it's also a matter
of money-- I don't
• 48:23 - 48:26
want to use the math
department copier-- I'd
• 48:26 - 48:30
like to make a stack of notes.
• 48:30 - 48:33
So that's why I'm collecting
these notes, to bring them back
• 48:33 - 48:34
to you.
• 48:34 - 48:34
• 48:34 - 48:36
I'm not going to
sell them to you.
• 48:36 - 48:38
I'm [INAUDIBLE].
• 48:38 - 48:42
So that you can have those
with you whenever you want,
• 48:42 - 48:45
or put them in a spiral,
punch holes in them,
• 48:45 - 48:49
and have them for
review at any time.
• 48:49 - 48:51
Reminds me of what that
was-- that was in the notes.
• 48:51 - 48:55
a over a squared plus b squared.
• 48:55 - 48:57
So who can tell me, a
and b really quickly,
• 48:57 - 49:01
so we don't waste too
much time, Mr. a is--?
• 49:01 - 49:06
• 49:06 - 49:07
STUDENT: So this is another way
• 49:07 - 49:08
STUDENT: 2.
• 49:08 - 49:08
PROFESSOR TODA: 2.
• 49:08 - 49:13
STUDENT: So is this another
way of defining k in k of s?
• 49:13 - 49:14
PROFESSOR TODA: Actually--
• 49:14 - 49:17
STUDENT: That's the general
curvature for [INAUDIBLE].
• 49:17 - 49:21
PROFESSOR TODA: You know
what is the magic thing?
• 49:21 - 49:23
Even if-- the curvature
is an invariant.
• 49:23 - 49:27
It doesn't depend the
reparametrization.
• 49:27 - 49:30
There is a way maybe I'm going
to teach you, although this
• 49:30 - 49:32
is not in the book.
• 49:32 - 49:36
What are the formulas
corresponding
• 49:36 - 49:42
to the [INAUDIBLE] t and v that
depend on curvature and torsion
• 49:42 - 49:44
and the speed along the curve.
• 49:44 - 49:49
And if you analyze the notion
of curvature, [INAUDIBLE],
• 49:49 - 49:52
no matter what your
parameter will be, t, s, tau,
• 49:52 - 49:57
God knows what, k will
still be the same number.
• 49:57 - 49:59
So k is viewed as an
invariant with respect
• 49:59 - 50:01
to the parametrization.
• 50:01 - 50:04
STUDENT: So then that a over
a squared plus b squared,
• 50:04 - 50:06
that's another way of finding k?
• 50:06 - 50:07
PROFESSOR TODA: Say it again?
• 50:07 - 50:09
STUDENT: So using a over
a squared plus b squared
• 50:09 - 50:11
is another way of finding k?
• 50:11 - 50:12
PROFESSOR TODA: No.
• 50:12 - 50:14
Somebody gave you k.
• 50:14 - 50:17
And then you say, if it's
a standard parametrization,
• 50:17 - 50:25
and then I get 2/5,
can I be sure a is 2?
• 50:25 - 50:28
I'm sure a is 2 from nothing.
• 50:28 - 50:33
This is what makes me aware
that a is 2 the first place.
• 50:33 - 50:37
of the cylinder.
• 50:37 - 50:39
This is x squared, x and y.
• 50:39 - 50:42
You see, x squared plus
y squared is a squared.
• 50:42 - 50:44
This is where I get a from.
• 50:44 - 50:45
a is 2.
• 50:45 - 50:47
I replace it in here
and I say, all righty,
• 50:47 - 50:52
so I only have one
choice. a is 2 and b is?
• 50:52 - 50:53
STUDENT: [INAUDIBLE]
• 50:53 - 50:57
• 50:57 - 51:00
PROFESSOR TODA: But can b
plus-- So what I'm saying,
• 51:00 - 51:01
a is 2, right?
• 51:01 - 51:04
We know that from this.
• 51:04 - 51:08
If I block in here I have 4
and somebody says plus minus 1.
• 51:08 - 51:10
No.
• 51:10 - 51:11
b is always positive.
• 51:11 - 51:13
So you remember the
last time we discussed
• 51:13 - 51:17
parametrization.
• 51:17 - 51:20
But somebody will say,
but what if I put a minus?
• 51:20 - 51:23
What if I'm going
to put a minus?
• 51:23 - 51:24
That's an excellent question.
• 51:24 - 51:27
What's going to happen
if you put minus t?
• 51:27 - 51:28
[INTERPOSING VOICES]
• 51:28 - 51:29
PROFESSOR TODA: Exactly.
• 51:29 - 51:31
In the opposite direction.
• 51:31 - 51:36
up, you go down.
• 51:36 - 51:37
All right.
• 51:37 - 51:41
Now, I'm gonna-- what else?
• 51:41 - 51:43
Ah, I said, let's do this.
• 51:43 - 51:48
Let's prove that the
angle is a constant,
• 51:48 - 51:51
the angle that's
• 51:51 - 51:56
vector of the train with the
horizontal plane is a constant.
• 51:56 - 51:58
Is this hard?
• 51:58 - 51:58
Nah.
• 51:58 - 51:59
Yes, sir?
• 51:59 - 52:04
STUDENT: Are we still going
to find R of t given only k?
• 52:04 - 52:06
PROFESSOR TODA: But didn't we?
• 52:06 - 52:07
We did.
• 52:07 - 52:14
R of t was 2 cosine
t, 2 sine t, and t.
• 52:14 - 52:16
All right?
• 52:16 - 52:17
OK, so we are done.
• 52:17 - 52:19
What did I say?
• 52:19 - 52:22
I said that let's
prove-- it's a proof.
• 52:22 - 52:27
Let's prove that the angle made
by the velocity to the train--
• 52:27 - 52:31
to the train?-- to the direction
of motion, which is the helix.
• 52:31 - 52:37
And the horizontal
plane is a constant.
• 52:37 - 52:38
Is this hard?
• 52:38 - 52:40
How are we going to do that?
• 52:40 - 52:43
Now I start waking up,
because I was very tired.
• 52:43 - 52:44
STUDENT: [INAUDIBLE]
• 52:44 - 52:45
PROFESSOR TODA: Excuse me.
• 52:45 - 52:47
STUDENT: [INAUDIBLE]
• 52:47 - 53:01
PROFESSOR TODA: So you see,
the helix contains this point.
• 53:01 - 53:04
And I'm looking at
the velocity vector
• 53:04 - 53:06
that is standard to the helix.
• 53:06 - 53:09
And I'll call that R prime.
• 53:09 - 53:11
And then you say,
yea, but how am I
• 53:11 - 53:14
going to compute that angle?
• 53:14 - 53:16
What is that angle?
• 53:16 - 53:18
STUDENT: It's a function of b.
• 53:18 - 53:21
• 53:21 - 53:22
PROFESSOR TODA: It will be.
• 53:22 - 53:25
But we have to do it rigorously.
• 53:25 - 53:28
So what's going to happen
for me to draw that angle?
• 53:28 - 53:30
First of all, I should
take-- from the tip
• 53:30 - 53:33
of the vector I should
draw perpendicular
• 53:33 - 53:36
to the horizontal plane
passing through the point.
• 53:36 - 53:37
And I'll get P prime.
• 53:37 - 53:38
God knows why.
• 53:38 - 53:41
I don't know why, I don't know
why. [? Q. ?] And this is PR,
• 53:41 - 53:43
and P-- not PR.
• 53:43 - 53:47
PR is too much
• 53:47 - 53:51
OK, so then you would
take PQ and then
• 53:51 - 53:53
you would measure this angle.
• 53:53 - 53:55
Well, you have to be a
little bit smarter than that,
• 53:55 - 53:58
because you can
prove something else.
• 53:58 - 54:03
This is the complement of
another angle that you love.
• 54:03 - 54:07
And using chapter 9 you can
do that angle in no time.
• 54:07 - 54:16
• 54:16 - 54:21
So this is the
complement of the angle
• 54:21 - 54:24
formed by the velocity vector
of prime with the normal.
• 54:24 - 54:27
• 54:27 - 54:30
But not the normal principle
normal to the curve,
• 54:30 - 54:32
but the normal to the plane.
• 54:32 - 54:35
And what is the
normal to the plane?
• 54:35 - 54:39
Let's call the principal normal
n to the curve big N bar.
• 54:39 - 54:42
So in order to avoid confusion,
I'll write this little n.
• 54:42 - 54:43
• 54:43 - 54:45
Do you guys know-- like
they do in mechanics.
• 54:45 - 54:48
If you have two normals,
they call that 1n.
• 54:48 - 54:51
1 is little n, and
stuff like that.
• 54:51 - 54:53
So this is the complement.
• 54:53 - 54:55
If I were able to prove
that that complement
• 54:55 - 55:00
is a constant-- this is the
Stanford [? property-- ?] then
• 55:00 - 55:01
I will be happy.
• 55:01 - 55:03
Is it hard?
• 55:03 - 55:04
No, for god's sake.
• 55:04 - 55:07
Who is little n?
• 55:07 - 55:11
Little n would be-- is
that the normal to a plane
• 55:11 - 55:12
that you love?
• 55:12 - 55:13
• 55:13 - 55:14
STUDENT: xy plane.
• 55:14 - 55:17
PROFESSOR TODA: Your
plane is horizontal plane.
• 55:17 - 55:17
STUDENT: xy.
• 55:17 - 55:19
PROFESSOR TODA: Yes, xy plane.
• 55:19 - 55:22
Or xy plane shifted,
shifted, shifted, shifted.
• 55:22 - 55:23
That's the normal change?
• 55:23 - 55:24
No.
• 55:24 - 55:25
Who is the normal?
• 55:25 - 55:26
STUDENT: [INAUDIBLE]
• 55:26 - 55:27
PROFESSOR TODA: [INAUDIBLE].
• 55:27 - 55:28
STUDENT: 0, 0, 1.
• 55:28 - 55:29
PROFESSOR TODA: 0, 0, 1.
• 55:29 - 55:30
OK.
• 55:30 - 55:32
When I put 0 I was [INAUDIBLE].
• 55:32 - 55:34
So this is k.
• 55:34 - 55:36
• 55:36 - 55:38
All right.
• 55:38 - 55:40
And what is our prime?
• 55:40 - 55:42
I was-- that was
a piece of cake.
• 55:42 - 55:47
We did it last time minus a
sine t, a equals sine t and b.
• 55:47 - 55:51
• 55:51 - 55:54
Let's find that angle.
• 55:54 - 55:55
Well, I don't know.
• 55:55 - 55:58
You have to teach me, because
you have chapter 9 fresher
• 55:58 - 56:02
in your memory than I have it.
• 56:02 - 56:04
Are you taking attendance also?
• 56:04 - 56:07
Are you writing your name down?
• 56:07 - 56:08
Oh, no problem whatsoever.
• 56:08 - 56:09
STUDENT: We didn't get it.
• 56:09 - 56:11
PROFESSOR TODA:
You didn't get it.
• 56:11 - 56:12
Circulate it.
• 56:12 - 56:17
All right, so who is going
to help me with the angle?
• 56:17 - 56:20
What is the angle between
two vectors, guys?
• 56:20 - 56:24
That should be review from what
we just covered in chapter 9.
• 56:24 - 56:28
Let me call them
u and v. And who's
• 56:28 - 56:30
going to tell me how
I get that angle?
• 56:30 - 56:32
STUDENT: [INAUDIBLE] is equal
to the inverse cosine of the dot
• 56:32 - 56:33
product of [? the magnitude. ?]
• 56:33 - 56:35
PROFESSOR TODA: Do you
like me to write arc
• 56:35 - 56:36
cosine or cosine [INAUDIBLE].
• 56:36 - 56:38
Doesn't matter.
• 56:38 - 56:40
Arc cosine of--
• 56:40 - 56:41
STUDENT: The dot products.
• 56:41 - 56:47
PROFESSOR TODA: The dot
product between u and v.
• 56:47 - 56:49
STUDENT: Over magnitude.
• 56:49 - 56:52
PROFESSOR TODA: Divided by the
product of their magnitudes.
• 56:52 - 56:55
Look, I will change the
order, because you're not
• 56:55 - 56:56
going to like it.
• 56:56 - 56:57
Doesn't matter.
• 56:57 - 56:58
OK?
• 56:58 - 57:03
So the angle phi between
my favorite vectors
• 57:03 - 57:08
here is going to be
simply the dot product.
• 57:08 - 57:10
That's a blessing.
• 57:10 - 57:10
It's a constant.
• 57:10 - 57:12
STUDENT: So you're
doing the dot product
• 57:12 - 57:13
between the normal [INAUDIBLE]?
• 57:13 - 57:15
PROFESSOR TODA:
Between this and that.
• 57:15 - 57:18
So this is u and this
is v. So the dot product
• 57:18 - 57:22
would be 0 plus v.
So the dot product
• 57:22 - 57:29
is arc cosine of v, which,
thank god, is a constant.
• 57:29 - 57:30
I don't have to do
anything anymore.
• 57:30 - 57:33
I'm done with the proof
bit, because arc cosine
• 57:33 - 57:36
of a constant will
be a constant.
• 57:36 - 57:37
OK?
• 57:37 - 57:38
All right.
• 57:38 - 57:41
So I have v over what?
• 57:41 - 57:45
What is the length
of this vector?
• 57:45 - 57:47
1. [INAUDIBLE].
• 57:47 - 57:51
What's the length
of that vector?
• 57:51 - 57:56
Square root of a
squared plus b squared.
• 57:56 - 57:56
All right?
• 57:56 - 58:02
• 58:02 - 58:05
STUDENT: How did
you [INAUDIBLE].
• 58:05 - 58:08
PROFESSOR TODA: So now
let me ask you one thing.
• 58:08 - 58:11
• 58:11 - 58:14
What kind of function
is arc cosine?
• 58:14 - 58:16
Of course I said arc cosine
of a constant is a constant.
• 58:16 - 58:18
What kind of a
function is arc cosine?
• 58:18 - 58:22
I'm doing review with you
because I think it's useful.
• 58:22 - 58:26
Arc cosine is defined on
what with values in what?
• 58:26 - 58:30
• 58:30 - 58:33
STUDENT: Repeat the question?
• 58:33 - 58:34
PROFESSOR TODA: Arc cosine.
• 58:34 - 58:36
Or cosine inverse,
like Ryan prefers.
• 58:36 - 58:38
Cosine inverse is
the same thing.
• 58:38 - 58:40
It's a function defined
by where to where?
• 58:40 - 58:43
Cosine is defined
from where to where?
• 58:43 - 58:46
From R to minus 1.
• 58:46 - 58:48
It's a cosine of t.
• 58:48 - 58:50
t could be any real number.
• 58:50 - 58:52
The range is minus 1, 1.
• 58:52 - 58:53
Close the interval.
• 58:53 - 58:55
STUDENT: So it's-- so
I just wonder why--
• 58:55 - 58:57
PROFESSOR TODA: Minus
1 to 1, close interval.
• 58:57 - 58:58
• 58:58 - 59:03
Because it cannot go back to R.
It has to be a 1 to 1 function.
• 59:03 - 59:06
You cannot have an inverse
function if you don't take
• 59:06 - 59:09
a restriction of a
function to be 1 to 1.
• 59:09 - 59:12
And we took that
restriction of a function.
• 59:12 - 59:15
And do you remember what it was?
• 59:15 - 59:16
[INTERPOSING VOICES]
• 59:16 - 59:18
PROFESSOR TODA: 0 to pi.
• 59:18 - 59:20
Now, on this one
I'm really happy.
• 59:20 - 59:23
several people-- people
• 59:23 - 59:27
come to my office to get
all sorts of transcripts,
• 59:27 - 59:28
[INAUDIBLE].
• 59:28 - 59:31
And in trigonometry
• 59:31 - 59:32
so you took trigonometry.
• 59:32 - 59:33
So do you remember that?
• 59:33 - 59:34
He didn't remember that.
• 59:34 - 59:35
• 59:35 - 59:40
the sine inverse?
• 59:40 - 59:45
How was my restriction so that
would be a 1 to 1 function?
• 59:45 - 59:47
It's got to go
from minus 1 to 1.
• 59:47 - 59:48
What is the range?
• 59:48 - 59:49
[INTERPOSING VOICES]
• 59:49 - 59:51
PROFESSOR TODA: Minus pi over 2.
• 59:51 - 59:53
• 59:53 - 59:54
Good.
• 59:54 - 59:56
That's a very good thing.
• 59:56 - 59:59
You were in high school
when you learned that?
• 59:59 - 60:00
Here at Lubbock High?
• 60:00 - 60:01
STUDENT: Yes.
• 60:01 - 60:02
PROFESSOR TODA: Great.
• 60:02 - 60:04
Good job, Lubbock High.
• 60:04 - 60:06
But many students, I caught
them, who wanted credit
• 60:06 - 60:08
for trig who didn't know that.
• 60:08 - 60:10
Good.
• 60:10 - 60:20
So since arc cosine is a
function that is of 0, pi,
• 60:20 - 60:25
for example, what if my--
let me give you an example.
• 60:25 - 60:27
What was last time, guys?
• 60:27 - 60:31
a was 1. b was 1.
• 60:31 - 60:32
For one example.
• 60:32 - 60:34
In that case, 1 with 5b.
• 60:34 - 60:36
for the example you just did?
• 60:36 - 60:38
PROFESSOR TODA: No last time.
• 60:38 - 60:40
STUDENT: A was 3 and b was--
• 60:40 - 60:44
PROFESSOR TODA: So what would
that be, in this case 5?
• 60:44 - 60:47
STUDENT: That would be
b over the square root--
• 60:47 - 60:48
STUDENT: 3 over pi.
• 60:48 - 60:50
• 60:50 - 60:52
PROFESSOR TODA: a is 1 and b
is 1, like we did last time.
• 60:52 - 60:55
STUDENT: [INAUDIBLE]
2, which is--
• 60:55 - 60:57
PROFESSOR TODA: Plug
in 1 is a, b is 1.
• 60:57 - 60:58
What is this?
• 60:58 - 60:59
STUDENT: It's just pi over 4.
• 60:59 - 61:00
PROFESSOR TODA: Pi over 4.
• 61:00 - 61:07
So pi will be our cosine, of
1 over square root 2, which
• 61:07 - 61:12
is 45 degree angle, which is--
you said pi over 4, right?
• 61:12 - 61:15
[INAUDIBLE].
• 61:15 - 61:20
So exactly, you would
have that over here.
• 61:20 - 61:23
This is where the
cosine [INAUDIBLE].
• 61:23 - 61:28
Now you see, guys, the way we
have, the way I assume a and b,
• 61:28 - 61:31
the way anybody-- the
book also introduces
• 61:31 - 61:33
a and b to be positive numbers.
• 61:33 - 61:37
Can you tell me what kind
of angle phi will be,
• 61:37 - 61:40
not only restricted to 0 pi?
• 61:40 - 61:41
Well, a is positive.
• 61:41 - 61:42
b is positive.
• 61:42 - 61:44
a doesn't matter.
• 61:44 - 61:47
The whole thing
will be positive.
• 61:47 - 61:51
Arc cosine of a
positive number--
• 61:51 - 61:52
STUDENT: Between
0 and pi over 2.
• 61:52 - 61:53
PROFESSOR TODA: That is.
• 61:53 - 61:56
Yeah, so it has to be
between 0 and pi over 2.
• 61:56 - 61:58
So it's going to be
• 61:58 - 62:00
Does that make sense?
• 62:00 - 62:03
Yes, think with the
• 62:03 - 62:05
or the eyes of your imagination.
• 62:05 - 62:06
OK.
• 62:06 - 62:08
You have a cylinder.
• 62:08 - 62:10
And you are moving
along that cylinder.
• 62:10 - 62:12
And this is how you turn.
• 62:12 - 62:14
You turn with that little train.
• 62:14 - 62:17
Du-du-du-du-du, you go up.
• 62:17 - 62:20
When you turn the
velocity vector and you
• 62:20 - 62:23
look at the-- mm.
• 62:23 - 62:24
STUDENT: The normal.
• 62:24 - 62:25
PROFESSOR TODA: The normal!
• 62:25 - 62:25
Thank you.
• 62:25 - 62:30
The z axis, you always have an
angle between 0 and pi over 2.
• 62:30 - 62:32
So it makes sense.
• 62:32 - 62:34
I'm going to go ahead and
erase the whole thing.
• 62:34 - 62:41
• 62:41 - 62:48
So we reviewed, more or less, s
of t, integration, derivation,
• 62:48 - 62:52
moving from position vector
to velocity to acceleration
• 62:52 - 62:56
and back, acceleration to
velocity to position vector,
• 62:56 - 62:58
the meaning of arclength.
• 62:58 - 63:01
There are some things I
would like to tell you,
• 63:01 - 63:08
because Ryan asked me a few more
• 63:08 - 63:12
The curvature
formula depends very
• 63:12 - 63:17
much on the type of formula
you used for the curve.
• 63:17 - 63:19
So you say, wait,
wait, wait, Magdelena,
• 63:19 - 63:21
you told us-- you
are confusing us.
• 63:21 - 63:24
You told us that the
curvature is uniquely
• 63:24 - 63:34
defined as the magnitude
of the acceleration vector
• 63:34 - 63:37
when the law of motion
is an arclength.
• 63:37 - 63:39
And that is correct.
• 63:39 - 63:43
So suppose my original law of
motion was R of t [INAUDIBLE]
• 63:43 - 63:48
time, any time, t,
any time parameter.
• 63:48 - 63:49
I'm making a face.
• 63:49 - 63:53
But then from that we switch
to something beautiful,
• 63:53 - 63:56
which is called the
arclength parametrization.
• 63:56 - 63:58
Why am I so happy?
• 63:58 - 64:05
Because in this parametrization
the magnitude of the speed
• 64:05 - 64:07
is 1.
• 64:07 - 64:18
And I define k to
be the magnitude
• 64:18 - 64:20
of R double prime of s, right?
• 64:20 - 64:22
The acceleration only in
the arclength [? time ?]
• 64:22 - 64:23
parameterization.
• 64:23 - 64:25
And then this was
the definition.
• 64:25 - 64:30
• 64:30 - 64:37
A. Can you prove-- what?
• 64:37 - 64:40
Can you prove the
following formula?
• 64:40 - 64:52
• 64:52 - 64:59
T prime of s equals
k times N of s.
• 64:59 - 65:03
This is famous for people
who do-- not for everybody.
• 65:03 - 65:06
But imagine you have
an engineer who does
• 65:06 - 65:08
research of the laws of motion.
• 65:08 - 65:13
Maybe he works for
the railways and he's
• 65:13 - 65:17
looking at skew
curves, or he is one
• 65:17 - 65:20
of those people who
project the ski slopes,
• 65:20 - 65:25
or all sorts of winter sports
slope or something, that
• 65:25 - 65:29
involve a lot of
curvatures and torsions.
• 65:29 - 65:31
That guy has to know
the Frenet formula.
• 65:31 - 65:34
So this is the famous
first Frenet formula.
• 65:34 - 65:40
• 65:40 - 65:47
Frenet was a mathematician
who gave the name to the TNB
• 65:47 - 65:48
vectors, the trihedron.
• 65:48 - 65:50
You have the T was what?
• 65:50 - 65:53
The T was the tangent
[INAUDIBLE] vector.
• 65:53 - 65:58
The N was the
principal unit normal.
• 65:58 - 66:01
In those videos that I'm
watching that I also sent you--
• 66:01 - 66:02
I like most of them.
• 66:02 - 66:06
more than everything.
• 66:06 - 66:09
Also I like the one that
was made by Dr. [? Gock ?]
• 66:09 - 66:13
But Dr. [? Gock ?] made a
little bit of a mistake.
• 66:13 - 66:14
A conceptual mistake.
• 66:14 - 66:17
We all make mistakes by
• 66:17 - 66:18
or goofy mistake.
• 66:18 - 66:21
But he said this is
the normal vector.
• 66:21 - 66:23
This is not-- it's the
principle normal vectors.
• 66:23 - 66:25
There are many normals.
• 66:25 - 66:27
There is only one
tangent direction,
• 66:27 - 66:29
but in terms of normals
there are many that
• 66:29 - 66:31
are-- all of these are normals.
• 66:31 - 66:35
All the perpendicular in
the plane-- [INAUDIBLE]
• 66:35 - 66:40
so this is my law of motion,
T. All this plane is normal.
• 66:40 - 66:42
So any of these
vectors is a normal.
• 66:42 - 66:45
The one we choose and
defined as T prime
• 66:45 - 66:47
over T prime [INAUDIBLE]
absolute values
• 66:47 - 66:49
called the principal normal.
• 66:49 - 66:51
It's like the principal
of a high school.
• 66:51 - 66:53
He is important.
• 66:53 - 66:58
So T and B-- B goes
down, or goes-- down.
• 66:58 - 67:05
Well, yeah, because B is T cross
N. So when you find the Frenet
• 67:05 - 67:10
Trihedron, TNB, it's like that.
• 67:10 - 67:16
T, N, and B. What's special,
why do we call it the frame,
• 67:16 - 67:18
is that every
[? payer ?] of vectors
• 67:18 - 67:20
are mutually orthogonal.
• 67:20 - 67:22
And they are all unit vectors.
• 67:22 - 67:26
This is the famous Frenet frame.
• 67:26 - 67:28
Now, Mr. Frenet was a smart guy.
• 67:28 - 67:32
He found-- I don't know whether
• 67:32 - 67:33
or not.
• 67:33 - 67:34
Doesn't matter.
• 67:34 - 67:38
He found a bunch of formulas,
of which this is the first one.
• 67:38 - 67:42
And it's called a
first Frenet formula.
• 67:42 - 67:44
That's one thing
• 67:44 - 67:47
And then I'm going to give you
more formulas for curvatures,
• 67:47 - 67:50
depending on how you
• 67:50 - 68:09
So for example, base B
based on the definition one
• 68:09 - 68:19
can prove that for a curve
that is not parametrizing
• 68:19 - 68:23
arclength-- you say, ugh,
• 68:23 - 68:24
in arclength.
• 68:24 - 68:27
This time you're
assuming, I want to know!
• 68:27 - 68:29
I'm coming to this
• 68:29 - 68:32
I want to know, what is
the formula directly?
• 68:32 - 68:34
Is there a direct
formula that comes
• 68:34 - 68:39
from here for the curvature?
• 68:39 - 68:41
Yeah, but it's a lot
more complicated.
• 68:41 - 68:45
When I was a freshman, maybe
a freshman or a sophomore,
• 68:45 - 68:48
I don't remember, when
• 68:48 - 68:53
that, I did not memorize it.
• 68:53 - 68:57
Then when I started working
as a faculty member,
• 68:57 - 69:02
I saw that I am supposed
to ask it from my students.
• 69:02 - 69:06
So this is going to be
R prime plus product
• 69:06 - 69:12
R double prime in magnitude
over R prime cubed.
• 69:12 - 69:15
So how am I supposed
to remember that?
• 69:15 - 69:16
It's not so easy.
• 69:16 - 69:18
Are you cold there?
• 69:18 - 69:19
It's cold there.
• 69:19 - 69:23
I don't know how
• 69:23 - 69:25
Velocity times acceleration.
• 69:25 - 69:27
This is what I try
to teach myself.
• 69:27 - 69:30
I was old already, 26 or 27.
• 69:30 - 69:33
Velocity times
acceleration, cross product,
• 69:33 - 69:36
take the magnitude,
divide by the speed, cube.
• 69:36 - 69:37
Oh my god.
• 69:37 - 69:41
So I was supposed to know
that when I was 18 or 19.
• 69:41 - 69:45
Now, I was teaching majors
of mechanical engineering.
• 69:45 - 69:46
They knew that by heart.
• 69:46 - 69:48
I didn't, so I had to learn it.
• 69:48 - 69:51
So if one is too
lazy or it's simply
• 69:51 - 69:55
inconvenient to try to
reparametrize from R of T
• 69:55 - 70:01
being arclength parametrization
R of s and do that thing here,
• 70:01 - 70:05
one can just plug in and
find the curvature like that.
• 70:05 - 70:08
For example, guys,
• 70:08 - 70:13
if I have A cosine, [INAUDIBLE],
and I do this with respect
• 70:13 - 70:17
to T, can I get k
without-- k will not
• 70:17 - 70:19
depend on T or s or tau.
• 70:19 - 70:21
It will always be the same.
• 70:21 - 70:23
I will still get A
over A squared plus B
• 70:23 - 70:25
squared, no matter what.
• 70:25 - 70:29
So even if I use this
formula for my helix,
• 70:29 - 70:31
I'm going to get the same thing.
• 70:31 - 70:33
I'll get A over A
squared plus B squared,
• 70:33 - 70:35
because curvature
is an invariant.
• 70:35 - 70:39
There is another invariant
that's-- the other invariant,
• 70:39 - 70:41
of course, in space
is called torsion.
• 70:41 - 70:44
We want to talk a little
• 70:44 - 70:49
So is this hard?
• 70:49 - 70:49
No.
• 70:49 - 70:50
It shouldn't be hard.
• 70:50 - 70:55
And you guys should be able
to help me on that, hopefully.
• 70:55 - 70:57
How do we prove that?
• 70:57 - 70:58
STUDENT: N is G
prime [INAUDIBLE].
• 70:58 - 71:02
• 71:02 - 71:03
PROFESSOR TODA:
That's right, proof.
• 71:03 - 71:06
And that's a very good
start, wouldn't you say?
• 71:06 - 71:09
So what were the definitions?
• 71:09 - 71:14
Let me start from
the definition of T.
• 71:14 - 71:17
That's going to be-- I
am in hard planes, right?
• 71:17 - 71:21
So you say, wait, why do
you write it as a quotient?
• 71:21 - 71:22
You're being silly.
• 71:22 - 71:25
You are in arclength, Magdalena.
• 71:25 - 71:25
I am.
• 71:25 - 71:26
I am.
• 71:26 - 71:30
I just pretend that
I cannot see that.
• 71:30 - 71:32
So if I'm in
arclength, that means
• 71:32 - 71:36
that the denominator is 1.
• 71:36 - 71:37
So I'm being silly.
• 71:37 - 71:44
So R prime of s is
T. Say it again.
• 71:44 - 71:49
R prime of s is T. OK.
• 71:49 - 71:54
Now, did we know that
T and N are orthogonal?
• 71:54 - 72:01
• 72:01 - 72:04
How did we know that T
and N were orthogonal?
• 72:04 - 72:08
We proved that last
time, actually.
• 72:08 - 72:11
T and N are orthogonal.
• 72:11 - 72:13
How do I write
that? [INAUDIBLE].
• 72:13 - 72:16
• 72:16 - 72:22
Meaning that T is
perpendicular to N, right?
• 72:22 - 72:24
From the definition.
• 72:24 - 72:26
You said it right, Sandra.
• 72:26 - 72:28
But why is it from
the definition
• 72:28 - 72:31
conclusions and say, oh,
• 72:31 - 72:36
since I have T prime here, then
this is perpendicular to T?
• 72:36 - 72:37
Well, we did that last time.
• 72:37 - 72:39
STUDENT: Two parallel vectors.
• 72:39 - 72:41
PROFESSOR TODA: We did
it-- how did we do it?
• 72:41 - 72:42
We did this last.
• 72:42 - 72:46
We said T dot T equals 1.
• 72:46 - 72:48
Prime the whole thing.
• 72:48 - 72:54
T prime times T plus T times T
prime, T dot T prime will be 0.
• 72:54 - 72:57
So T and T prime are
perpendicular always.
• 72:57 - 72:58
Right?
• 72:58 - 73:03
OK, so the whole thing is a
colinear vector to T prime.
• 73:03 - 73:05
It's just T prime
times the scalar.
• 73:05 - 73:08
So he must be
perpendicular to T.
• 73:08 - 73:11
So T and N are perpendicular.
• 73:11 - 73:15
So I do have the
direction of motion.
• 73:15 - 73:19
I know that I must
have some scalar here.
• 73:19 - 73:23
• 73:23 - 73:27
How do I prove that this
scalar is the curvature?
• 73:27 - 73:31
• 73:31 - 73:36
So if I have-- if they
are colinear-- why are
• 73:36 - 73:37
they colinear?
• 73:37 - 73:42
T perpendicular to T
prime implies that T prime
• 73:42 - 73:46
is colinear to N. Say it again.
• 73:46 - 73:50
If T and T prime are
perpendicular to one another,
• 73:50 - 73:53
that means T prime is
calling it to the normal.
• 73:53 - 73:58
So here I may have
alph-- no alpha.
• 73:58 - 74:00
I don't know!
• 74:00 - 74:04
Alpha over [INAUDIBLE]
sounds like a curve.
• 74:04 - 74:05
Give me some function.
• 74:05 - 74:09
• 74:09 - 74:10
STUDENT: u of s?
• 74:10 - 74:11
PROFESSOR TODA: Gamma of s.
• 74:11 - 74:15
u of s, I don't know.
• 74:15 - 74:17
So how did I conclude that?
• 74:17 - 74:20
From T perpendicular to T prime.
• 74:20 - 74:22
Now from here on, you
have to tell me why
• 74:22 - 74:29
gamma must be exactly kappa.
• 74:29 - 74:34
Well, let's take
T prime from here.
• 74:34 - 74:38
T prime from here
will give me what?
• 74:38 - 74:41
T prime is our prime prime.
• 74:41 - 74:42
Say what?
• 74:42 - 74:43
Our prime prime.
• 74:43 - 74:45
What is our prime prime?
• 74:45 - 74:47
Our [? problem ?] prime of s.
• 74:47 - 74:49
STUDENT: You have one
too many primes inside.
• 74:49 - 74:50
PROFESSOR TODA: Oh my god.
• 74:50 - 74:50
Yeah.
• 74:50 - 74:53
• 74:53 - 74:54
So R prime prime.
• 74:54 - 74:58
So T prime in
absolute value will
• 74:58 - 75:03
be exactly R double prime of s.
• 75:03 - 75:05
Oh, OK.
• 75:05 - 75:10
Note that from here also T
prime of s in absolute value,
• 75:10 - 75:14
in magnitude, I'm sorry,
has to be gamma of s.
• 75:14 - 75:15
Why is that?
• 75:15 - 75:17
Because the magnitude of N is 1.
• 75:17 - 75:21
N is unique vector
by definition.
• 75:21 - 75:25
So these two guys
have to coincide.
• 75:25 - 75:27
So R double prime,
the best thing
• 75:27 - 75:29
that I need to do,
it must coincide
• 75:29 - 75:30
with the scalar gamma of s.
• 75:30 - 75:33
So who is the
mysterious gamma of s?
• 75:33 - 75:36
He has no chance
but being this guy.
• 75:36 - 75:39
But this guy has a name.
• 75:39 - 75:42
This guy, he's the curvature
[? cap ?] of s by definition.
• 75:42 - 75:46
• 75:46 - 75:49
Remember, Ryan, this
is the definition.
• 75:49 - 75:52
So by definition the
curvature was the magnitude
• 75:52 - 75:55
of the acceleration
in arclength.
• 75:55 - 75:56
OK.
• 75:56 - 75:58
Both of these guys are
T prime in magnitude.
• 75:58 - 76:02
So they must be equal
from here and here.
• 76:02 - 76:05
It implies that my
gamma must be kappa.
• 76:05 - 76:08
And I prove the formula.
• 76:08 - 76:09
OK.
• 76:09 - 76:11
How do you say
something is proved?
• 76:11 - 76:12
Because this is what we wanted.
• 76:12 - 76:16
We wanted to replace this
generic scalar function
• 76:16 - 76:20
to prove that this is
just the curvature.
• 76:20 - 76:21
QED.
• 76:21 - 76:24
• 76:24 - 76:27
That's exactly what
we wanted to prove.
• 76:27 - 76:29
Now, whatever scalar
function you have here,
• 76:29 - 76:30
that must be the curvature.
• 76:30 - 76:34
• 76:34 - 76:36
Very smart guy, this Mr. Frenet.
• 76:36 - 76:40
• 76:40 - 76:41
I'm now going to take a break.
• 76:41 - 76:44
If you want to go use the
bathroom really quickly,
• 76:44 - 76:45
feel free to do it.
• 76:45 - 76:48
• 76:48 - 76:49
I'm just going to
clean the board,
• 76:49 - 76:52
and I'll keep going
in a few minutes.
• 76:52 - 77:51
• 77:51 - 77:51
STUDENT: [INAUDIBLE]
• 77:51 - 77:56
• 77:56 - 77:57
PROFESSOR TODA: I
will do it-- well,
• 77:57 - 78:01
actually I want to do a
different example, simple one,
• 78:01 - 78:06
which is a plain curve, and show
that the curvature has a very
• 78:06 - 78:11
pretty formula that you
could [INAUDIBLE] memorize,
• 78:11 - 78:14
that in essence is the same.
• 78:14 - 78:17
But it depends on
y equals f of x.
• 78:17 - 78:19
[INAUDIBLE] So if
somebody gives you
• 78:19 - 78:22
a plane called y
equals f of x, can you
• 78:22 - 78:25
write that curvature
[INAUDIBLE] function of f?
• 78:25 - 78:27
And you can.
• 78:27 - 78:31
And again, I was deep in
that when I was 18 or 19
• 78:31 - 78:32
as a freshman.
• 78:32 - 78:36
But unfortunately for me I
didn't learn it at that time.
• 78:36 - 78:41
And several years later when
I started teaching engineers,
• 78:41 - 78:44
well, they are
mostly mechanical.
• 78:44 - 78:47
And mechanical
engineering [INAUDIBLE].
• 78:47 - 78:50
They knew those, and they needed
those in every research paper.
• 78:50 - 78:54
So I had to learn it
together with them.
• 78:54 - 78:58
• 78:58 - 79:01
STUDENT: Can you do a really
ugly one, like [INAUDIBLE]?
• 79:01 - 79:05
PROFESSOR TODA: I can
do some ugly ones.
• 79:05 - 80:37
• 80:37 - 80:48
And once you know the
general parametrization,
• 80:48 - 80:52
it will give you a curvature.
• 80:52 - 80:53
• 80:53 - 80:55
Let's see what you remember.
• 80:55 - 81:00
Um-- don't look at the notes.
• 81:00 - 81:03
A positive function,
absolute-- actually,
• 81:03 - 81:07
magnitude of what vector?
• 81:07 - 81:08
STUDENT: R prime.
• 81:08 - 81:17
PROFESSOR TODA: R prime velocity
plus acceleration speed cubed.
• 81:17 - 81:19
Right?
• 81:19 - 81:19
OK.
• 81:19 - 81:24
of what we just learned
• 81:24 - 81:30
and find-- you find
with me, of course, not
• 81:30 - 81:34
as professor and student,
but like a group of students
• 81:34 - 81:35
together.
• 81:35 - 81:46
Let's find a simple
formula corresponding
• 81:46 - 81:52
to the curvature
of a plane curve.
• 81:52 - 82:00
• 82:00 - 82:05
And the plane curve
could be [INAUDIBLE]
• 82:05 - 82:09
in two different ways,
just because I want
• 82:09 - 82:15
you to practice more on that.
• 82:15 - 82:18
Either given as a general
parametrization-- guys,
• 82:18 - 82:20
what is the general
parametrization
• 82:20 - 82:25
for a plane curve?
• 82:25 - 82:26
x of t, y of t, right?
• 82:26 - 82:29
x equals x of t.
• 82:29 - 82:30
y equals y of t.
• 82:30 - 82:34
So one should not have
to do that all the time,
• 82:34 - 82:37
not have to do that for a
simplification like a playing
• 82:37 - 82:38
card.
• 82:38 - 82:42
We have to find another
formula that's pretty, right?
• 82:42 - 82:43
Well, maybe it's not as pretty.
• 82:43 - 82:45
But when is it really pretty?
• 82:45 - 82:49
I bet it's going to be really
pretty if you have a plane
• 82:49 - 82:55
curve even as you're used
to in an explicit form--
• 82:55 - 82:57
I keep going.
• 82:57 - 83:00
No stop. [INAUDIBLE].
• 83:00 - 83:01
I think it's better.
• 83:01 - 83:03
We make better use
of time this way.
• 83:03 - 83:07
Or y equals f of x.
• 83:07 - 83:13
• 83:13 - 83:18
This is an explicit way to
write the equation of a curve.
• 83:18 - 83:21
• 83:21 - 83:23
OK, so what do we need to do?
• 83:23 - 83:26
That should be really easy.
• 83:26 - 83:33
R of t being the first case of
our general parametrization,
• 83:33 - 83:41
x equals x of t, y equals y of
t will be-- who tells me, guys,
• 83:41 - 83:43
that-- this is in your hands.
• 83:43 - 83:48
Now you convinced me
that, for whatever reason,
• 83:48 - 83:50
you [INAUDIBLE].
• 83:50 - 83:52
You became friends
with these curves.
• 83:52 - 83:53
I don't know when.
• 83:53 - 83:55
I guess in the process
of doing homework.
• 83:55 - 83:56
Am I right?
• 83:56 - 84:00
I think you did not quite like
them before or the last week.
• 84:00 - 84:02
But I think you're
friends with them now.
• 84:02 - 84:07
x of t, y of t.
• 84:07 - 84:08
Let people talk.
• 84:08 - 84:13
• 84:13 - 84:14
STUDENT: 0.
• 84:14 - 84:16
PROFESSOR TODA: So.
• 84:16 - 84:16
Great.
• 84:16 - 84:21
And then R prime of t will be
x prime of t, y prime of t,
• 84:21 - 84:22
and 0.
• 84:22 - 84:24
I assume this to
be always non-zero.
• 84:24 - 84:26
I have a regular curve.
• 84:26 - 84:31
R double prime will be--
x double prime where
• 84:31 - 84:34
double prime-- we
did the review today
• 84:34 - 84:36
of the lasting acceleration.
• 84:36 - 84:40
here, are they nice or mean?
• 84:40 - 84:43
I hope they are not so mean.
• 84:43 - 84:46
The cross product is
a friendly fellow.
• 84:46 - 84:49
You have i, j, k, and
then the second row
• 84:49 - 84:51
would be x prime, y prime, 0.
• 84:51 - 84:55
The last row would be x double
prime, y double prime, 0.
• 84:55 - 84:59
And it's a piece of cake.
• 84:59 - 85:02
• 85:02 - 85:04
OK, piece of cake,
piece of cake.
• 85:04 - 85:09
But I want to know
• 85:09 - 85:16
So you have exactly 15 seconds
• 85:16 - 85:23
Who is R prime plus R double
prime as a [? coordinate. ?]
• 85:23 - 85:25
[INTERPOSING VOICES]
• 85:25 - 85:29
• 85:29 - 85:30
PROFESSOR TODA: Good.
• 85:30 - 85:35
x prime, y double prime minus x
double prime, y prime times k.
• 85:35 - 85:38
And it doesn't matter
when I take the magnitude,
• 85:38 - 85:41
because magnitude of k is 1.
• 85:41 - 85:42
So I discovered some.
• 85:42 - 85:47
This is how mathematicians like
to discover new formulas based
• 85:47 - 85:49
on the formulas they
[? knew. ?] They
• 85:49 - 85:50
have a lot of satisfaction.
• 85:50 - 85:51
Look what I got.
• 85:51 - 85:57
Of course, they in general have
more complicated things to do,
• 85:57 - 85:59
and they have to
check and recheck.
• 85:59 - 86:06
But every piece of a
computation is a challenge.
• 86:06 - 86:10
And that gives
people satisfaction.
• 86:10 - 86:15
And when they make a mistake, it
brings a lot of tears as well.
• 86:15 - 86:21
So what-- could be written
on the bottom, what's
• 86:21 - 86:25
the speed cubed?
• 86:25 - 86:27
Speed is coming from this guy.
• 86:27 - 86:32
So the speed of the velocity,
the magnitude of the velocity
• 86:32 - 86:33
is the speed.
• 86:33 - 86:35
And that-- going
to give you square.
• 86:35 - 86:37
I'm not going to write
down [INAUDIBLE].
• 86:37 - 86:39
Square root of x squared,
x prime squared times
• 86:39 - 86:43
y prime squared,
and I cube that.
• 86:43 - 86:46
Many people, and I saw
that in engineering, they
• 86:46 - 86:50
don't like to put that
square root anymore.
• 86:50 - 86:54
And they just write x prime
squared plus y prime squared
• 86:54 - 86:55
to the what power?
• 86:55 - 86:56
STUDENT: 3/2.
• 86:56 - 86:57
PROFESSOR TODA: 3/2.
• 86:57 - 87:02
So this is very useful
for engineering styles,
• 87:02 - 87:05
when you have to deal
with plane curves, motions
• 87:05 - 87:09
in plane curves.
• 87:09 - 87:14
But now what do you
have in the case,
• 87:14 - 87:19
in the happy case, when
you have y equals f of x?
• 87:19 - 87:21
I'm going to do
that in a second.
• 87:21 - 87:26
• 87:26 - 87:29
I want to keep this
formula on the board.
• 87:29 - 87:38
• 87:38 - 87:41
What's the simplest
parametrization?
• 87:41 - 87:43
Because that's why we
need it, to look over
• 87:43 - 87:47
parametrizations
again and again.
• 87:47 - 87:52
R of t for this plane
curve will be-- what is t?
• 87:52 - 87:54
x is t, right?
• 87:54 - 87:56
x is t, y is f of t.
• 87:56 - 87:57
Piece of cake.
• 87:57 - 88:00
So you have t and f of t.
• 88:00 - 88:03
And how many of you watched
the videos that I sent you?
• 88:03 - 88:06
• 88:06 - 88:09
Do you prefer Khan
• 88:09 - 88:13
prefer the guys, [INAUDIBLE]
guys who are lecturing?
• 88:13 - 88:16
The professors who are lecturing
in front of a board or in front
• 88:16 - 88:18
of a-- what is that?
• 88:18 - 88:21
A projector screen?
• 88:21 - 88:23
I like all of them.
• 88:23 - 88:25
I think they're very good.
• 88:25 - 88:28
I think you can learn
a lot from three
• 88:28 - 88:30
or four different
instructors at the same time.
• 88:30 - 88:32
That's ideal.
• 88:32 - 88:36
I guess that you have
this chance only now
• 88:36 - 88:37
in the past few years.
• 88:37 - 88:41
Because 20 years ago, if you're
• 88:41 - 88:46
or just you couldn't stand
them, you had no other chance.
• 88:46 - 88:48
There was no
• 88:48 - 88:51
no way to learn from others.
• 88:51 - 89:00
R prime of t would
be 1 f prime of t.
• 89:00 - 89:03
out x, because x is t.
• 89:03 - 89:04
I don't care.
• 89:04 - 89:07
R double prime of t would
be 0, f double prime of x.
• 89:07 - 89:12
So I feel that, hey, I know
what's going to come up.
• 89:12 - 89:15
• 89:15 - 89:18
to write it down.
• 89:18 - 89:20
This is going to be Mr. x prime.
• 89:20 - 89:23
This is going to be
replacing Mr. y prime.
• 89:23 - 89:26
This is going to replace
Mr. a double prime.
• 89:26 - 89:29
This is going to be replacing
Mr. y double prime of x.
• 89:29 - 89:31
Oh, OK, all right.
• 89:31 - 89:39
So k, our old friend from
here will become what?
• 89:39 - 89:42
And I'd better shut up,
because I'm talking too much.
• 89:42 - 89:45
STUDENT: [INAUDIBLE]
double prime [INAUDIBLE].
• 89:45 - 89:48
PROFESSOR TODA: That is
the absolute value, mm-hmm.
• 89:48 - 89:54
[? n ?] double prime
of x, and nothing else.
• 89:54 - 89:55
Right, guys?
• 89:55 - 89:56
Are you with me?
• 89:56 - 89:57
Divided by--
• 89:57 - 89:58
STUDENT: [INAUDIBLE]
• 89:58 - 90:00
PROFESSOR TODA: Should
• 90:00 - 90:01
I love square roots.
• 90:01 - 90:02
• 90:02 - 90:12
So you go 1 plus f
prime squared cubed.
• 90:12 - 90:16
So that's going to
be-- any questions?
• 90:16 - 90:18
Are you guys with me?
• 90:18 - 90:21
That's going to be the
formula that I'm going
• 90:21 - 90:22
to use in the next example.
• 90:22 - 90:26
• 90:26 - 90:30
In case somebody
wants to know-- I got
• 90:30 - 90:32
this question from one of you.
• 90:32 - 90:35
Suppose we get a
parametrization of a circle
• 90:35 - 90:37
in the midterm or the final.
• 90:37 - 90:44
Somebody says, I have x
of t, just like we did it
• 90:44 - 90:48
today, a cosine t plus 0.
• 90:48 - 90:52
And y of t equals
a sine t plus y 0.
• 90:52 - 90:54
What is this, guys?
• 90:54 - 91:04
This is a circle, a center
at 0, y, 0, and radius a.
• 91:04 - 91:09
• 91:09 - 91:15
Can use a better formula-- that
anticipated my action today--
• 91:15 - 91:19
to actually prove that k
is going to be [? 1/a? ?]
• 91:19 - 91:19
Precisely.
• 91:19 - 91:21
Can we do that in the exam?
• 91:21 - 91:23
Yes.
• 91:23 - 91:25
So while I told
you a long time ago
• 91:25 - 91:29
that engineers and
mathematicians observed
• 91:29 - 91:31
hundreds of years
ago-- actually,
• 91:31 - 91:33
somebody said, no,
you're not right.
• 91:33 - 91:35
• 91:35 - 91:38
inverse proportionality
• 91:38 - 91:42
in Egypt, which makes sense
if you look at the pyramids.
• 91:42 - 91:48
So one look at the radius,
it says if the radius is 2,
• 91:48 - 91:51
then the curvature
is not very bent.
• 91:51 - 91:53
So the curvature's inverse
proportion [INAUDIBLE]
• 91:53 - 91:53
• 91:53 - 91:57
So if this is 2, we said
the curvature's 1/2.
• 91:57 - 92:02
If you take a big
circle, the bigger
• 92:02 - 92:04
smaller the bending
• 92:04 - 92:08
of the arc of the circle,
the smaller of the curvature.
• 92:08 - 92:11
Apparently the ancient
• 92:11 - 92:12
They Egyptians knew that.
• 92:12 - 92:13
The Greeks knew that.
• 92:13 - 92:15
But I think they
never formalized it--
• 92:15 - 92:17
not that I know.
• 92:17 - 92:19
• 92:19 - 92:25
So if you are asked to
do this in any exam,
• 92:25 - 92:27
do you think that
would be a problem?
• 92:27 - 92:28
Of course we would do review.
• 92:28 - 92:32
Because people are going to
forget this formula, or even
• 92:32 - 92:33
the definition.
• 92:33 - 92:36
You can compute k
for this formula.
• 92:36 - 92:39
And we are going to
get k to the 1/a.
• 92:39 - 92:41
This is a piece
of cake, actually.
• 92:41 - 92:44
You may not believe me, but
once you plug in the equations
• 92:44 - 92:46
it's very easy.
• 92:46 - 92:48
Or you can do it
from the definition
• 92:48 - 92:51
that gives you k of s.
• 92:51 - 92:53
You'll reparametrize
this in arclength.
• 92:53 - 92:55
You can do that as well.
• 92:55 - 92:58
And you still get 1/a.
• 92:58 - 93:00
The question that
I got by email,
• 93:00 - 93:01
and I get a lot of email.
• 93:01 - 93:05
I told you, that
keeps me busy a lot,
• 93:05 - 93:08
• 93:08 - 93:11
I really like the emails
I get from students,
• 93:11 - 93:14
because I get emails from
all sorts of sources--
• 93:14 - 93:15
Got some spam also.
• 93:15 - 93:22
Anyway, what I'm trying to say,
I got this question last time
• 93:22 - 93:25
saying, if on the midterm
we get such a question,
• 93:25 - 93:28
can we say simply, curvature's
• 93:28 - 93:31
Is that enough?
• 93:31 - 93:34
Depends on how the
problem was formulated.
• 93:34 - 93:39
Most likely I'm going to make
it through that or show that.
• 93:39 - 93:43
Even if you state something,
like, yes, it's 1/a,
• 93:43 - 93:46
with a little argument,
it's inverse proportional
• 93:46 - 93:50
still give partial credit.
• 93:50 - 93:54
For any argument that
is valid, especially
• 93:54 - 93:56
if it's based on
empirical observation,
• 93:56 - 93:59
I do give some extra
credit, even if you didn't
• 93:59 - 94:03
use the specific formula.
• 94:03 - 94:05
Let's see one example.
• 94:05 - 94:08
Let's take y equals e to the x.
• 94:08 - 94:12
• 94:12 - 94:16
No, let's take e
to the negative x.
• 94:16 - 94:17
Doesn't matter.
• 94:17 - 94:21
• 94:21 - 94:26
y equals e to the negative x.
• 94:26 - 94:31
And let's make x
between 0 and 1.
• 94:31 - 94:35
• 94:35 - 94:37
I'll say, write the curvature.
• 94:37 - 94:41
• 94:41 - 94:45
Write the equation or the
formula of the curvature.
• 94:45 - 94:50
• 94:50 - 94:55
And I know it's 2 o'clock
• 94:55 - 94:58
This was a question that one of
you had during the short break
• 94:58 - 94:59
we took.
• 94:59 - 95:00
Can we do such a problem?
• 95:00 - 95:02
Like she said.
• 95:02 - 95:03
Yes, I [INAUDIBLE]
to the negative
• 95:03 - 95:06
x because I want
to catch somebody
• 95:06 - 95:07
not knowing the derivative.
• 95:07 - 95:09
I don't know why I'm doing this.
• 95:09 - 95:11
Right?
• 95:11 - 95:15
So if I were to draw that, OK,
try and draw that, but not now.
• 95:15 - 95:19
Now, what formula
are you going to use?
• 95:19 - 95:22
Of course, you could
do this in many ways.
• 95:22 - 95:25
All those formulas are
equivalent for the curvature.
• 95:25 - 95:27
What's the simplest
way to do it?
• 95:27 - 95:30
Do y prime.
• 95:30 - 95:34
Minus it to the minus x.
• 95:34 - 95:37
Note here in this problem that
even if you mess up and forget
• 95:37 - 95:40
the minus sign, you still
• 95:40 - 95:46
But I may subtract a few points
if I see something nonsensical.
• 95:46 - 95:47
y double prime equals--
• 95:47 - 95:49
[INTERPOSING VOICES]
• 95:49 - 95:52
--plus e to the minus x.
• 95:52 - 95:57
And what is the
curvature k of t?
• 95:57 - 95:59
STUDENT: y prime over--
• 95:59 - 96:01
PROFESSOR TODA: Oh, I
didn't say one more thing.
• 96:01 - 96:05
I want the curvature, but
I also want the curvature
• 96:05 - 96:08
in three separate moments,
in the beginning, in the end,
• 96:08 - 96:09
and in the middle.
• 96:09 - 96:11
STUDENT: Don't we
need to parametrize it
• 96:11 - 96:15
so we can [INAUDIBLE]
x prime [INAUDIBLE]?
• 96:15 - 96:17
PROFESSOR TODA: No.
• 96:17 - 96:18
Did I erase it?
• 96:18 - 96:19
STUDENT: Yeah, you did.
• 96:19 - 96:21
PROFESSOR TODA: [INAUDIBLE].
• 96:21 - 96:24
And one of my colleagues
said, Magda, you are smart,
• 96:24 - 96:28
but you are like one
of those people who,
• 96:28 - 96:30
in the anecdotes
• 96:30 - 96:32
gets out of their office
and starts walking
• 96:32 - 96:33
and stops a student.
• 96:33 - 96:35
Was I going this
way or that way?
• 96:35 - 96:36
And that's me.
• 96:36 - 96:38
• 96:38 - 96:42
I should not have erased that.
• 96:42 - 96:44
I'm going to go
• 96:44 - 96:48
because I'm a goofball.
• 96:48 - 96:55
So the one that I wanted to use
k of x will be f double prime.
• 96:55 - 96:57
STUDENT: And cubed.
• 96:57 - 96:58
PROFESSOR TODA: Cubed!
• 96:58 - 96:59
Thank you.
• 96:59 - 97:03
• 97:03 - 97:10
So that 3/2, remember it,
[INAUDIBLE] 3/2 [INAUDIBLE]
• 97:10 - 97:11
square root cubed.
• 97:11 - 97:14
Now, for this one, is it hard?
• 97:14 - 97:15
No.
• 97:15 - 97:16
That's a piece of cake.
• 97:16 - 97:19
I said I like it in
general, but I also
• 97:19 - 97:23
like it-- find the curvature
of this curve in the beginning.
• 97:23 - 97:24
You travel on me.
• 97:24 - 97:28
From time 0 to 1
o'clock, whatever.
• 97:28 - 97:28
One second.
• 97:28 - 97:33
That's saying this is in seconds
to make it more physical.
• 97:33 - 97:40
I want the k at 0, I want k
at 1/2, and I want k at 1.
• 97:40 - 97:42
And I'd like you to
compare those values.
• 97:42 - 97:46
• 97:46 - 97:49
And I'll give you one
• 97:49 - 97:51
But let me start working.
• 97:51 - 97:53
So you say you help me on that.
• 97:53 - 97:55
[INAUDIBLE]
• 97:55 - 98:03
Minus x over square
root of 1 plus--
• 98:03 - 98:04
STUDENT: [INAUDIBLE]
• 98:04 - 98:05
PROFESSOR TODA: Right.
• 98:05 - 98:08
So can I write this differently,
a little bit differently?
• 98:08 - 98:10
Like what?
• 98:10 - 98:12
I don't want to square
each of the minus 2x.
• 98:12 - 98:14
Can I do that?
• 98:14 - 98:20
And then the whole thing
I can say to the 3/2
• 98:20 - 98:25
or I can use the square root,
• 98:25 - 98:29
Now, what is k of 0?
• 98:29 - 98:29
STUDENT: 0.
• 98:29 - 98:33
Or 1.
• 98:33 - 98:34
PROFESSOR TODA: Really?
• 98:34 - 98:36
STUDENT: 1/2.
• 98:36 - 98:37
3/2.
• 98:37 - 98:39
PROFESSOR TODA: So
let's take this slowly.
• 98:39 - 98:44
Because we can all make
mistakes, goofy mistakes.
• 98:44 - 98:45
That doesn't mean
we're not smart.
• 98:45 - 98:47
We're very smart, right?
• 98:47 - 98:51
But it's just a matter of
book-keeping and paying
• 98:51 - 98:53
attention, being attentive.
• 98:53 - 98:55
OK.
• 98:55 - 99:00
When I take 0 and replace--
this is drying fast.
• 99:00 - 99:03
I'm trying to draw it.
• 99:03 - 99:10
I have 1 over 1
plus 1 to the 3/2.
• 99:10 - 99:15
I have a student in one exam
who was just-- I don't know.
• 99:15 - 99:17
He was rushing.
• 99:17 - 99:21
He didn't realize that
he had to take it slowly.
• 99:21 - 99:23
He was extremely smart, though.
• 99:23 - 99:30
1 over-- you have
that 1 plus 1 is 2.
• 99:30 - 99:34
2 to the 1/2 would be
square root of 2 cubed.
• 99:34 - 99:36
It would be exactly
2 square root of 2.
• 99:36 - 99:40
And more you can write
this as rationalized.
• 99:40 - 99:42
Now, I have a question for you.
• 99:42 - 99:43
[INAUDIBLE]
• 99:43 - 99:47
I'm When we were kids, if you
remember-- you are too young.
• 99:47 - 99:48
Maybe you don't remember.
• 99:48 - 99:53
But I remember when I was a kid,
my teacher would always ask me,
• 99:53 - 99:54
• 99:54 - 99:57
• 99:57 - 100:00
Put the rational number
in the denominator.
• 100:00 - 100:03
Why do you think that was?
• 100:03 - 100:05
For hundreds of years
people did that.
• 100:05 - 100:07
STUDENT: [INAUDIBLE]
• 100:07 - 100:12
PROFESSOR TODA: Because they
didn't have a calculator.
• 100:12 - 100:16
So we used to, even I used to be
able to get the square root out
• 100:16 - 100:18
by hand.
• 100:18 - 100:21
Has anybody taught you how to
compute square root by hand?
• 100:21 - 100:22
You know that.
• 100:22 - 100:22
Who taught you?
• 100:22 - 100:23
STUDENT: I don't remember it.
• 100:23 - 100:25
teacher taught us.
• 100:25 - 100:26
PROFESSOR TODA:
There is a technique
• 100:26 - 100:29
of taking groups of twos
and then fitting the--
• 100:29 - 100:31
and they still teach that.
• 100:31 - 100:33
I was amazed, they
still teach that
• 100:33 - 100:35
in half of the Asian countries.
• 100:35 - 100:39
And it's hard, but kids
• 100:39 - 100:45
have that practice, which some
of us learned and forgot about.
• 100:45 - 100:50
So imagine that how people would
have done this, and of course,
• 100:50 - 100:51
square root of 2 is easy.
• 100:51 - 100:54
1.4142, blah blah blah.
• 100:54 - 100:54
Divide by 2.
• 100:54 - 100:56
You can do it by hand.
• 100:56 - 100:58
At least a good approximation.
• 100:58 - 101:02
But imagine having a nasty
square root there to compute,
• 101:02 - 101:06
and then you would divide
by that natural number.
• 101:06 - 101:09
You have to rely on your
own computation to do it.
• 101:09 - 101:11
There were no calculators.
• 101:11 - 101:14
• 101:14 - 101:15
How is that?
• 101:15 - 101:16
What is that?
• 101:16 - 101:20
• 101:20 - 101:24
e to the minus 1.
• 101:24 - 101:27
That's a little bit
harder to compute, right?
• 101:27 - 101:29
1 plus [INAUDIBLE].
• 101:29 - 101:32
What is that going to be?
• 101:32 - 101:34
Minus 2.
• 101:34 - 101:37
Replace it by 1 to the 3/2.
• 101:37 - 101:42
I would like you to go
home and do the following.
• 101:42 - 101:46
[INAUDIBLE]-- Not now, not now.
• 101:46 - 101:48
We stay a little
bit longer together.
• 101:48 - 101:52
k of 0, k of 1/2, and k of 1.
• 101:52 - 101:53
Which one is bigger?
• 101:53 - 102:00
• 102:00 - 102:03
that, how much extra credit
• 102:03 - 102:04
should I give you?
• 102:04 - 102:06
One point?
• 102:06 - 102:08
One point if you turn this in.
• 102:08 - 102:11
Um, yeah.
• 102:11 - 102:13
Four, [? maybe ?] two points.
• 102:13 - 102:19
Compare all these
three values, and find
• 102:19 - 102:33
the maximum and the
minimum of kappa of t,
• 102:33 - 102:38
kappa of x, for
the interval where
• 102:38 - 102:47
x is in the interval 0, 1.
• 102:47 - 102:49
0, closed 1.
• 102:49 - 102:50
Close it.
• 102:50 - 102:54
because it's extra credit.
• 102:54 - 102:59
One question was, by email,
can I ask my tutor to help me?
• 102:59 - 103:02
As long as your tutor doesn't
• 103:02 - 103:04
you are in good shape.
• 103:04 - 103:07
understand some constants,
• 103:07 - 103:08
spend time with you.
• 103:08 - 103:13
But they should not write
• 103:13 - 103:13
OK?
• 103:13 - 103:17
So it's not a big deal.
• 103:17 - 103:22
Not I want to tell you one
secret that I normally don't
• 103:22 - 103:27
tell my Calculus 3 students.
• 103:27 - 103:29
But the more I get
to know you, the more
• 103:29 - 103:34
I realize that you are worth
• 103:34 - 103:35
STUDENT: [INAUDIBLE]
• 103:35 - 103:38
PROFESSOR TODA: No.
• 103:38 - 103:42
There is a beautiful
theory that engineers
• 103:42 - 103:48
use when they start the motions
of curves and parametrizations
• 103:48 - 103:51
in space.
• 103:51 - 103:53
And that includes
the Frenet formulas.
• 103:53 - 103:56
• 103:56 - 103:59
know the first one.
• 103:59 - 104:05
And I was debating, I was just
reviewing what I taught you,
• 104:05 - 104:07
and I was happy with
what I taught you.
• 104:07 - 104:11
And I said, they know
• 104:11 - 104:13
velocity, acceleration.
• 104:13 - 104:16
They know how to get back
and forth from one another.
• 104:16 - 104:17
They know our claim.
• 104:17 - 104:18
They know how to
[? reparameterize our ?]
• 104:18 - 104:20
claims.
• 104:20 - 104:25
They know the [INAUDIBLE]
• 104:25 - 104:27
the first Frenet formula.
• 104:27 - 104:28
They know the curvature.
• 104:28 - 104:30
What else can I teach them?
• 104:30 - 104:34
I want to show you--
• 104:34 - 104:38
is this all that we should know?
• 104:38 - 104:42
This is all that a regular
student should know in Calculus
• 104:42 - 104:44
3, but there is more.
• 104:44 - 104:45
And you are honor students.
• 104:45 - 104:50
And I want to show you some
beautiful equations here.
• 104:50 - 104:55
So do you remember that
if I introduce r of s
• 104:55 - 105:04
as a curving arclength,
that is a regular curve.
• 105:04 - 105:11
I said there is a certain famous
formula that is T prime of s
• 105:11 - 105:14
called-- leave space.
• 105:14 - 105:15
Leave a little bit of space.
• 105:15 - 105:16
You'll see why.
• 105:16 - 105:18
It's a surprise.
• 105:18 - 105:23
k times-- why
don't I say k of s?
• 105:23 - 105:26
Because I want to point
out that k is an invariant.
• 105:26 - 105:29
Even if you have
another parameter,
• 105:29 - 105:30
would be the same function.
• 105:30 - 105:39
But yes, as a function of s,
would be k times N bar, bar.
• 105:39 - 105:41
More bars because
they are free vectors.
• 105:41 - 105:43
They are not bound
to a certain point.
• 105:43 - 105:45
They're not married
to a certain point.
• 105:45 - 105:49
They are free to shift
by parallelism in space.
• 105:49 - 105:54
However, I'm going to review
them as bound at the point
• 105:54 - 105:55
where they are.
• 105:55 - 105:58
So they-- no way they
are married to the point
• 105:58 - 106:04
that they belong to.
• 106:04 - 106:07
Maybe the [? bend ?]
will change.
• 106:07 - 106:09
I don't know how it's
going to change like crazy.
• 106:09 - 106:18
• 106:18 - 106:19
Something like that.
• 106:19 - 106:27
At every point you have a T, an
N, and it's a 90 degree angle.
• 106:27 - 106:31
Then you have the binormal,
which makes a 90 degree
• 106:31 - 106:33
angle-- [INAUDIBLE].
• 106:33 - 106:37
So the way you should
imagine these corners
• 106:37 - 106:39
would be something
like that, right?
• 106:39 - 106:41
90-90-90.
• 106:41 - 106:43
It's just hard to draw them.
• 106:43 - 106:52
Between the vectors you have--
If you draw T and N, am I
• 106:52 - 106:53
right, that is coming out?
• 106:53 - 106:54
No.
• 106:54 - 106:56
I have to switch them.
• 106:56 - 106:58
T and N. Now, am I right?
• 106:58 - 106:59
Now I'm thinking of
the [? faucet. ?]
• 106:59 - 107:02
If I move T-- yeah,
now it's coming out.
• 107:02 - 107:08
So this is not getting
into the formula.
• 107:08 - 107:09
So this is the first formula.
• 107:09 - 107:10
You say, so what?
• 107:10 - 107:11
You've taught that.
• 107:11 - 107:12
We proved it together.
• 107:12 - 107:14
What do you want from us?
• 107:14 - 107:18
I want to teach you
two more formulas.
• 107:18 - 107:19
N prime.
• 107:19 - 107:22
• 107:22 - 107:24
And I'd like you to
leave more space here.
• 107:24 - 107:27
• 107:27 - 107:31
So you have like an empty field
here and an empty field here
• 107:31 - 107:32
[INAUDIBLE].
• 107:32 - 107:36
If you were to compute
T prime, the magic thing
• 107:36 - 107:40
is that T prime is a vector.
• 107:40 - 107:41
N prime is a vector.
• 107:41 - 107:43
B prime is a vector.
• 107:43 - 107:44
They're all vectors.
• 107:44 - 107:49
They are the derivatives
of the vectors T and NB.
• 107:49 - 107:51
And you say, why would I
• 107:51 - 107:52
of the vectors T and NB?
• 107:52 - 107:54
I'll tell you in a second.
• 107:54 - 107:58
So if you were to
compute in prime,
• 107:58 - 108:00
you're going to get here.
• 108:00 - 108:04
Minus k of s times T of s.
• 108:04 - 108:07
Leave room.
• 108:07 - 108:10
Leave room, because there
is no component that
• 108:10 - 108:14
depends on N. No such component
that that depends on N.
• 108:14 - 108:15
This is [INAUDIBLE].
• 108:15 - 108:17
There is nothing in
N. And then in the end
• 108:17 - 108:29
you'll say, plus tau of s
times B. There is missing--
• 108:29 - 108:30
something is.
• 108:30 - 108:33
And finally, if
you take B prime,
• 108:33 - 108:35
there is nothing
here, nothing here.
• 108:35 - 108:43
In the middle you have
minus tau of s times N of s.
• 108:43 - 108:46
• 108:46 - 108:50
And now you know that nobody
else but you knows that.
• 108:50 - 108:54
The other regular sections
don't know these formulas.
• 108:54 - 108:57
• 108:57 - 109:02
this bunch of equations?
• 109:02 - 109:04
Say, oh, wait a minute.
• 109:04 - 109:07
First of all, why did
you put it like that?
• 109:07 - 109:08
Looks like a cross.
• 109:08 - 109:09
It is a cross.
• 109:09 - 109:13
It is like one is shaped in the
name of the Father, of the Son,
• 109:13 - 109:14
and so on.
• 109:14 - 109:17
So does it have anything
to do with religion?
• 109:17 - 109:18
No.
• 109:18 - 109:23
memorize better the equations.
• 109:23 - 109:27
These are the famous
Frenet equations.
• 109:27 - 109:30
• 109:30 - 109:34
You only saw the first one.
• 109:34 - 109:35
What do they represent?
• 109:35 - 109:38
• 109:38 - 109:40
If somebody asks you, what is k?
• 109:40 - 109:43
What it is k of s?
• 109:43 - 109:44
What's the curvature?
• 109:44 - 109:45
You go to a party.
• 109:45 - 109:47
There are only nerds.
• 109:47 - 109:47
It's you.
• 109:47 - 109:50
calculus or some people
• 109:50 - 109:55
from Physics, and they say, OK,
have you heard of the Frenet
• 109:55 - 109:57
motion, Frenet
formulas, and you say,
• 109:57 - 109:59
• 109:59 - 110:02
What if they ask you, what
is the curvature of k?
• 110:02 - 110:08
You say, curvature measures
how a curve is bent.
• 110:08 - 110:12
And they say, yeah, but the
Frenet formula tells you
• 110:12 - 110:14
• 110:14 - 110:18
Not only k shows you
how bent the curve is.
• 110:18 - 110:27
But k is a measure of
how fast T changes.
• 110:27 - 110:28
And he sees why.
• 110:28 - 110:31
Practically, if you take
the [INAUDIBLE] to the bat,
• 110:31 - 110:37
this is the speed of T. So how
fast the teaching will change.
• 110:37 - 110:40
That will be magnitude,
will be just k.
• 110:40 - 110:42
Because magnitude of N is 1.
• 110:42 - 110:49
So note that k of s is
the length of T prime.
• 110:49 - 111:04
This measures the change
in T. So how fast T varies.
• 111:04 - 111:09
• 111:09 - 111:11
What does the torsion represent?
• 111:11 - 111:17
Well, how fast the
binormal varies.
• 111:17 - 111:21
But if you want to
think of a helix,
• 111:21 - 111:26
and it's a little
bit hard to imagine,
• 111:26 - 111:30
the curvature measures how
bent a certain curve is.
• 111:30 - 111:34
And it measures how
bent a plane curve is.
• 111:34 - 111:39
For example, for the circle you
have radius a, 1/a, and so on.
• 111:39 - 111:41
But there must be
also a function that
• 111:41 - 111:46
shows you how a curve twists.
• 111:46 - 111:50
Because you have not
just a plane curve where
• 111:50 - 111:52
• 111:52 - 111:59
But in the space curve you
care how the curves twist.
• 111:59 - 112:03
How fast do they move
away from a certain plane?
• 112:03 - 112:11
Now, if I were to draw-- is
it hard to memorize these?
• 112:11 - 112:11
No.
• 112:11 - 112:14
I memorized them easily
based on the fact
• 112:14 - 112:20
that everything looks
like a decomposition
• 112:20 - 112:24
of a vector in terms of
T, N, and B. So in my mind
• 112:24 - 112:28
it was like, I take any vector
I want, B. And this is T,
• 112:28 - 112:33
this is N, and this is B.
Just the weight was IJK.
• 112:33 - 112:37
of J, I have N. Instead of K,
• 112:37 - 112:40
I have B. They are
still unit vectors.
• 112:40 - 112:43
So locally at the
point I have this frame
• 112:43 - 112:44
and I have any vector.
• 112:44 - 112:47
This vector-- I'm a physicist.
• 112:47 - 112:51
So let's say I'm going to
represent that as v1 times
• 112:51 - 112:54
the T plus v2
• 112:54 - 112:58
we'll use that N plus
B3 times-- that's
• 112:58 - 113:00
the last element of the bases.
• 113:00 - 113:04
Instead of k I have v.
So it's the same here.
• 113:04 - 113:06
You try to pick a
vector and decompose
• 113:06 - 113:10
that in terms of T, N, and B.
Will I put that on the final?
• 113:10 - 113:11
No.
• 113:11 - 113:13
But I would like you to
remember it, especially
• 113:13 - 113:17
if you are an engineering
major or physics major,
• 113:17 - 113:20
that there is this
kind of Frenet frame.
• 113:20 - 113:26
For those of you who are taking
a-- for differential equations,
• 113:26 - 113:29
and built-in systems
• 113:29 - 113:32
of equations, systems of
differential equations.
• 113:32 - 113:33
I'm not going to get there.
• 113:33 - 113:38
But suppose you don't know
differential equations,
• 113:38 - 113:42
but you know a little
bit of linear algebra.
• 113:42 - 113:45
And I know you know how
to multiply matrices.
• 113:45 - 113:47
You know how I know
you multiply matrices,
• 113:47 - 113:50
no matter how much
mathematics you learn.
• 113:50 - 113:53
And most of you, you are not in
general algebra this semester.
• 113:53 - 113:55
Only two of you are
in general algebra.
• 113:55 - 114:03
When I took a C++ course,
the first homework I got was
• 114:03 - 114:07
to program a matrix
multiplication.
• 114:07 - 114:08
I have to give in matrices.
• 114:08 - 114:11
I have to program that in C++.
• 114:11 - 114:15
And freshmen knew that.
• 114:15 - 114:20
So that means you know how
to write this as a matrix
• 114:20 - 114:21
multiplication.
• 114:21 - 114:23
Can anybody help me?
• 114:23 - 114:26
So T, N, B is the magic triple.
• 114:26 - 114:29
T, N, B's the magic corner.
• 114:29 - 114:32
T, N, and B are the Three
Musketeers who are all
• 114:32 - 114:34
orthogonal to one another.
• 114:34 - 114:38
And then I do derivative
with respect to s.
• 114:38 - 114:42
If I want to be
elegant, I'll put d/ds.
• 114:42 - 114:44
OK.
• 114:44 - 114:47
How am I going to
fill in this matrix?
• 114:47 - 114:51
So somebody who wants to know
• 114:51 - 114:52
this would be a--
• 114:52 - 114:53
STUDENT: 0, k, 0.
• 114:53 - 114:54
PROFESSOR TODA: Very good.
• 114:54 - 115:05
0, k, 0, minus k 0
tau, 0 minus tau 0.
• 115:05 - 115:07
This is called the
skew symmetric matrix.
• 115:07 - 115:12
• 115:12 - 115:15
Such matrices are very
important in robotics.
• 115:15 - 115:17
If you've ever been
to a robotics team,
• 115:17 - 115:20
like one of those
projects, you should
• 115:20 - 115:23
know that when we study
motions of-- let's say
• 115:23 - 115:27
that my arm performs
two rotations in a row.
• 115:27 - 115:30
All these motions
are described based
• 115:30 - 115:35
on some groups of rotations.
• 115:35 - 115:40
And if I go into details,
it's going to be really hard.
• 115:40 - 115:46
But practically
in such a setting
• 115:46 - 115:50
we have to deal with matrices
that either have determined
• 115:50 - 115:54
one, like all rotations
actually have,
• 115:54 - 115:58
or have some other
properties, like this guy.
• 115:58 - 116:00
What's the determinant
of this guy?
• 116:00 - 116:02
What do you guys think?
• 116:02 - 116:03
Just look at it.
• 116:03 - 116:03
STUDENT: 0?
• 116:03 - 116:04
PROFESSOR TODA: 0.
• 116:04 - 116:06
It has determinant 0.
• 116:06 - 116:08
And moreover, it
looks in the mirror.
• 116:08 - 116:11
So this comes from
a group of motion,
• 116:11 - 116:15
which is little s over 3,
the linear algebra, actually.
• 116:15 - 116:17
So when k is looking
in the mirror,
• 116:17 - 116:21
it becomes minus k tau,
is becoming minus tau.
• 116:21 - 116:24
It is antisymmetric
or skew symmetric.
• 116:24 - 116:27
Skew symmetric or
antisymmetric is the same.
• 116:27 - 116:30
STUDENT: Antisymmetric,
skew symmetric matrix.
• 116:30 - 116:32
PROFESSOR TODA: Skew
symmetric or antisymmetric
• 116:32 - 116:33
is exactly the same thing.
• 116:33 - 116:34
They are synonyms.
• 116:34 - 116:37
• 116:37 - 116:40
So it looks in the mirror
and picks up the minus sign,
• 116:40 - 116:42
has 0 in the bag.
• 116:42 - 116:43
What am I going to put here?
• 116:43 - 116:44
• 116:44 - 116:47
So when Ryan gave
me this, he meant
• 116:47 - 116:50
that he knew what I'm going
to put here, as a vector,
• 116:50 - 116:54
as a column vector.
• 116:54 - 116:55
STUDENT: [INAUDIBLE]
• 116:55 - 116:56
PROFESSOR TODA: No, no no.
• 116:56 - 116:57
How do I multiply?
• 116:57 - 116:59
TNB, right?
• 116:59 - 117:01
So guys, how do you
multiply matrices?
• 117:01 - 117:05
You go first row
and first column.
• 117:05 - 117:06
So you go like this.
• 117:06 - 117:14
0 times T plus k times 10
plus 0 times B. Here it is.
• 117:14 - 117:15
So I'm teaching you
a little bit more
• 117:15 - 117:19
than-- if you are going to
stick with linear algebra
• 117:19 - 117:21
and stick with
differential equations,
• 117:21 - 117:25
this is a good introduction
to more of those mathematics.
• 117:25 - 117:26
Yes, sir?
• 117:26 - 117:28
STUDENT: Why don't
you use Cramer's rule?
• 117:28 - 117:29
PROFESSOR TODA: Uh?
• 117:29 - 117:31
STUDENT: Why don't you
use the Cramer's rule?
• 117:31 - 117:33
PROFESSOR TODA:
The Cramer's rule?
• 117:33 - 117:35
STUDENT: Yeah. [INAUDIBLE].
• 117:35 - 117:36
PROFESSOR TODA: No.
• 117:36 - 117:44
First of all, Crarmer's rule is
to solve systems of equations
• 117:44 - 117:48
that don't involve derivatives,
like a linear system
• 117:48 - 117:52
like Ax equals B.
I'm going to have,
• 117:52 - 117:57
for example, 3x1
plus 2x3 equals 1.
• 117:57 - 118:01
5x1 plus x2 plus x3
equals something else.
• 118:01 - 118:03
So for that I can
use Cramer's rule.
• 118:03 - 118:05
But look at that!
• 118:05 - 118:06
This is really complicated.
• 118:06 - 118:08
It's a dynamical system.
• 118:08 - 118:12
At every moment of time
the vectors are changing.
• 118:12 - 118:13
So it's a crazy [INAUDIBLE].
• 118:13 - 118:19
Like A of t times
something, so some vector
• 118:19 - 118:23
that is also depending on
time equals the derivative
• 118:23 - 118:25
of that vector that [INAUDIBLE].
• 118:25 - 118:32
So that's a OD system that
you should learn in 3351.
• 118:32 - 118:33
So I don't know what
• 118:33 - 118:35
but most of you in
engineering will
• 118:35 - 118:44
take my class, 2316 in algebra,
OD1 3350 where they teach you
• 118:44 - 118:45
• 118:45 - 118:48
These are all differential
equations, all three of them.
• 118:48 - 118:51
In 3351 you learn
• 118:51 - 118:54
which is a system of
differential equation.
• 118:54 - 118:57
And then you
practically say, now I
• 118:57 - 119:00
know everything I need to
know in math, and you say,
• 119:00 - 119:01
goodbye math.
• 119:01 - 119:03
If you guys wanted
• 119:03 - 119:06
of course I would be very
happy to learn that, hey, I
• 119:06 - 119:09
like math, I'd like
to be a double major.
• 119:09 - 119:12
I'd like to be not just an
engineering, but also math
• 119:12 - 119:15
major if you really like it.
• 119:15 - 119:18
have a minor.
• 119:18 - 119:20
Many of you have a
• 119:20 - 119:23
Like for that minor
you only need--
• 119:23 - 119:24
STUDENT: One extra math course.
• 119:24 - 119:26
PROFESSOR TODA: One
extra math course.
• 119:26 - 119:31
For example, with 3350 you
don't need 3351 for a minor.
• 119:31 - 119:31
Why?
• 119:31 - 119:34
Because you are taking the
probability in stats anyway.
• 119:34 - 119:35
You have to.
• 119:35 - 119:39
They force you to do that, 3342.
• 119:39 - 119:45
So if you take 3351 it's on top
of the minor that we give you.
• 119:45 - 119:46
I know because that's what I do.
• 119:46 - 119:48
I look at the degree plans.
• 119:48 - 119:52
And I work closely to the
• 119:52 - 119:54
She has all the [INAUDIBLE].
• 119:54 - 119:55
STUDENT: So is [INAUDIBLE]?
• 119:55 - 120:00
• 120:00 - 120:01
PROFESSOR TODA: You mean double?
• 120:01 - 120:02
Double degree?
• 120:02 - 120:04
We have this already in place.
• 120:04 - 120:05
We've had it for many years.
• 120:05 - 120:07
It's an excellent plan.
• 120:07 - 120:10
162 hours it is now.
• 120:10 - 120:13
It used to be 159.
• 120:13 - 120:18
Double major, computer
science and mathematics.
• 120:18 - 120:23
And I could say they were
some of the most successful
• 120:23 - 120:27
in terms of finding jobs.
• 120:27 - 120:29
What would you take
on top of that?
• 120:29 - 120:31
Well, as a math major you
have a few more courses
• 120:31 - 120:33
to take one top of that.
• 120:33 - 120:37
science with the mathematics,
• 120:37 - 120:40
for example, by taking
numerical analysis.
• 120:40 - 120:42
If you love computers
and you like calculus
• 120:42 - 120:47
and you want to put
together all the information
• 120:47 - 120:49
you have in both, then
numerical analysis
• 120:49 - 120:50
• 120:50 - 120:55
And they require that in
both computer science degree
• 120:55 - 120:58
if you are a double major,
• 120:58 - 121:03
So the good thing is that some
things count for both degrees.
• 121:03 - 121:07
And so with those 160
hours you are very happy.
• 121:07 - 121:10
Oh, I'm done, I got
a few more hours.
• 121:10 - 121:12
Many math majors
• 121:12 - 121:14
They're not supposed to.
• 121:14 - 121:16
They are supposed
to stop at 120.
• 121:16 - 121:20
So why not go the extra 20 hours
and get two degrees in one?
• 121:20 - 121:21
STUDENT: It's a semester.
• 121:21 - 121:22
PROFESSOR TODA: Yeah.
• 121:22 - 121:23
Of course, it's a lot more work.
• 121:23 - 121:26
But we have people
who like-- really they
• 121:26 - 121:30
are nerdy people who loved
computer science from when
• 121:30 - 121:32
they were three or four.
• 121:32 - 121:33
And they also like math.
• 121:33 - 121:37
And they say, OK,
I want to do both.
• 121:37 - 121:42
OK, a little bit more
and I'll let you go.
• 121:42 - 121:45
Now I want you to ask
me other questions
• 121:45 - 121:48
homework, anything that
• 121:48 - 121:59
you feel you need a little bit
• 121:59 - 122:01
• 122:01 - 122:12
• 122:12 - 122:13
Yes?
• 122:13 - 122:14
STUDENT: I just have one.
• 122:14 - 122:16
In WeBWork, what
is the easiest way
• 122:16 - 122:18
to take the square
root of something?
• 122:18 - 122:19
STUDENT: sqrt.
• 122:19 - 122:22
PROFESSOR TODA: sqrt
is what you type.
• 122:22 - 122:25
But of course you can
also go to the caret 1/2.
• 122:25 - 122:28
• 122:28 - 122:29
Something non-technical?
• 122:29 - 122:34
Any question, yes sir,
from the homework?
• 122:34 - 122:39
Or in relation to [INAUDIBLE]?
• 122:39 - 122:42
STUDENT: I don't understand
why is the tangent unit vector,
• 122:42 - 122:44
it's just the slope off
of that line, right?
• 122:44 - 122:45
The drunk bug?
• 122:45 - 122:47
Whatever line the
drunk bug is on?
• 122:47 - 122:49
PROFESSOR TODA: So it
would be the tangent
• 122:49 - 122:52
to the directional
motion, which is a curve.
• 122:52 - 122:55
• 122:55 - 122:58
And normalized to
have length one.
• 122:58 - 123:02
Because otherwise our
prime is-- you may say,
• 123:02 - 123:04
why do you need T to be unitary?
• 123:04 - 123:07
• 123:07 - 123:11
OK, computations become
• 123:11 - 123:14
is 1 or 5 or 9.
• 123:14 - 123:18
If the speed is a constant,
everything else becomes easier.
• 123:18 - 123:20
So that's one reason.
• 123:20 - 123:22
STUDENT: And why
is the derivative
• 123:22 - 123:24
of T then perpendicular?
• 123:24 - 123:27
Why does it always turn into--
• 123:27 - 123:28
PROFESSOR TODA:
Perpendicular to T?
• 123:28 - 123:31
We've done that last time,
but I'm glad to do it again.
• 123:31 - 123:34
And I forgot what we
wrote in the book,
• 123:34 - 123:37
and I also saw in
the book this thing
• 123:37 - 123:43
that if you have R, in
absolute value, constant--
• 123:43 - 123:45
and I've done that
with you guys--
• 123:45 - 123:52
prove that R and R prime had
every point perpendicular.
• 123:52 - 123:55
So if you have-- we've
done that before.
• 123:55 - 123:57
Now, what do you do then?
• 123:57 - 124:01
T [INAUDIBLE] T is 1.
• 124:01 - 124:04
The scalar [INAUDIBLE]
the product.
• 124:04 - 124:10
T prime times T plus
T prime T prime.
• 124:10 - 124:12
So 0.
• 124:12 - 124:17
And T is perpendicular
to T prime,
• 124:17 - 124:21
because that means T
or T prime equals 0.
• 124:21 - 124:28
• 124:28 - 124:30
When you run in a
circle, you say--
• 124:30 - 124:34
OK, let's run in a circle.
• 124:34 - 124:41
I say, this is my T. I can feel
that there is something that's
• 124:41 - 124:43
trying to bend me this way.
• 124:43 - 124:44
That is my acceleration.
• 124:44 - 124:49
And I have to-- but I don't
know-- how familiar are you
• 124:49 - 124:51
with the winter sports?
• 124:51 - 124:54
• 124:54 - 124:58
In many winter sports, the
Frenet Trihedron is crucial.
• 124:58 - 125:01
Imagine that you have
one of those slopes,
• 125:01 - 125:05
and all of the sudden the
torsion becomes too weak.
• 125:05 - 125:07
That means it becomes dangerous.
• 125:07 - 125:10
That means that the
vehicle you're in,
• 125:10 - 125:15
the snow vehicle or any kind
• 125:15 - 125:21
if the torsion of your body
moving can become too big,
• 125:21 - 125:22
that will be a problem.
• 125:22 - 125:25
So you have to redesign
that some more.
• 125:25 - 125:27
And this is what they do.
• 125:27 - 125:29
You know there have
been many accidents.
• 125:29 - 125:32
And many times they say,
even in Formula One,
• 125:32 - 125:38
the people who project
a certain racetrack,
• 125:38 - 125:42
like a track in
Indianapolis or Montecarlo
• 125:42 - 125:44
or whatever, they
have to have in mind
• 125:44 - 125:48
that Frenet frame every second.
• 125:48 - 125:51
So there are
simulators showing how
• 125:51 - 125:53
the Frenet frame is changing.
• 125:53 - 125:56
There are programs that
measure the curvature
• 125:56 - 126:00
in a torsion for those
simulators at every point.
• 126:00 - 126:03
Neither the curvature
nor the torsion
• 126:03 - 126:04
can exceed a certain value.
• 126:04 - 126:07
Otherwise it becomes dangerous.
• 126:07 - 126:10
You say, oh, I thought
only the speed is a danger.
• 126:10 - 126:11
Nope.
• 126:11 - 126:15
It's also the way that the
motion, if it's a skew curve,
• 126:15 - 126:17
it's really complicated.
• 126:17 - 126:20
Because you twist and turn
and bend in many ways.
• 126:20 - 126:22
And it can become
really dangerous.
• 126:22 - 126:23
Speed is not [INAUDIBLE].
• 126:23 - 126:26
• 126:26 - 126:30
STUDENT: So the torsion was
the twists in the track?
• 126:30 - 126:32
PROFESSOR TODA: The
torsion is the twist.
• 126:32 - 126:35
And by the way, keep your idea.
• 126:35 - 126:37
something more?
• 126:37 - 126:43
When you twist-- suppose you
have something like a race car.
• 126:43 - 126:47
And the race car is at
the walls of the track.
• 126:47 - 126:58
And here's-- when you have
a very abrupt curvature
• 126:58 - 127:04
and torsion, and you can have
that in Formula One as well,
• 127:04 - 127:10
why do they build one wall
a lot higher than the other?
• 127:10 - 127:14
Because the poor car-- I
don't know how passionate you
• 127:14 - 127:20
One or car races--
• 127:20 - 127:25
the poor car is going
to be close to the wall.
• 127:25 - 127:28
It's going to bend like that,
that wall would be round.
• 127:28 - 127:33
And as a builder, you have to
build the wall really high.
• 127:33 - 127:36
Because that kind of high
speed, high velocity,
• 127:36 - 127:39
high curvature, the poor
car's going szhhhhh-- then
• 127:39 - 127:42
again on a normal track.
• 127:42 - 127:45
Imagine what happens if the
wall is not high enough.
• 127:45 - 127:48
The wheels of the car
will go up and get over.
• 127:48 - 127:50
And it's going to be a disaster.
• 127:50 - 127:53
• 127:53 - 127:57
So that engineer ha to study
all the parametric equations
• 127:57 - 128:01
and the Frenet frame and
deep down make a simulator,
• 128:01 - 128:04
compute how tall the walls
should be in order for the car
• 128:04 - 128:10
not to get over on the other
side or get off the track.
• 128:10 - 128:12
It's really complicated stuff.
• 128:12 - 128:15
It's all mathematics
and physics,
• 128:15 - 128:19
but all the applications are
run by engineers and-- yes, sir?
• 128:19 - 128:22
STUDENT: What's the difference
[INAUDIBLE] centrifugal force?
• 128:22 - 128:24
PROFESSOR TODA: The
centrifugal force
• 128:24 - 128:26
is related to our double prime.
• 128:26 - 128:32
Our double prime is related
to N and T at the same time.
• 128:32 - 128:36
So at some point, let me ask you
one last question and I'm done.
• 128:36 - 128:39
• 128:39 - 128:43
What's the relationship between
acceleration or double prime?
• 128:43 - 128:46
And are they the same thing?
• 128:46 - 128:50
And when are they
not the same thing?
• 128:50 - 128:52
Because you say, OK,
practically the centrifugal--
• 128:52 - 128:54
STUDENT: They're
the same on a curve.
• 128:54 - 128:56
PROFESSOR TODA:
They are the same--
• 128:56 - 128:57
STUDENT: Like on a circle.
• 128:57 - 128:59
PROFESSOR TODA: On a circle!
• 128:59 - 129:00
And you are getting so close.
• 129:00 - 129:01
It's hot, hot, hot.
• 129:01 - 129:08
On a circle and on a helix they
are the same up to a constant.
• 129:08 - 129:11
So what do you think the
• 129:11 - 129:12
N was what, guys?
• 129:12 - 129:15
N was-- remind me again.
• 129:15 - 129:18
That was T prime over
absolute value of T prime.
• 129:18 - 129:22
But that doesn't mean,
does not equal, in general,
• 129:22 - 129:26
does not equal to
R double prime.
• 129:26 - 129:28
When is it equal?
• 129:28 - 129:30
In general it's not equal.
• 129:30 - 129:31
When is it equal?
• 129:31 - 129:35
If you are in aclength, you
• 129:35 - 129:37
It's wonderful.
• 129:37 - 129:40
In arclength, T is R prime of s.
• 129:40 - 129:46
And in arclength that means T
prime is R double prime of s.
• 129:46 - 129:49
And in arclength
I just told you,
• 129:49 - 129:50
T prime is the first
Frenet formula.
• 129:50 - 129:56
It'll be curvature times the N.
• 129:56 - 130:02
So the acceleration
practically and the N
• 130:02 - 130:07
will be the same in arclength,
up to a scalar multiplication.
• 130:07 - 130:12
is not even constant?
• 130:12 - 130:13
• 130:13 - 130:17
Because the acceleration
R double prime and N
• 130:17 - 130:20
are not colinear.
• 130:20 - 130:24
So if I were to draw-- and
that's my last picture--
• 130:24 - 130:27
let me give you a
wild motion here.
• 130:27 - 130:32
You start slow and then you go
crazy and fast and slow down.
• 130:32 - 130:36
Just like most of the
physical models from the bugs
• 130:36 - 130:39
and the flies and so on.
• 130:39 - 130:45
In that kind of crazy motion you
have a T and N at every point.
• 130:45 - 130:45
[INAUDIBLE]
• 130:45 - 130:48
• 130:48 - 130:51
[? v ?] will be down.
• 130:51 - 130:53
And T is here.
• 130:53 - 130:57
So can you draw arc
double prime for me?
• 130:57 - 130:59
It will still be
towards the inside.
• 130:59 - 131:04
But it's still going to
coincide with N. Maybe this one.
• 131:04 - 131:13
What's the magic thing is
that T, N, and R double prime
• 131:13 - 131:16
are in the same plane always.
• 131:16 - 131:18
That's another
secret other students
• 131:18 - 131:20
don't know in Calculus 3.
• 131:20 - 131:22
That same thing is
called osculating plane.
• 131:22 - 131:26
• 131:26 - 131:31
We have a few magic
names for these things.
• 131:31 - 131:37
So T and N, the plane that
is-- how shall I say that?
• 131:37 - 131:37
I don't know.
• 131:37 - 131:44
The plane given by T and N
is called osculating plane.
• 131:44 - 131:47
• 131:47 - 131:49
The acceleration is
always on that plane.
• 131:49 - 131:52
So imagine T and N are
• 131:52 - 131:56
R double prime is
in the same plane.
• 131:56 - 131:57
OK?
• 131:57 - 131:59
Now, can you guess
the other two names?
• 131:59 - 132:04
So this is T, this
is N. And B is up.
• 132:04 - 132:05
This is my body's direction.
• 132:05 - 132:06
T and N, look at me.
• 132:06 - 132:10
T, N, and B. I'm the
Frenet Trihedron.
• 132:10 - 132:13
Which one is the
osculating plane?
• 132:13 - 132:17
It's the horizontal xy plane.
• 132:17 - 132:21
OK, do you know-- maybe you're
a mechanical engineering major,
• 132:21 - 132:24
and after that I
will let you go.
• 132:24 - 132:26
No extra credit,
• 132:26 - 132:29
Maybe I'm going to start asking
questions and give you \$1.
• 132:29 - 132:31
I used to do that a lot
in differential equations,
• 132:31 - 132:35
whoever gets it first,
• 132:35 - 132:36
give her a dollar.
• 132:36 - 132:42
Until a point when they asked
me to teach Honors 3350 when
• 132:42 - 132:44
I started having three or four
• 132:44 - 132:45
at the same time.
• 132:45 - 132:49
And that was a
significant expense,
• 132:49 - 132:52
because I had to give \$4
away at the same time.
• 132:52 - 132:54
STUDENT: I feel like
you should've just
• 132:54 - 132:55
split it between--
• 132:55 - 132:58
PROFESSOR TODA: So that's
normal and binormal.
• 132:58 - 133:01
This is me, the binormal,
and this is the normal.
• 133:01 - 133:03
Does anybody know the
name of this plane,
• 133:03 - 133:06
between normal and bionormal?
• 133:06 - 133:08
This would be this plane.
• 133:08 - 133:11
STUDENT: The skew [INAUDIBLE].
• 133:11 - 133:12
PROFESSOR TODA:
Normal and binormal.
• 133:12 - 133:14
They call that normal plane.
• 133:14 - 133:17
• 133:17 - 133:23
So it's tricky if you are not
a mechanical engineering major.
• 133:23 - 133:28
But some of you are maybe
and will learn that later.
• 133:28 - 133:30
Any other questions for me?
• 133:30 - 133:34
Now, in my office I'm
going to do review.
• 133:34 - 133:38
I was wondering
if you have time,
• 133:38 - 133:40
I don't know if you have
time to come to my office,
• 133:40 - 133:43
but should you have any kind
of homework related question,
• 133:43 - 133:46
I'll be very happy
• 133:46 - 133:49
3:00 to 5:00.
• 133:49 - 133:51
Now, one time I
• 133:51 - 133:53
• 133:53 - 133:56
He came to my office and
he left with no homework.
• 133:56 - 133:58
We finished all of them.
• 133:58 - 133:58
And I felt guilty.
• 133:58 - 134:01
But at the same, he
said, well, no, it's
• 134:01 - 134:03
better I came to you instead
of going to my tutor.
• 134:03 - 134:05
It was fine.
• 134:05 - 134:09
So we can try some
problems together today
• 134:09 - 134:12
if you want between 3:00 and
5:00, if you have the time.
• 134:12 - 134:14
Some of you don't have the time.
• 134:14 - 134:15
All right?
• 134:15 - 134:16
If you don't have
the time today,
• 134:16 - 134:19
and you would like to
be helped [INAUDIBLE],
• 134:19 - 134:21
click Email Instructor.
• 134:21 - 134:24
I'm going to get the
questions [INAUDIBLE].
• 134:24 - 134:26
You're welcome to
• 134:26 - 134:27
at any time over there.
• 134:27 - 134:38
• 134:38 - 134:41
[CLASSROOM CHATTER]
• 134:41 - 135:12
• 135:12 - 135:14
PROFESSOR TODA: I have
somebody who's taking notes.
• 135:14 - 135:15
STUDENT: Yeah, I know.
• 135:15 - 135:16
And that's why I was like--
• 135:16 - 135:17
PROFESSOR TODA: He's
going to make a copy
• 135:17 - 135:18
and I'll give you a copy.
• 135:18 - 135:19
STUDENT: Yeah.
• 135:19 - 135:24
My Cal 1 teacher,
Dr. [INAUDIBLE].
• 135:24 - 135:24
STUDENT: Thank you.
• 135:24 - 135:25
PROFESSOR TODA: Yes, yeah.
• 135:25 - 135:27
Have a nice day.
• 135:27 - 135:29
when I don't take notes.
• 135:29 - 135:35
Because he felt like
I was not, I guess--
• 135:35 - 135:37
Title:
TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 3
Description:

Integration of a Vector Fuction, Position Velocity and Acceleration, Frenet Formula

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Video Language:
English
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 3