## TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 2

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PROFESSOR: Plus 1.
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And next would be
between-- this is where
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most people have the problem.
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They thought x is
any real number.
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No-- no, no, no, no, no.
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You wanted a segment.
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x has the values
between this value,
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whatever value's on this
axis and that value.
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So x equals 1, x equals
2 are the end points.
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How do you write a
parameterized equation?
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very much on the web work
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homework on that problem
for such a function.
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Well, you say, wait a minute.
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Magdalena, this is
a linear function.
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It's a piece of cake.
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I have just x plus 1.
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I know how to deal with that.
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you something else.
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Rather than writing
the explicit equation
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in Cartesian coordinates x and
y, tell me what time it is.
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And then I'm going
to travel in time.
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I want to travel in time, in
space-time, on the segment,
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right?
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So why if x equals
x plus 1 has what
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is that-- what
parameterization has infinitely
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many parameterization?
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Somebody will say, ha, you told
us that it has infinitely many.
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Why do you insist on one?
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Which one is the most natural
and the easiest to grasp?
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STUDENT: Zero to one.
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PROFESSOR: Zero to one is
not a parameterization.
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STUDENT: Times zero one.
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PROFESSOR: So, so, so what
is the parametric equation
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of a curve in general?
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If I have a curve, y equals--
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X equals x of t
and y equals y of t
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represent the two
parametric questions that
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give that curve's
equation in plane--
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in plane where
the i of t belongs
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to a certain interval i.
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That's the mysterious interval.
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I don't really care
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In my case, which one is the
most natural parametrization,
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guys?
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Take x to be time.
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Say again, Magdalena.
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Take x to be time.
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And that will make
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I take x to be time.
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And then y would be time plus 1.
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And I'm happy.
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So the way they asked you to
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was as r of t equals-- and
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interactive field for you.
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You say, OK, t?
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T what?
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And I'm not going to
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Why?
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Because r of t, which is the
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or curve would give you the
position vector, which is what?
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Wait a second.
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Let me finish. x of t
times i plus y of t times
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j is the definition
I gave last time.
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STUDENT: Where'd you get
r of t and what is it?
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discussed it last time.
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So since I'm
reviewing today, just
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reviewing today
chapter 10, I really
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don't mind going over with you.
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mind this is the first
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and the last time I'm
going to review things
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with you last time.
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So what did you say a position
vector is for a curve?
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the drunken bug,
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we say the drunken bug is
following a trajectory.
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He or she is struggling in time.
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I have a given frame xyz
system of coordinates-- system
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of axes of coordinates
with a certain origin.
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Thank God for this origin
because you cannot refer
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to a position vector
unless you have a frame--
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an original frame, a position
frame, initial frame.
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So r of t represents the vector
that originates at the origin o
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and ends exactly at the position
of your particle at time t.
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If you want, if
you hate bugs, this
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is just the particle from
physics that travels in time t.
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So--
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STUDENT: OK, so the r of t
is represented in the parent
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equation
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PROFESSOR: Yes, sir.
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Exactly.
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In a plane where z
is 0-- so you imagine
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the z-axis coming at z0.
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This is the xy plane.
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And I'm very happy
I have on the floor.
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This bug is on the floor.
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He doesn't want to know
what's the dimension.
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So what's he going to do?
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He's going to say plus 0 times
k that I don't care about
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because the position
vector will be given by--
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STUDENT: So--
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PROFESSOR: --or
for a plane curve.
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STUDENT: So if this
was in 3D space
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so it was like z equals--
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is equal to 2y plus x plus 1,
then it would be-- then how
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would we do that?
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PROFESSOR: Let me remind us in
general the way I pointed it
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out last.
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R of t in general as
a position vector,
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we said many things about it.
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We said it is a smooth function.
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What does it mean
differential role
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with derivative continuous?
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What did-- actually, that's c1.
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What else did they say?
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He said it's a regular.
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It's a regular vector function.
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What does it mean?
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It never stops, not
even for a second.
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Well, the velocity
of that is zero.
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When we introduced
it-- all right,
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I cannot teach the whole
thing all over again,
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but I'll be happy to
do review just today.
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It's going to be x of ti
plus y of tj plus z over k.
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That is a way to
write it like that.
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Or the simpler way to write
it as x of t, y of t, z of t.
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Now, if it involves
using different notation,
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I want to warn you about that.
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Some people like to put
braces like angular brackets.
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Or some people like
because it's a vector.
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And that's the way they define
vector Some people like just
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round parentheses.
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This is more practically.
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These are the coordinates
of a position vector
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with respect to the ijk frame.
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So since we talked
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some simple examples
have been given.
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One of them was
a circling plane,
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another circling plane
of a different speed,
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a segment of a line.
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This is the segment of a line.
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What else have we discussed?
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something wilder,
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which was the helix
at different speeds?
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All right, so very good
question for him was-- so
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is this x of tt?
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Yes.
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Is this y of tt plus 1?
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Yes.
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Is this z of t 0 in my case?
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Precisely
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STUDENT: So if you
gave value to z,
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what would you chose to
make t parameterized?
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PROFESSOR: OK, t in
general, if you are moving,
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you have an infinite motion
that comes from nowhere,
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goes nowhere, right?
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OK, then you can say
t is between minus
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infinity plus infinity.
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STUDENT: But what I'm saying--
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PROFESSOR: But-- but in
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you think oh, I know
where I'm starting.
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So to that equals
to 1, t must be 1.
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So I start my
movement at 1 second
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and I end my movement at 2
seconds where x will be 2,
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and y will be 3.
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STUDENT: Well, I mean--
so you said x equals t.
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You took that from
the y equals x plus 1.
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variable t, what would you--
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PROFESSOR: It's not
a third variable.
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It's the time parameter.
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So I work in three
variables-- xyz in space.
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Those are my space coordinates.
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The space coordinates
are function of time.
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So mathematics sometimes
becomes physics.
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Thank God we are sisters,
even step-sisters.
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X is a function of t.
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Y is a function of t.
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Z is a function of t.
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Right?
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question or maybe
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I didn't quite understand the--
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STUDENT: Well, I understand
how to parameterize
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the idea of a plane.
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How do you do it
in space though?
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PROFESSOR: In space-- in
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So if you want to ride this
not in plane but in space,
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ti plus t plus 1j plus 0k,
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for this example,
anywhere in r3.
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We live in r3.
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All righty?
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We live in r3.
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OK, let me give
you more examples.
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Because I think I'm
running out of time.
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But I still have to
cover the material,
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eventually get somewhere.
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However, I want you to see
more examples that will help
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you grasp this notion better.
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So guys, imagine that
we have space r3-- that
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could be rn-- in
which I have an origin
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and I have a [INAUDIBLE].
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And somebody gives
me a position vector
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for a motion that's
a regular curve.
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And that's x of tri plus
y is tj plus z of tk.
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And since his question
is a very valid one,
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let's see what happens
in a later case.
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So I'm going to deviate a
little from my lesson plan.
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And I say let us be
flexible and compare
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that with the inner curve.
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Because in the
process of comparison,
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you learn a lot more.
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If I were to be right above
my [INAUDIBLE] like that.
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So this is the spacial curve in
our three imaginary trajectory
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run of a bug or a particle.
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As we said, this is the
planar curve-- planar,
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parametrized curve in r2.
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What's different?
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What do we know about them?
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We clearly know section 10.2.
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What I hate in general
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is if they are way
too structured.
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Mathematics cannot be talking
sections where you say, oh,
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velocity and acceleration.
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But section 10.4 is
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and principle normal.
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Well, they are related.
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So it's only natural when
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acceleration and velocity
that from acceleration, you
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have a induced line to tangent
unit vector-- tangent unit
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vector.
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And later on, you're going
to compare acceleration
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with a normal principal vector.
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Sometimes, they
are the same thing.
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Sometimes, they are
not the same thing.
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It's a good idea to see
when they are the same thing
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and when they are not.
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So in section 10.4, we
will focus practically
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or t, n, and v, the Frenet
frame and its consequences
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talked about that a little bit.
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In 10.2, practically,
we didn't cover much.
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velocity, acceleration.
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However, I would like
to review that for you.
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Because I don't want
to risk losing you.
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I'm going to lose
some of you anyway.
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Two people said this
course is too hard for me.
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I'm going to drop.
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You are free to drop and I
think it's better for you
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to drop than struggle.
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But as long as you can still
learn and you can follow,
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you shouldn't drop.
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So try to see exactly
how much you can handle.
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If you can handle just the
regular section of calc three,
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go to that regular section.
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If you can handle more, if
you are good at mathematics,
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if you have always
been considered bright
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in mathematics in high
school, let us stay here.
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Otherwise, go.
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Don't stay.
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All right, so the
velocities are prime of t.
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The acceleration is
our double prime of t.
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We have done that last time.
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We were very happy.
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What would happen in a
planar curve seen on 2?
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The same thing, of course,
except the last component
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is not there.
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It's part of ti
plus y prime of tj.
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And there is a 0k in both cases.
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So all these are factors.
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At times, I'm not going
to point that out anymore.
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The derivation goes
component-wise.
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So if you forgot how to derive
or you want to drink and derive
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or something, then you
don't belong in this class.
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So again, make sure you know
the derivations and integrations
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really well.
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I'm going to work
some examples out just
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But if you have struggled with
differentiation and integration
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in Calc 1, then you do not
do belong in this class.
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All right, let's
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what the speed was.
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The speed was the absolute
value or the magnitude.
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It's not an absolute
value, but it's a magnitude
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of the velocity factor.
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This is the speed.
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And the same in this case.
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If I want to write an explicit
formula because somebody
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asked me by email, can I write
an explicit formula, of course.
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That's a piece of cake and you
should know that from before.
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X prime of t squared plus
y prime of t squared plus z
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prime of t squared
under the square root.
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I was not going to insist
on the planar curve.
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Of course the planar curve will
have a speed that all of you
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And that's going to be
square root of x prime of t
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squared plus y root
prime of t squared.
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You should do your own thinking
to see what the particular case
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will become.
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However, I want to
see if you understood
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what derives from
that in the sense
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that you should know the
length of a arc of a curve.
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What is the length
of an arc of a curve?
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Well, we have to look back
at Calculus 2 a little bit
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and remember that the length of
an arc of a curve in Calculus 2
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was given by, what?
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So you say, well, yeah.
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That was a long time ago.
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Well, some of you
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remember that as being integral
from a to b of square root of 1
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plus 1 prime of x squared dx.
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And you were freaking
out thinking, oh my god,
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I don't see how this
formula from Calc 2,
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the arc of a curve, had
you travel between time
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equals a and time equals b
will relate to this formula.
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So what happened in Calc 2?
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In Calc 2, hopefully,
you have a good teacher.
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And hopefully,
you've learned a lot.
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This is between a and b, right?
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What did they teach
you in Calc 2?
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They taught you that
you have to take
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integral from a to b
of square root of 1
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plus y prime of x squared ds.
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Why?
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• 16:24 - 16:24
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You should do that.
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You should ask why every time.
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They make you swallow a
formula via memorization
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without understanding
this is the speed.
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And now I'm coming
with the good news.
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I have a proof of that.
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I know what speed
means when I'm moving
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along the arc of
a curve in plane.
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OK, so what is the distance
travelled between time equals A
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and time equals B?
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It's going to be integral form
a to be of the speed, right?
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This is the same one I'm
driving from-- level two--
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Amarillo or anywhere else.
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There.
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Now, what they showed
you and they fooled you
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into memorizing that is just
a consequence of this formula
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because of what he said.
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Why?
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The most usual
parameterization is
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going to be y of t equals t--
I'm sorry, x of t equals vxst
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and y of t equals y of t.
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So, again x is time.
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In many linear curves, you can
take x to be time, thank God.
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will be t comma y of t.
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Because x is t.
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And x prime of t will be 1.
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Y prime of t will
be y prime of t.
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When you take them, squish
them, square them, sum them up,
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you get exactly this one.
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But you notice
this is the speed.
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What is this the speed?
• 17:56 - 18:03
Of some value over prime
of t, which is speed.
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You see that what they forced
you to memorize in Calc 2
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is nothing but the speed.
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And I could change
the parameterization
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to something more general.
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Now, can I do this
parameterization for a circle?
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No.
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Why not?
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I could, but then
I'd have to split
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into the upper
part and lower part
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because the circle
is not a graph.
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So I take t between
this and that
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and then I have square root
of 1 minus t squared on top.
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And underneath, I have
minus square root of 1
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minus t squared.
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So I split the poor circle
into a graph and another graph.
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And I do it separately.
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And I can still apply that.
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But only a fool
would do that, right?
• 18:49 - 18:53
So what does a smart
mathematician do?
• 18:53 - 18:55
A smart mathematician
will say, OK,
• 18:55 - 19:00
for the circle, x is
cosine t, y is sine t.
• 19:00 - 19:02
And that is the
parameterization I'm
• 19:02 - 19:04
going to use for this formula.
• 19:04 - 19:06
And I get speed 1.
• 19:06 - 19:09
And I'm going to
be happy, right?
• 19:09 - 19:11
So it's a lot easier
to understand what
• 19:11 - 19:13
a general parameterization is.
• 19:13 - 19:19
What is the length of an arc
of a curve for a curving space?
• 19:19 - 19:21
There's the bug.
• 19:21 - 19:22
Time equals t0.
• 19:22 - 19:24
He's buzzing.
• 19:24 - 19:26
And after 10 seconds,
he will be at the end.
• 19:26 - 19:31
So it goes, [BUZZING] jump.
• 19:31 - 19:35
OK, how much did he travel?
• 19:35 - 19:42
Integral from a to b of square
root of x prime of t squared
• 19:42 - 19:44
plus y prime of t
squared plus z prime of t
• 19:44 - 19:50
squared-- no matter what that
position vector x of ty of t0
• 19:50 - 19:51
give us.
• 19:51 - 19:56
So you take the coordinates
of the velocity vector.
• 19:56 - 19:57
You look at them.
• 19:57 - 19:58
You square them.
• 19:58 - 19:59
• 19:59 - 20:01
You put them under
the square root.
• 20:01 - 20:03
That's going to be the speed.
• 20:03 - 20:06
And displacement is
integral of speed.
• 20:06 - 20:09
When you guys learned
• 20:09 - 20:11
oversimplified the things.
• 20:11 - 20:13
say in physics?
• 20:13 - 20:16
Space equals speed times time.
• 20:16 - 20:17
Say it again.
• 20:17 - 20:20
He said space traveled
is speed times time.
• 20:20 - 20:24
But he assumed the speed
is constant or constant
• 20:24 - 20:27
on portions-- like,
speedswise constant.
• 20:27 - 20:29
Well, if it's a
constant, the speed
• 20:29 - 20:31
will get the heck out of here.
• 20:31 - 20:35
And then the space will
be speed times b minus a.
• 20:35 - 20:38
But b minus a is delta t.
• 20:38 - 20:41
In mathematics, in physics,
we say b minus a is delta t.
• 20:41 - 20:45
That's the interval of time that
the bug travels or the particle
• 20:45 - 20:46
travels.
• 20:46 - 20:48
So he or she was right.
• 20:48 - 20:51
Space is speed times
time, but it's not like
• 20:51 - 20:54
that unless the
speed is constant.
• 20:54 - 20:56
So he oversimplified
• 20:56 - 20:58
of mathematics and physics.
• 20:58 - 20:59
Now you see the truth.
• 20:59 - 21:04
Space is integral of speed.
• 21:04 - 21:06
OK, now we understand.
• 21:06 - 21:10
And I promised you last
time that after reviewing,
• 21:10 - 21:14
I didn't even say I would review
anything from 10.2 and 10.4.
• 21:14 - 21:15
I promised you more.
• 21:15 - 21:18
I promised you that I'm going
to compute something that's
• 21:18 - 21:24
out of 10.4 which is called
a curvature of a helix
• 21:24 - 21:25
in particular.
• 21:25 - 21:30
Because we looked at curvature
of a parametric curve
• 21:30 - 21:31
in general.
• 21:31 - 21:37
I want to organize the material
of review from 10.2 and 10.4
• 21:37 - 21:40
in a big problem just like
you will have in the exams,
• 21:40 - 21:42
in the midterm,
and in the final.
• 21:42 - 21:43
I don't want to scare you.
• 21:43 - 21:46
I just want to
prepare you better
• 21:46 - 21:50
for the kind of multiple
questions we are going to have.
• 21:50 - 21:55
So let me give you a
funny looking curve.
• 21:55 - 21:59
I want you to think about
it and tell me what it is.
• 21:59 - 22:02
a and b are positive numbers.
• 22:02 - 22:07
a cosine ba sine t bt will be
some sort of funny trajectory.
• 22:07 - 22:10
familiar to that.
• 22:10 - 22:13
Last time, I gave you an example
where a was 4-- oh my god,
• 22:13 - 22:15
I don't even remember.
• 22:15 - 22:16
You'll need to help me.
• 22:16 - 22:19
[INAUDIBLE]
• 22:19 - 22:21
STUDENT: 4, 4, 3.
• 22:21 - 22:25
PROFESSOR: I took those because
they are Pythagorean numbers.
• 22:25 - 22:26
So what does it mean?
• 22:26 - 22:29
3 squared plus 4 squared
equals 5 squared.
• 22:29 - 22:32
I wanted the sum of them
to be a perfect square.
• 22:32 - 22:33
So I was playing games.
• 22:33 - 22:37
You can do that for any a and b.
• 22:37 - 22:38
Now, what do I want?
• 22:38 - 22:44
A-- like in 10.2 where
you write r prime of t,
• 22:44 - 22:47
rewrite that double prime of t.
• 22:47 - 22:50
So it's a complex problem.
• 22:50 - 22:53
In b, I want you to
find t and r prime
• 22:53 - 22:56
of t over-- who
remembers the formula?
• 22:56 - 22:58
I shouldn't have
spoon-fed you that.
• 22:58 - 22:59
STUDENT: Absolute--
• 22:59 - 23:01
PROFESSOR: Absolute
magnitude, actually.
• 23:01 - 23:04
It's more correct to
say magnitude, right?
• 23:04 - 23:04
Very good.
• 23:04 - 23:09
And what else did I
spoon-feed you last name?
• 23:09 - 23:10
I spoon-fed you n.
• 23:10 - 23:14
Let's compute n as well
as part of the problem
• 23:14 - 23:21
t prime t over t
prime of t magnitude.
• 23:21 - 23:24
STUDENT: So you're looking
for the tangent unit vector.
• 23:24 - 23:25
PROFESSOR: Tangent unit vector?
• 23:25 - 23:27
STUDENT: And then
you're looking for--
• 23:27 - 23:28
PROFESSOR: Yes, sir.
• 23:28 - 23:31
And-- OK, don't you
like me to also give you
• 23:31 - 23:34
grid, how much everything
• 23:34 - 23:35
would be worth.
• 23:35 - 23:37
Imagine you're taking an exam.
• 23:37 - 23:40
Why not put yourself
in an exam mode
• 23:40 - 23:44
so you don't freak out
during the actual exam?
• 23:44 - 23:48
C will be another
question, something smart.
• 23:48 - 24:02
Let's see-- reparameterize an
arc length to a plane, a curve,
• 24:02 - 24:05
rho of s.
• 24:05 - 24:09
Why not r of s like some
people call-- use it
• 24:09 - 24:10
and some books use it?
• 24:10 - 24:12
Because if you're
reparameterizing s,
• 24:12 - 24:13
it's going to be the
same physical limits
• 24:13 - 24:16
but a different function.
• 24:16 - 24:20
So if you remember the
diagram I wrote before,
• 24:20 - 24:24
little r is a function that
comes from integral i time
• 24:24 - 24:29
integral 2r3 and rho would
be coming from a j to r3.
• 24:29 - 24:33
And what is the
relationship between them?
• 24:33 - 24:36
This is t goes to s and
this is s goes to t.
• 24:36 - 24:39
What is d I'm asking you?
• 24:39 - 24:41
Well, if you're d
and c, of course
• 24:41 - 24:45
you know what the arc
length parameter will be.
• 24:45 - 24:50
It's going to be integral
from 0 to t or any t0 here
• 24:50 - 24:55
of the speed-- of the speed
of the original function here
• 24:55 - 24:56
of t.
• 24:56 - 25:02
The tau-- maybe tau is better
than the dummy variable t.
• 25:02 - 25:05
And e I want.
• 25:05 - 25:07
You say, how much
more do you want?
• 25:07 - 25:08
I want a lot.
• 25:08 - 25:09
I'm a greedy person.
• 25:09 - 25:14
I want the curvature
of the curve.
• 25:14 - 25:18
And you have to remind me.
• 25:18 - 25:20
Some of you are very good
students, better than me.
• 25:20 - 25:24
I mean, I'm still behind
with a research course
• 25:24 - 25:25
that I have--
research paper i have
• 25:25 - 25:30
to read in two days in biology.
• 25:30 - 25:36
But this curvature of the
curve had a very simple formula
• 25:36 - 25:37
that we all love.
• 25:37 - 25:40
For mathematicians, it's a
piece of cake to remember it.
• 25:40 - 25:43
K-- that's what I like
• 25:43 - 25:45
I don't need a good memory.
• 25:45 - 25:48
Now I remember why I didn't
go to medical school--
• 25:48 - 25:51
because my father
told me, well, you
• 25:51 - 25:54
should be able to remember all
the bones in a person's body.
• 25:54 - 25:56
And I said, dad, do you
know all these names?
• 25:56 - 25:56
Yes, of course.
• 25:56 - 25:57
And he started telling me.
• 25:57 - 26:01
Well, I realized that I
would never remember those.
• 26:01 - 26:07
But I remember this
formula which is r rho.
• 26:07 - 26:10
In this case, if
our r is Greek rho,
• 26:10 - 26:13
it's got to be rho
double prime of what?
• 26:13 - 26:16
of S. Is this
correct, what I wrote?
• 26:16 - 26:16
No.
• 26:16 - 26:18
What's missing?
• 26:18 - 26:23
The acceleration and arc length
but in magnitude because that's
• 26:23 - 26:24
a vector, of course.
• 26:24 - 26:27
This is the scalar function.
• 26:27 - 26:29
Anything else you
want, Magdalena?
• 26:29 - 26:30
Oh, that's enough.
• 26:30 - 26:34
All right, so I want
to know everything
• 26:34 - 26:38
this curve from 10.2 and 10.4
• 26:38 - 26:40
sections.
• 26:40 - 26:42
10.3-- skip 10.5.
• 26:42 - 26:44
• 26:44 - 26:45
Yes sir.
• 26:45 - 26:48
STUDENT: For the
parameter on v, is it a t?
• 26:48 - 26:50
And what's the integral?
• 26:50 - 26:51
What's on the bottom.
• 26:51 - 26:54
PROFESSOR: Ah, that value
erased when I wrote that one.
• 26:54 - 26:56
It was there-- t0.
• 26:56 - 27:01
t0 as my initial moment in time.
• 27:01 - 27:03
I would like my
initial moment in time
• 27:03 - 27:06
to be 0 just to make
my things easier.
• 27:06 - 27:08
this problem together?
• 27:08 - 27:12
I think we have just
• 27:12 - 27:14
we need to do everything.
• 27:14 - 27:18
First of all, you have to tell
me what kind of curve this is.
• 27:18 - 27:20
Of course you know because
you were here last time.
• 27:20 - 27:23
Don't skip classes because
you are missing everything out
• 27:23 - 27:25
and then you will have
to drop or withdraw.
• 27:25 - 27:27
So don't skip class.
• 27:27 - 27:31
What was that from last time?
• 27:31 - 27:34
It was a helix.
• 27:34 - 27:35
I'm going to try and redraw it.
• 27:35 - 27:38
I know I'm wasting
my time, but I would
• 27:38 - 27:44
try to draw a better curve.
• 27:44 - 27:46
Ah, what's the equation
of the cylinder?
• 27:46 - 27:50
[CLASS MURMURS]
• 27:50 - 27:51
PROFESSOR: Huh?
• 27:51 - 27:53
What's the equation
of the cylinder?
• 27:53 - 27:56
that you are all
• 27:56 - 28:01
familiar with on which on my
beautiful helix is sitting on.
• 28:01 - 28:03
I taught you the
trick last time.
• 28:03 - 28:04
Don't forget it.
• 28:04 - 28:10
STUDENT: a over 4 cosine of
t squared plus 8 over 4 sine
• 28:10 - 28:11
of t squared.
• 28:11 - 28:14
• 28:14 - 28:16
PROFESSOR: So we do
that-- very good.
• 28:16 - 28:19
X is going to be-- let
me right that down.
• 28:19 - 28:20
X is cosine.
• 28:20 - 28:23
Y is a sine t.
• 28:23 - 28:25
And that's exactly
• 28:25 - 28:26
And z is bt.
• 28:26 - 28:30
And then what I need to do
is square these guys out
• 28:30 - 28:32
as you said very well.
• 28:32 - 28:33
• 28:33 - 28:35
He's not in the picture here.
• 28:35 - 28:39
X squared plus y squared will be
a squared, which means I better
• 28:39 - 28:43
go ahead and draw a circle
of radius a on the bottom
• 28:43 - 28:45
and then build
my-- oh my god, it
• 28:45 - 28:50
looks horrible-- the cylinder
based on that circle.
• 28:50 - 28:51
Guys, it's now straight.
• 28:51 - 28:52
I'm sorry.
• 28:52 - 28:55
I mean, I can do
better than that.
• 28:55 - 28:59
OK, good.
• 28:59 - 29:03
So I'm starting at what point?
• 29:03 - 29:06
I'm starting at a0
0 time t equals 0.
• 29:06 - 29:08
We discussed that last time.
• 29:08 - 29:09
I'm not going to repeat.
• 29:09 - 29:12
I'm starting here,
and two of you
• 29:12 - 29:14
told me that if t
equals phi over two,
• 29:14 - 29:18
I'm going to be here
and so on and so forth.
• 29:18 - 29:22
If I ask you one more thing
for extra credit, what
• 29:22 - 29:31
is the length of the trajectory
traveled by the bug, whatever
• 29:31 - 29:38
that is, between time t equals
0 and time t equals phi over 2.
• 29:38 - 29:40
I'd say that's extra credit.
• 29:40 - 29:52
So, oh my god, 20%, 20%, 20%,
20%, 20%, and 10% for this one.
• 29:52 - 29:57
And if you think why does she
• 29:57 - 29:59
and points, you will
care and I care.
• 29:59 - 30:03
Because I want you to see how
you are going to be assessed.
• 30:03 - 30:05
If you have no idea how
you're going to assessed,
• 30:05 - 30:09
then you're going to be
happy and i will be unhappy.
• 30:09 - 30:12
All right, so for 20%
credit on this problem,
• 30:12 - 30:16
we want to see r prime of t
will be, r double prime of t
• 30:16 - 30:16
will be.
• 30:16 - 30:18
That's going to be
a piece of cake.
• 30:18 - 30:21
And of course, it's maybe the
reward is too big for that,
• 30:21 - 30:23
but that's life.
• 30:23 - 30:32
Minus a sine t a equals time
t and d, d as in infinity.
• 30:32 - 30:34
So I have an infinite
cylinder on which
• 30:34 - 30:37
I draw an infinite
helix coming from hell
• 30:37 - 30:39
• 30:39 - 30:44
So between minus infinity and
plus infinity, there's a guy.
• 30:44 - 30:48
I'm going to draw a
beautiful infinite helix.
• 30:48 - 30:50
And this is what I posted here.
• 30:50 - 30:53
What's the acceleration
of this helix?
• 30:53 - 31:00
Minus a cosine t
minus 5 sine t and 0.
• 31:00 - 31:03
Question, quick
question for you.
• 31:03 - 31:07
Will-- you guys are fast.
• 31:07 - 31:11
Maybe I shouldn't
• 31:11 - 31:15
the speed is right now.
• 31:15 - 31:18
So why would I do something
that's not on the final, right?
• 31:18 - 31:20
So let's see.
• 31:20 - 31:23
T, you will have to compute
the speed when you get to here.
• 31:23 - 31:26
But wait until we get there.
• 31:26 - 31:27
What is mister t?
• 31:27 - 31:30
Mister t will be
the tangent vector.
• 31:30 - 31:35
So the velocity is going like
a crazy guy, long vector.
• 31:35 - 31:39
The normal unit vector says,
I'm the tangent unit vector.
• 31:39 - 31:43
I'm always perpendicular
to the direction.
• 31:43 - 31:44
I'm of length 1.
• 31:44 - 31:47
STUDENT: I thought the tangent
was parallel to the direction.
• 31:47 - 31:49
PROFESSOR: Yes, the
direction of motion is this.
• 31:49 - 31:51
Look at me.
• 31:51 - 31:53
This is my direction of motion.
• 31:53 - 31:54
And the tangent is--
• 31:54 - 31:55
STUDENT: You said it was--
• 31:55 - 31:57
PROFESSOR: --in the
direction of motion.
• 31:57 - 31:58
STUDENT: But you said
it was perpendicular.
• 31:58 - 31:59
PROFESSOR: I said perpendicular?
• 31:59 - 32:03
Because I was thinking
• 32:03 - 32:04
And I apologize.
• 32:04 - 32:06
So thank you for correcting me.
• 32:06 - 32:08
So t is the tangent unit vector.
• 32:08 - 32:13
• 32:13 - 32:15
I'm going along the
direction of motion.
• 32:15 - 32:18
And it's going to be
perpendicular to t.
• 32:18 - 32:22
And that's the principal
normal unit vector--
• 32:22 - 32:24
principal normal unit vector.
• 32:24 - 32:27
And you're going to tell
me what I'm having here.
• 32:27 - 32:28
Because I don't know.
• 32:28 - 32:31
• 32:31 - 32:37
T is minus a sine
t a equals sine t
• 32:37 - 32:41
and v divided by the speed.
• 32:41 - 32:44
That's why I was
• 32:44 - 32:47
you'll need later on anyway.
• 32:47 - 32:50
need it here, right?
• 32:50 - 32:55
Because the denominator of this
expression will be the speed.
• 32:55 - 32:58
Magnitude of r
prime-- what is that?
• 32:58 - 33:01
Piece of cake--
square root of the sum
• 33:01 - 33:06
of the squares of square root
of a squared plus b squared.
• 33:06 - 33:06
Piece of cake.
• 33:06 - 33:07
I love it.
• 33:07 - 33:09
So what do I notice?
• 33:09 - 33:12
That although I'm going
on a funny curve which
• 33:12 - 33:16
is a parametrized helix,
I expect some-- maybe
• 33:16 - 33:18
I expected something
wild in terms of speed.
• 33:18 - 33:20
Well, the speed is constant.
• 33:20 - 33:27
STUDENT: [INAUDIBLE] the square
root of negative a sine t
• 33:27 - 33:27
squared--
• 33:27 - 33:30
PROFESSOR: And what are those?
• 33:30 - 33:33
A squared sine squared plus c
squared cosine squared plus b
• 33:33 - 33:36
squared, right?
• 33:36 - 33:37
And what sine squared
plus cosine squared
• 33:37 - 33:38
is 1 [INAUDIBLE].
• 33:38 - 33:41
So you get a squared
plus b squared.
• 33:41 - 33:46
Good-- now let's
go on and do the n.
• 33:46 - 33:53
The n will be t prime
over magnitude of t prime.
• 33:53 - 33:56
When you do t prime,
you'll say, wait a minute.
• 33:56 - 34:00
I have square root of a squared
plus b squared on the bottom.
• 34:00 - 34:05
On the top, I have minus equals
sine t minus a sine t and 0.
• 34:05 - 34:06
We have time to finish?
• 34:06 - 34:07
I think.
• 34:07 - 34:09
I hope so.
• 34:09 - 34:18
Divided by-- divided by the
magnitude of this fellow.
• 34:18 - 34:21
I will say, oh, wait a minute.
• 34:21 - 34:24
The magnitude of this fellow
is simply the magnitude
• 34:24 - 34:26
of this over this magnitude.
• 34:26 - 34:30
• 34:30 - 34:34
And we've seen last time this is
the magnitude of this vector a,
• 34:34 - 34:35
right?
• 34:35 - 34:36
Good.
• 34:36 - 34:39
be n is going to be a unit
• 34:39 - 34:42
vector, very nice friend
of yours, minus cosine t
• 34:42 - 34:44
minus sine t0.
• 34:44 - 34:50
Can you draw a conclusion about
how I should draw this vector?
• 34:50 - 34:52
You see the component in k is 0.
• 34:52 - 34:56
So this vector
cannot be like that--
• 34:56 - 34:58
cannot be inclined with
respect to the horizontal.
• 34:58 - 34:58
Yes sir.
• 34:58 - 35:00
STUDENT: So what happens
to-- down there-- square root
• 35:00 - 35:02
of a squared plus b squared?
• 35:02 - 35:03
PROFESSOR: They simplify.
• 35:03 - 35:04
This is division.
• 35:04 - 35:05
STUDENT: Oh, OK.
• 35:05 - 35:08
PROFESSOR: So this simplifies
with that and a simplifies
• 35:08 - 35:10
with a.
• 35:10 - 35:12
I should leave some
things as an exercise,
• 35:12 - 35:16
but this is an obvious one so I
don't have to explain anything.
• 35:16 - 35:19
Minus cosine t
minus sine t-- if do
• 35:19 - 35:22
you guys imagine what that is?
• 35:22 - 35:26
• 35:26 - 35:32
So if you have an acceleration
that's pointing inside
• 35:32 - 35:36
like from a centrifugal force,
the corresponding acceleration
• 35:36 - 35:39
would go pointing
inside, not outside.
• 35:39 - 35:44
That's going to be exactly
minus cosine t minus sine t0.
• 35:44 - 35:48
So the way I should draw the
n would not be just any n,
• 35:48 - 35:53
but should be at every
point a beautiful vector
• 35:53 - 35:56
that's horizontal and is
moving along the helix.
• 35:56 - 35:58
My elbow is moving
along the helix.
• 35:58 - 35:59
See my elbow?
• 35:59 - 36:00
Where's my elbow moving?
• 36:00 - 36:01
I'm trying.
• 36:01 - 36:03
I swear, I won't do it that way.
• 36:03 - 36:07
So this is the helix and this
is the acceleration, which
• 36:07 - 36:13
is acceleration and the normal
unit vector in this case
• 36:13 - 36:13
are co-linear.
• 36:13 - 36:15
They are not
co-linear in general.
• 36:15 - 36:19
But if the speed is a
constant, they are co-linear.
• 36:19 - 36:21
The n and the acceleration.
• 36:21 - 36:21
Yes, sir?
• 36:21 - 36:25
STUDENT: How do you know it's
pointing in the central axis?
• 36:25 - 36:26
I thought it was--
• 36:26 - 36:27
PROFESSOR: Good question.
• 36:27 - 36:28
Good question.
• 36:28 - 36:29
Well, yeah.
• 36:29 - 36:30
Let's see now.
• 36:30 - 36:31
Plug in t equals 0.
• 36:31 - 36:32
What do you have?
• 36:32 - 36:36
Minus cosine 0 minus 1 0, 0.
• 36:36 - 36:40
So you guys would have to
draw the vector minus 1, 0, 0.
• 36:40 - 36:42
That's minus i, right?
• 36:42 - 36:48
So when I start here, this
is my n-- from here to here,
• 36:48 - 36:51
from the particle to the insid.
• 36:51 - 36:53
So I go on that.
• 36:53 - 36:55
All right, so this is the
normal principal vector.
• 36:55 - 36:57
• 36:57 - 37:00
STUDENT: Isn't the normal
principal vector is the-- is it
• 37:00 - 37:01
the derivative of
t, or is just--
• 37:01 - 37:03
PROFESSOR: It was
by definition--
• 37:03 - 37:05
it's in your notes-- t prime
over the magnitude of the--
• 37:05 - 37:09
STUDENT: So then did
you-- why didn't you
• 37:09 - 37:11
take a derivative of t prime?
• 37:11 - 37:12
PROFESSOR: I did.
• 37:12 - 37:13
STUDENT: Yeah, I know.
• 37:13 - 37:16
I see you took a
derivative of t of--
• 37:16 - 37:19
PROFESSOR: This is t prime.
• 37:19 - 37:20
STUDENT: OK.
• 37:20 - 37:24
PROFESSOR: And this is
magnitude of t prime.
• 37:24 - 37:26
Why don't you try
this at home, like,
• 37:26 - 37:30
slowly until you're sure
this is what yo got?
• 37:30 - 37:32
So I did-- I did
the derivative of i.
• 37:32 - 37:34
STUDENT: I saw that.
• 37:34 - 37:36
PROFESSOR: This
is a [INAUDIBLE].
• 37:36 - 37:38
STUDENT: You said you were--
• 37:38 - 37:40
PROFESSOR: So when we
have t times a function
• 37:40 - 37:43
and we prime the
product, k goes out.
• 37:43 - 37:45
Lucky for us--
imagine how life would
• 37:45 - 37:47
be if it weren't like that.
• 37:47 - 37:49
So the constant
that falls out is
• 37:49 - 37:52
1 over square root
of what I derived.
• 37:52 - 37:56
And then I have to derive
this whole function also.
• 37:56 - 37:59
So I would suggest to
everybody, not just to yo--
• 37:59 - 38:02
go home and see if
you can redo this
• 38:02 - 38:03
• 38:03 - 38:05
Close the damn notes.
• 38:05 - 38:09
Open and then you look at--
it's line by line, line by line
• 38:09 - 38:11
all the derivations.
• 38:11 - 38:14
Because you guys will have to
do that yourselves in the exam,
• 38:14 - 38:17
either midterm or final anyway.
• 38:17 - 38:24
Reparameterizing arc lengths
to obtain a curve-- I
• 38:24 - 38:26
still have that to
finish the problem.
• 38:26 - 38:32
Reparameterizing arc length
to obtain a curve rho of s.
• 38:32 - 38:33
How do we do that?
• 38:33 - 38:34
Who is s?
• 38:34 - 38:37
First of all, you should
• 38:37 - 38:39
reparameterize.
• 38:39 - 38:40
So you say, hey, teacher.
• 38:40 - 38:42
You try to fool me, right?
• 38:42 - 38:46
I want s to be grabbed
as a parameter first.
• 38:46 - 38:50
And then I will reparameterize
the way you want me to do that.
• 38:50 - 38:53
So s of t will be
integral from 0
• 38:53 - 38:56
to t square root of a
prime a squared times
• 38:56 - 39:00
b squared b tau-- d tau, yes.
• 39:00 - 39:01
S of t will be, what?
• 39:01 - 39:03
Who's helping me on that?
• 39:03 - 39:05
Because I want you to be awake.
• 39:05 - 39:06
Are you guys awake?
• 39:06 - 39:07
[CLASS MURMURS]
• 39:07 - 39:09
PROFESSOR: The
square root of that
• 39:09 - 39:14
is a constant gets out times t.
• 39:14 - 39:19
So what did I tell you when
it comes to these functions?
• 39:19 - 39:22
I have to turn my
• 39:22 - 39:24
This was the alpha t or s of t.
• 39:24 - 39:32
And this was t of s, which
is the inverse function.
• 39:32 - 39:33
I'm not going to
write anything stupid.
• 39:33 - 39:37
But this is practically the
inverse function of s of t.
• 39:37 - 39:39
I told you it was easiest t do.
• 39:39 - 39:40
Put it here.
• 39:40 - 39:43
T has to be replaced
by, in terms of s,
• 39:43 - 39:46
by a certain expression.
• 39:46 - 39:48
So who is t?
• 39:48 - 39:52
And you will do that
in no time in the exam.
• 39:52 - 39:56
T pulled out from
there will be just
• 39:56 - 40:01
s over square root a
squared plus b squared
• 40:01 - 40:05
s over square root a
squared plus b squared
• 40:05 - 40:09
and s over square root.
• 40:09 - 40:11
OK?
• 40:11 - 40:13
So can I keep the
notation out of s?
• 40:13 - 40:15
No.
• 40:15 - 40:19
It's not mathematically
correct to keep r of s.
• 40:19 - 40:21
Why do the books
sometimes by using
• 40:21 - 40:23
multiplication keep r of s?
• 40:23 - 40:27
Because the books are
not always rigorous.
• 40:27 - 40:29
But I'm trying to be rigorous.
• 40:29 - 40:31
This is an honors class.
• 40:31 - 40:35
So How do I rewrite
the whole thing?
• 40:35 - 40:40
r of t, who is a function
of s, t as a function of s
• 40:40 - 40:46
was again s over square root
a squared plus b squared
• 40:46 - 40:49
will be renamed rho of s.
• 40:49 - 40:51
And what is that?
• 40:51 - 40:55
That is a of cosine
of parentheses
• 40:55 - 41:01
s over square root a
squared r b squared, comma,
• 41:01 - 41:06
a sine of s over square root
a squared plus b squared
• 41:06 - 41:13
and b times s over square
root a squared plus b squared.
• 41:13 - 41:15
So what have I done?
• 41:15 - 41:16
Did I get my 20%?
• 41:16 - 41:17
Yes.
• 41:17 - 41:17
Why?
• 41:17 - 41:19
Because I reparameterized
the curve.
• 41:19 - 41:22
Did I get my other 20%?
• 41:22 - 41:26
Yes, because I told
people who s of t was.
• 41:26 - 41:33
So 20% for this box and
20% for this expression.
• 41:33 - 41:35
So what have I done?
• 41:35 - 41:39
On the same physical curve, I
have slowed down, thank God.
• 41:39 - 41:42
You say, finally, she's
slowing down, right?
• 41:42 - 41:43
I've changed this speed.
• 41:43 - 41:46
• 41:46 - 41:51
On the contrary, if a would
be 4 and be would be 3,
• 41:51 - 41:56
I increase my speed
multiple five times, right?
• 41:56 - 42:00
So you can go back and
forth between s and t.
• 42:00 - 42:03
What does s do compared to t?
• 42:03 - 42:05
It increases the
speed five times.
• 42:05 - 42:05
Yes sir.
• 42:05 - 42:07
STUDENT: So when
you reparameterize,
• 42:07 - 42:09
it's just basically the
integral from 0 to t
• 42:09 - 42:12
of whatever
[INAUDIBLE] of tau is.
• 42:12 - 42:14
PROFESSOR: Exactly.
• 42:14 - 42:19
So my suggestion to all
of you-- it took me a year
• 42:19 - 42:21
to understand how
to reparameterize
• 42:21 - 42:25
because I was not smart enough
to get it as a freshman.
• 42:25 - 42:26
I got an A in that class.
• 42:26 - 42:28
I didn't understand anything.
• 42:28 - 42:32
As a sophomore, I really--
because sometimes, you know,
• 42:32 - 42:36
you can get an A without
understanding things in there.
• 42:36 - 42:39
As a sophomore, I
said, OK, what the heck
• 42:39 - 42:40
was that reparameterization?
• 42:40 - 42:43
I have to understand that
because it bothers me.
• 42:43 - 42:43
I went back.
• 42:43 - 42:45
I took the book.
• 42:45 - 42:48
reparameterization.
• 42:48 - 42:51
Our book, I think,
does a very good job
• 42:51 - 42:52
when it comes to
reparameterizing.
• 42:52 - 42:58
So if you open the 10.2 and
10.4, you have to skip-- well,
• 42:58 - 43:00
am I telling you to skip 10.3?
• 43:00 - 43:01
• 43:01 - 43:04
If you're interested in
dancing and all sorts of,
• 43:04 - 43:08
like, how the bullet
will be projected
• 43:08 - 43:11
in this motion or that
motion, you can learn that.
• 43:11 - 43:14
Those are plane curves that
are interested in physics
• 43:14 - 43:15
and mathematics.
• 43:15 - 43:19
But 10.3 is not part of
them and they are required.
• 43:19 - 43:20
• 43:20 - 43:22
You understand this much better.
• 43:22 - 43:23
Yes, ma'am.
• 43:23 - 43:25
STUDENT: Will the midterm
or the final just be, like,
• 43:25 - 43:27
a series problems, or
will it be anything--
• 43:27 - 43:30
PROFESSOR: This is going to
be like that-- 15 problems
• 43:30 - 43:30
like that.
• 43:30 - 43:32
STUDENT: Will it be
anything, like, super
• 43:32 - 43:33
in depth like the extra credit?
• 43:33 - 43:35
PROFESSOR: That-- isn't
that in-depth enough?
• 43:35 - 43:37
OK, this is going
to be like that.
• 43:37 - 43:41
So I would say at this
point, the way I feel,
• 43:41 - 43:45
I feel that I am ready to
put extra credit there.
• 43:45 - 43:49
My policy is that
• 43:49 - 43:53
So even if at this point,
you say extra credit.
• 43:53 - 43:55
And you put it at
the end for me.
• 43:55 - 43:57
Say, look, I'm doing
the extra credit here.
• 43:57 - 44:00
Then I'll be ready and I'll
say, OK, what did she mean?
• 44:00 - 44:02
Length of the arc?
• 44:02 - 44:02
Which arc?
• 44:02 - 44:06
From here to here is
• 44:06 - 44:08
• 44:08 - 44:11
And that's going to be-- you
• 44:11 - 44:13
inside the actual problem.
• 44:13 - 44:14
I see it.
• 44:14 - 44:15
STUDENT: Yes.
• 44:15 - 44:16
PROFESSOR: Don't worry.
• 44:16 - 44:17
STUDENT: Would it
just be as like-- just
• 44:17 - 44:20
like the casual problem
on the test or midterm
• 44:20 - 44:23
or whatever-- would it
be, like, an extra credit
• 44:23 - 44:23
problem in itself?
• 44:23 - 44:25
I know there will
be extra credit,
• 44:25 - 44:26
but the kind of proving--
• 44:26 - 44:30
PROFESSOR: That is--
that is decided together
• 44:30 - 44:33
with the course coordinator.
• 44:33 - 44:35
The course coordinator
himself said
• 44:35 - 44:40
that he is encouraging us to
set up the scale so that if you
• 44:40 - 44:43
all the problems that
are written on the exam,
• 44:43 - 44:47
you get something like 120%
if everything is perfect.
• 44:47 - 44:48
STUDENT: OK, if we can--
• 44:48 - 44:50
PROFESSOR: So it's sort
of in-built in that-- yes.
• 44:50 - 44:51
STUDENT: If we can
do the web work,
• 44:51 - 44:53
is that a good indication of--
• 44:53 - 44:54
PROFESSOR: Wonderful.
• 44:54 - 44:56
That's exactly--
because the way we
• 44:56 - 44:58
write those problems
for the final,
• 44:58 - 45:02
we pull them out of the web work
problems we do for homework.
• 45:02 - 45:04
So a square root
of a squared times
• 45:04 - 45:08
b squared times pi over 2--
so what have I discovered?
• 45:08 - 45:11
If I would take a
piece of that paper
• 45:11 - 45:14
and I would measure from
this point to this point
• 45:14 - 45:18
how much I traveled in
inches from here to here,
• 45:18 - 45:21
that's exactly that square root
of- this would be like a 5.
• 45:21 - 45:25
That's 3.1415 divided by 2.
• 45:25 - 45:25
Yes, sir.
• 45:25 - 45:29
STUDENT: So in the
interval of a squared plus
• 45:29 - 45:31
b squared, I know
that that's supposed
• 45:31 - 45:34
to be the interval
the magnitude of r--
• 45:34 - 45:36
PROFESSOR: The speed--
integral of speed?
• 45:36 - 45:36
STUDENT: Right.
• 45:36 - 45:39
So which is the r prime, right?
• 45:39 - 45:40
PROFESSOR: Yes, sir.
• 45:40 - 45:44
STUDENT: OK, so r prime was--
• 45:44 - 45:44
PROFESSOR: Velocity.
• 45:44 - 45:47
STUDENT: --a sine--
or negative a sine t,
• 45:47 - 45:49
a cosine t, and then b?
• 45:49 - 45:52
So where did the square root
of a squared plus b squared
• 45:52 - 45:53
come from?
• 45:53 - 45:55
STUDENT: That's from the--
• 45:55 - 45:57
PROFESSOR: I just erased it.
• 45:57 - 46:03
OK, so you have minus i-- minus
a sine b equals sine p and d.
• 46:03 - 46:05
When you squared them,
what did you get?
• 46:05 - 46:06
He has the same thing.
• 46:06 - 46:07
STUDENT: So that's just--
• 46:07 - 46:09
PROFESSOR: The square of that
plus the square root of that
• 46:09 - 46:10
plus the square root of that.
• 46:10 - 46:15
STUDENT: So it's just like a 2D
representation of the top one.
• 46:15 - 46:16
STUDENT: This side--
• 46:16 - 46:19
• 46:19 - 46:21
PROFESSOR: I just need the
magnitude of r prime, which
• 46:21 - 46:23
is this p, right?
• 46:23 - 46:24
STUDENT: Right.
• 46:24 - 46:25
PROFESSOR: The
magnitude of this is
• 46:25 - 46:29
the speed, which is square root
of a squared plus b squared.
• 46:29 - 46:31
Is that clear?
• 46:31 - 46:31
STUDENT: Yes.
• 46:31 - 46:33
PROFESSOR: I can
go on if you want.
• 46:33 - 46:37
So a square root of-- the sum
of the squares of this, this,
• 46:37 - 46:41
and that is exactly
square of [INAUDIBLE].
• 46:41 - 46:42
Keep this in mind as an example.
• 46:42 - 46:45
It's an extremely important one.
• 46:45 - 46:49
It appears very frequently
in tests-- on tests.
• 46:49 - 46:53
And it's one of the
most beautiful examples
• 46:53 - 46:58
in applications of
mathematics to physics.
• 46:58 - 47:03
I have something
else that was there.
• 47:03 - 47:04
Yes ma'am
• 47:04 - 47:08
STUDENT: I was just going to
ask if you want to curvature.
• 47:08 - 47:08
PROFESSOR: Eh?
• 47:08 - 47:09
STUDENT: The letter--
• 47:09 - 47:10
PROFESSOR: Curvature?
• 47:10 - 47:11
STUDENT: Curvature.
• 47:11 - 47:13
PROFESSOR: That's
exactly what I want.
• 47:13 - 47:18
And when I said I had something
else for 20%, what was k?
• 47:18 - 47:23
K was rho double prime
of s in magnitude.
• 47:23 - 47:30
So I have to be smart enough
to look at that and rho of s.
• 47:30 - 47:33
And rho of s was
• 47:33 - 47:36
here-- that's going to be
probably the end of my lesson
• 47:36 - 47:36
today.
• 47:36 - 47:40
• 47:40 - 47:46
Since you have so many
questions, I will continue.
• 47:46 - 47:50
I should consider--
the chapter is finished
• 47:50 - 47:54
but I will continue with a
• 47:54 - 47:57
on Tuesday with more problems.
• 47:57 - 48:01
Because I have the feeling that
although we covered 10.1, 10.2,
• 48:01 - 48:04
10.4, you need a
lot more examples
• 48:04 - 48:06
until you feel comfortable.
• 48:06 - 48:08
Many of you not,
maybe 10 people.
• 48:08 - 48:10
They feel very comfortable.
• 48:10 - 48:10
They get it.
• 48:10 - 48:13
But I think nobody will be
hurt by more review and more
• 48:13 - 48:16
examples and more applications.
• 48:16 - 48:21
Now, who can help me
finish my goal for today?
• 48:21 - 48:23
Is this hard?
• 48:23 - 48:26
This is rho of s.
• 48:26 - 48:30
So you have to tell me with
the derivation, is it hard?
• 48:30 - 48:31
No.
• 48:31 - 48:38
Minus a sine of the
whole thing times 1
• 48:38 - 48:41
over square root of a squared
plus b squared because I'm
• 48:41 - 48:42
applying the chain rule, right?
• 48:42 - 48:44
Let me change color.
• 48:44 - 48:45
Who's the next guy?
• 48:45 - 48:50
A Cosine of s over square
root a squared plus b squared.
• 48:50 - 48:53
I'm now going to leave
you this as an exercise
• 48:53 - 48:56
because you're going to haunt
me back ask me why I got this.
• 48:56 - 48:59
So I want to make it very clear.
• 48:59 - 49:04
B times 1 over square root
a squared by b squared.
• 49:04 - 49:06
So are we happy with this?
• 49:06 - 49:07
Is this understood?
• 49:07 - 49:11
It's a simple derivation
of the philosophy.
• 49:11 - 49:13
We are not done.
• 49:13 - 49:15
We have to do the acceleration.
• 49:15 - 49:18
So the acceleration
with respect to s
• 49:18 - 49:22
of this curve where s was
the arc length parameter
• 49:22 - 49:24
is real easy to compute
in the same way.
• 49:24 - 49:26
What is different?
• 49:26 - 49:30
I'm not going to
write more explicitly
• 49:30 - 49:32
because this should be
visible for everybody.
• 49:32 - 49:35
STUDENT: x [INAUDIBLE].
• 49:35 - 49:39
PROFESSOR: Good,
minus a over-- I'll
• 49:39 - 49:42
wait for you to
simplify because I don't
• 49:42 - 49:43
want to pull two roots out.
• 49:43 - 49:44
STUDENT: A squared--
• 49:44 - 49:46
PROFESSOR: A squared
plus b squared.
• 49:46 - 49:47
And why is that, [INAUDIBLE]?
• 49:47 - 49:52
Because you have once and
twice from the chain rule.
• 49:52 - 49:56
So again, I hope you guys don't
have a problem with the chain
• 49:56 - 50:01
rule so I don't have to
send you back to Calculus 1.
• 50:01 - 50:06
A over a squared times b
squared with a minus-- why
• 50:06 - 50:06
with a minus?
• 50:06 - 50:07
Somebody explain.
• 50:07 - 50:09
STUDENT: Use the
derivative of cosine.
• 50:09 - 50:13
PROFESSOR: There's a cosine
and there's a minus sine.
• 50:13 - 50:16
From deriving, I have
a minus and a sine.
• 50:16 - 50:22
• 50:22 - 50:25
And finally, thank
God, the 0-- why 0?
• 50:25 - 50:31
Because I have a constant that
I'm deriving with respect to s.
• 50:31 - 50:33
Is it hard to see what's up?
• 50:33 - 50:36
What's going out?
• 50:36 - 50:41
What is the curvature
of the helix?
• 50:41 - 50:45
A beautiful, beautiful
function that
• 50:45 - 50:49
is known in most of
these math, calculus,
• 50:49 - 50:54
multivariable calculus and
differential geometry classes.
• 50:54 - 50:57
What did you get?
• 50:57 - 51:03
Square root of sum of the
squares of all these guys.
• 51:03 - 51:04
You process it.
• 51:04 - 51:06
That's very easy.
• 51:06 - 51:07
Shall I write it down?
• 51:07 - 51:10
Let me write it down
like a silly girl--
• 51:10 - 51:14
square root of a squared,
although I hate when I cannot
• 51:14 - 51:15
• 51:15 - 51:19
But let's say there's
this little baby thing.
• 51:19 - 51:22
• 51:22 - 51:25
Now I can say it's
a over a squared
• 51:25 - 51:27
plus b squared-- finally.
• 51:27 - 51:29
you a few questions
• 51:29 - 51:30
and then I'm going
to let you go.
• 51:30 - 51:33
It's a punishment
for one minute.
• 51:33 - 51:37
OK, if I have the
• 51:37 - 51:42
the beautiful helix with
a Pythagorean number
• 51:42 - 51:45
like 3 cosine t, 3
sine t, and 4t, what
• 51:45 - 51:48
is the curvature of that helix?
• 51:48 - 51:50
STUDENT: 3 over 5--
• 51:50 - 51:52
PROFESSOR: 3 over 5, excellent.
• 51:52 - 51:54
• 51:54 - 51:58
What if I changed the numbers
in web work or on the midterm
• 51:58 - 52:01
and I say it's going
to be-- it could even
• 52:01 - 52:02
be with a minus, guys.
• 52:02 - 52:05
It's just the way you travel
it would be different.
• 52:05 - 52:08
So whether I put
plus minus here,
• 52:08 - 52:10
you will try on
different examples.
• 52:10 - 52:13
Sometimes if we put
minus here or minus here,
• 52:13 - 52:15
it really doesn't matter.
• 52:15 - 52:18
Let's say we have
cosine t sine t and t.
• 52:18 - 52:22
What's the curvature of
that parametrized curve?
• 52:22 - 52:23
1 over--
• 52:23 - 52:24
STUDENT: 2.
• 52:24 - 52:27
PROFESSOR: 1 over 2-- excellent.
• 52:27 - 52:27
So you got it.
• 52:27 - 52:29
So I'm proud of you.
• 52:29 - 52:32
Now, I want to do more examples
until you feel confident
• 52:32 - 52:33
• 52:33 - 52:37
I know I got most of you to
the point where I want it.
• 52:37 - 52:39
But you need more
• 52:39 - 52:41
and you need to
see more examples.
• 52:41 - 52:43
the whole chapter.
• 52:43 - 52:48
I would-- if you don't have
time for 10.3, skip it.
• 52:48 - 52:50
10.5 is not going
to be required.
• 52:50 - 52:53
So if I were a student,
I'd go home, open the book,
• 52:53 - 52:56
10.4, close the book.
• 52:56 - 53:00
It's actually a lot less
than you think it is.
• 53:00 - 53:02
If you go over the most
important formulas,
• 53:02 - 53:04
for the homework.
• 53:04 - 53:06
The second homework is due when?
• 53:06 - 53:08
February 11.
• 53:08 - 53:10
You guys have plenty of time.
• 53:10 - 53:14
Rather than going to the
• 53:14 - 53:17
On Tuesday, you'll have plenty
of time for applications.
• 53:17 - 53:20
OK, have a wonderful weekend.
• 53:20 - 53:23
Don't forget to email when
you get in trouble, OK?
• 53:23 - 53:28
Title:
TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 2
Description:

Derivative of a Vector Function, Unit Tangent Vector, Principal Unit Normal Vector, Arc-length Parameterization and Curvature

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Video Language:
English
Duration:
53:28
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Sec 10.2 and 10.4 part 2