-
-
PROFESSOR: Any
questions about theory
-
that gave you headaches
regarding homework
-
you'd like to talk about?
-
Anything related
to what we covered
-
from chapter nine and today?
-
STUDENT: Can we
do some problems?
-
PROFESSOR: I can
fix from problems
-
like the ones in the
homework, but also I
-
can have you tell me what
bothers you in the homework.
-
STUDENT: Oh, I have [INAUDIBLE].
-
PROFESSOR: What bothered
me about my own homework
-
was that I realized that I
did not remind you something
-
I assume you should
know, which is
-
the equation of a sphere of
given center and given radius.
-
And since I trust you so much,
I said, OK they know about it.
-
And then somebody asked
me by email what that was,
-
and I said, oh, yeah.
-
I did not review that in class.
-
So review the equation
in r3 form that's x, y, z
-
of the sphere of radius r and
center p of coordinates x0, y0,
-
z0.
-
One of you asked me by email,
does-- of course you do,
-
and then if you know it,
can you help me-- can you
-
help remind what that was?
-
-
STUDENT: x minus x0--
-
PROFESSOR: x minus x0 squared
plus y minus y0 squared
-
plus z minus z0 squared
equals R squared.
-
OK?
-
When you ask, for
example, what is
-
the equation of a units sphere,
what do I mean by unit sphere?
-
STUDENT: Radius--
-
PROFESSOR: Radius 1, and
center 0, standard unit sphere,
-
will be.
-
There is a notation for that
in mathematics called s2.
-
I'll tell you why its called s2.
-
x squared plus y squared
plus z squared equals 1.
-
-
s2 stands for the dimension.
-
That means the number
of the-- the number
-
of degrees of freedom.
-
-
you have on a certain manifold.
-
-
What is a manifold?
-
It's a geometric structure.
-
I'm not going to
go into details.
-
It's a geometric structure
with some special properties.
-
-
I'm not talking about
other fields of algebra,
-
anthropology.
-
I'm just talking about geometry
and calculus math 3, which
-
is multivariable calculus.
-
Now, how do I think
of degrees of freedom?
-
Look at the table.
-
What freedom do I have to move
along one of these sticks?
-
I have one degree of
freedom in the sense
-
that it's given by a
parameter like time.
-
Right?
-
It's a 1-parameter
manifold in the sense
-
that maybe I have
a line, maybe I
-
have the trajectory of the
parking space in terms of time.
-
The freedom that the bag has
is to move according to time,
-
and that's considered only
one degree of freedom.
-
Now if you were on a
plane or another surface,
-
why would you have more
than one degrees of freedom?
-
Well, I can move towards
you, or I can move this way.
-
I can draw a grid the way
the x and y coordinate.
-
And those are my
degrees of freedom.
-
Practically, the basis
IJ gives me that kind
-
of two degrees of freedom.
-
Right?
-
If I'm in three coordinates, I
have without other constraints,
-
because I could be
in three coordinates
-
and constrained to be on
a cylinder, in which case
-
I still have two
degrees of freedom.
-
But if I am a bug
who is free to fly,
-
I have the freedom to go with
three degrees of freedom,
-
right?
-
I have three degrees of
freedom, but if the bug
-
is moving-- not flying,
moving on a surface,
-
then he has two
degrees of freedom.
-
So to again review, lines
and curves in general
-
are one dimensional
things, because you
-
have one degree of freedom.
-
Two dimensional
things are surfaces,
-
three dimensional things are
spaces, like the Euclidean
-
space, and we are not
going to go beyond,
-
at least for the time
being, we are not
-
going to go beyond that.
-
However, where anybody is
interested in relativity,
-
say or let's say four
dimensional spaces, or things
-
of x, y, z spatial coordinates
and t as a fourth coordinate,
-
then we can go into higher
dimensions, as well.
-
OK.
-
-
I want to ask you a question.
-
If somebody gives you on
WeBWorK or outside of WeBWorK,
-
on the first quiz or
on the final exam,
-
let's say you have
this equation,
-
x squared plus y squared plus
z squared plus 2x plus 2y
-
equals 9.
-
What is this identified as?
-
It's a quadric.
-
Why would this be a quadric?
-
Well, there is no x, y, y, z.
-
Those terms are missing.
-
But I have something of the
type of quadric x squared
-
plus By squared plus
c squared plus dxy
-
plus exz plus fyz plus, those
are, oh my God, so many.
-
Degree two.
-
Degree one I would
have ax plus by plus cz
-
plus a little d constant, and
whew, that was a long one.
-
Right?
-
Now, is this of the
type of a project?
-
Yes, it is.
-
Of course there are some terms
that are missing, good for us.
-
How are you going to try to
identify the type of quadric
-
by looking at this?
-
As you said very well,
I think it's-- you say,
-
I think of a sphere, maybe I can
complete the squares, you said.
-
How do we complete the squares?
-
x squared plus 2x plus
some missing number,
-
a magic number-- yes sir?
-
STUDENT: So, basically I'll
have to take x plus 2 times 4
-
will go outside.
-
It's like x min-- x plus 2--
-
PROFESSOR: Why x plus 2?
-
STUDENT: Because it's 2x--
-
STUDENT: It's 2x.
-
PROFESSOR: But if
I take x plus 2,
-
then that's going to give
me x squared plus 4x plus 4,
-
so it's not a good idea.
-
STUDENT: On the x plus 1
-
PROFESSOR: x plus 1.
-
So I'm going to complete
x plus 1 squared.
-
What did I invent
that wasn't there?
-
STUDENT: 1.
-
PROFESSOR: I invented
the 1, and I have
-
to compensate for my invention.
-
I added the 1, created
the 1 out of nothing,
-
so I have to compensate
by subtracting it.
-
How much is from here to here?
-
Is it exactly the
thing that I underlined
-
with a wiggly line, a
light wiggly line thing,
-
plus what is the
blue wiggly line,
-
the blue wiggly line
that doesn't show--
-
I have y plus 1 squared, and
again, I have to compensate
-
for what I invented.
-
I created a 1 out of nothing,
so this is y squared plus 2y.
-
-
The z squared is all by himself,
and he's crying, I'm so lonely,
-
I don't know, there is
nobody like me over there.
-
So in the end, I can rewrite
the whole thing as x plus 1
-
squared plus y plus 1
squared plus z squared, if I
-
want to work them out in
this format, equals what?
-
STUDENT: 10.
-
-
PROFESSOR: 11.
-
11 is the square
root of 11 squared.
-
Like my son said the other day.
-
So that the radius
would be square foot
-
11 of a sphere of what circle?
-
What is the-- or the
sphere of what center?
-
STUDENT: Minus 1--
-
PROFESSOR: Minus
1, minus 1, and 0.
-
So I don't want to insult you.
-
Of course you know how
to complete squares.
-
-
However, I have discovered in an
upper level class at some point
-
that my students didn't know
how to complete squares, which
-
was very, very heartbreaking.
-
All right, now.
-
-
Any questions regarding--
while I have a few of yours,
-
I'm going to wait
a little bit longer
-
until I give
everybody the chance
-
to complete the extra credit.
-
I have the question
by email saying,
-
you mentioned that
genius guy in your class.
-
This is a 1-sheeted hyperboloid.
-
x squared plus y squared minus
z squared minus 1 equals 0.
-
The question was, by
email, how in the world,
-
did he figure out what the two
families of generatrices are?
-
So you have one family
and another family,
-
and both together generate
the 1-sheeted hyperboloid.
-
-
Let me give you a little
bit more of a hint,
-
but I'm still going to stop.
-
So last time I said, he
noticed you can root together
-
the y squared minus 1 and the
x squared minus z squared,
-
and you can separate them.
-
So you're going to have x
squared minus z squared equals
-
1 minus y squared.
-
-
You can't hide the
difference of two squares
-
as product of sum
and difference.
-
x plus z times x minus z equals
1 plus y times 1 minus y.
-
So how can you
eventually arrange stuff
-
to be giving due
to the lines that
-
are sitting on the surface?
-
The lines that are
sitting on the surface
-
are infinitely many,
and I would like
-
at least a 1-parameter
family of such lines.
-
You can have choices.
-
One of the choices
would be-- this
-
is a product, of
two numbers, right?
-
So you can write it as an
equality of two fractions.
-
So you would have something
like x plus z on top, x minus
-
z below.
-
Observe that you are
creating singularities here.
-
So you have to take x minus
z case equals 0 separately,
-
and then you have, let's
say you have 1 minus y here,
-
and 1 plus y here.
-
What else do you have to impose
when you impose x minus z
-
equals 0.
-
You cannot have 7 over 0.
-
That is undefined.
-
but if you have 0 over
0, that's still possible.
-
So whenever you take x
minus z equals 0 separately,
-
that will imply that the
numerator corresponding to it
-
will also have to be 0.
-
And together these
guys are friends.
-
What are they?
-
2--
-
STUDENT: A system of equations.
-
PROFESSOR: It's a
system of equations.
-
They both represent planes, and
the intersection of two planes
-
is a line.
-
It's a particular line, which
is part of the family-- which
-
is part of a family.
-
-
OK.
-
Now, on the other hand, in case
you have 1 plus y equals 0--
-
so if it happens that you
have this extreme case
-
that the denominator
will be 0, you absolutely
-
have to impose x plus z to be 0,
and then you have another life.
-
It's not easy for
me to draw those,
-
but I could if you
asked me privately
-
to draw those and show you
what the lines look like.
-
OK?
-
All right.
-
So you have two special lines
that are part of that picture.
-
They are embedded
in the surface.
-
How do you find a
family of planes?
-
Oh my god, I only
had one choice,
-
but I could have
yet another choice
-
of how to pick the parameters.
-
Let's take lambda to be
a real number parameter.
-
-
And lambda could be
anything-- if lambda is 0,
-
what have I got to have, guys?
-
STUDENT: The top.
-
PROFESSOR: The top
guys will be 0,
-
and I still have 1 minus y
equals 0, a plane, intersected
-
with x plus z equals 0,
another plane, so still a line.
-
So lambda equals 0 will give
me yet another line, which
-
is not written big.
-
Are you guys with me?
-
Could lambda ever
go to infinity?
-
-
Lambda wants to go to
infinity, and when does lambda
-
go to infinity?
-
STUDENT: When the
bottoms would equal 0--
-
PROFESSOR: When both
the bottoms would be 0.
-
-
So this is-- I can call it L
infinity, the line of infinity.
-
You see?
-
But still those
would be two planes.
-
There's an intersection,
it's a line.
-
OK.
-
Can we write this family--
just one family of lines?
-
A line is always an intersection
of two planes, right?
-
So which are the planes
that I'm talking about?
-
x plus z equals
lambda times 1 plus y.
-
This is not in the book,
because, oh my God, this is
-
too hard for the book, right?
-
But it's a nice example to
look at in an honors class.
-
1 minus y equals
lambda times x minus z.
-
It's not in the book.
-
It's not in any book that I know
of at the level of calculus.
-
All right, OK.
-
What are these animals?
-
The first animal is a plane.
-
The second animal is a plane.
-
How many planes
are in the picture?
-
For each lambda, you have a--
for each lambda value in R,
-
you have a couple of planes
that intersect along your line.
-
This is the line L lambda.
-
And shut up, Magdalena,
you told people too much.
-
If you still want them to do
this for 2 extra credit points,
-
give them the chance
to finish the exercise.
-
So I zip my lips, but
only after I ask you,
-
how do you think
you are going to get
-
the other family of rulers?
-
The ruling guys are
two families, you see?
-
So this family is
going in one direction.
-
How am I going to
get two families?
-
-
I have another choice
that-- how did I take this?
-
More or less, I made my choice.
-
Just like having two
people that would
-
be prospective job candidates.
-
You pick one of them.
-
STUDENT: Now, we can put 1
minus y in the denominator.
-
The denominator in
place of 1 plus y.
-
PROFESSOR: So I could have
done-- I could have taken this,
-
and put 1 plus y here,
and 1 minus y here.
-
I'm going to let
you do the rest,
-
and get the second
family of generators
-
for the whole surface.
-
That's enough.
-
You're not missing your credit.
-
Just, you wanted help,
and I helped you.
-
And I'm not mad whatsoever
when you ask me things.
-
The email I got sounded like--
says, this is not in the book,
-
or in any book, or
on the internet.
-
How shall I approach this?
-
How shall I start thinking
about this problem?
-
This is a completely
legitimate question.
-
How do I start on this problem?
-
OK.
-
On the homework-- maybe it's
too easy-- you have two or three
-
examples involving spheres.
-
Those will be too easy for you.
-
I only gave you a very thin
among of homework this time.
-
You Have plenty of time until
Monday at 1:30 or something PM.
-
-
I would like to draw
a little bit more,
-
because in this homework
and the next homework,
-
I'm building something special
called the Frenet Trihedron.
-
And I told you a little bit
about this Frenet Trihedron,
-
but I didn't tell you much.
-
-
Many textbooks in
multivariable calculus
-
don't say much about it,
which I think is a shame.
-
-
You have a position
vector that gives you
-
the equation of a regular curve.
-
-
x of t, y of t, z of t.
-
Again, what was a regular curve?
-
I'm just doing review of
what we did last time.
-
A very nice curve
that is differentiable
-
and whose derivative is
continuous everywhere
-
on the interval.
-
But moreover, the r prime
of t never becomes 0.
-
So continuously differentiable,
and r prime of t
-
never becomes 0 for any--
do you know this name,
-
any for every or for any?
-
OK.
-
This is the symbolistics
of mathematics.
-
You know because you
are as nerdy as me.
-
But everybody else doesn't.
-
You guys will learn.
-
This is what
mathematicians like.
-
You see, mathematicians hate
writing lots of words down.
-
If we liked writing essays
and lots of blah, blah, blah,
-
we would do something else.
-
We wouldn't do mathematics.
-
We would do debates,
we would do politics,
-
we would do other things.
-
Mathematicians like
ideas, but when
-
it comes to writing
them down, they
-
want to right them down in
the most compact way possible.
-
That's why they created
sort of their own language,
-
and they have all sorts
of logical quantifiers.
-
And it's like your
secret language
-
when it comes to your
less nerdy friends.
-
So you go for every--
for any or for every--
-
do you know this sign?
-
-
There exists.
-
-
And do you know this thing?
-
Because one of the-- huh?
-
STUDENT: Is that factorial?
-
PROFESSOR: Factorial,
but in logic,
-
that means there exists
a unique-- a unique.
-
So there exists a unique.
-
There exists a unique number.
-
There is a unique number.
-
So we have our own language.
-
Of course, empty set,
everybody knows that.
-
And it's used in
mathematical logic a lot.
-
You know most of the symbols
from unit intersection,
-
or, and.
-
I'm going to use some
of those as well.
-
Coming back to the
Frenet Trihedron,
-
we have that velocity
vector at every point.
-
We are happy with it.
-
We have our prime of t
that is referred from 0.
-
I said I want to
make it uniform,
-
and then I divided
by the magnitude,
-
and I have this wonderful t
vector we just talked about.
-
Mr. t is r prime over the
magnitude of r prime, which
-
is called it's peak right?
-
We divide by its peak.
-
What's the name of t, again?
-
STUDENT: Tangent unit--
-
PROFESSOR: Tangent
unit vector, very good.
-
How did you remember
that so quickly?
-
Tangent unit vector.
-
There is also another
guy who is famous.
-
I wanted to make him
green, but let's see
-
if I can make him blue.
-
t is defined-- should I
write the f on top of here?
-
Do you know what that is?
-
STUDENT: I thought n
was the normal vector.
-
PROFESSOR: t prime
divided by the length of--
-
STUDENT: Wait.
-
I thought the vector
n was the normal.
-
PROFESSOR: n-- there
are many normals.
-
It's a very good thing, because
we don't say that in the book.
-
OK, this is the t along my r.
-
Now when I go through a point,
this is the normal plane,
-
right?
-
There are many normals to
the surface-- to the curve.
-
Which one am I taking?
-
All of them are perpendicular
to the direction, right?
-
STUDENT: tf.
-
PROFESSOR: So I take
this one, or this one,
-
or this one, or this one, or
this one, or this one, there.
-
I have to make up my mind.
-
And that's how people came up
with the so-called principal
-
unit normal.
-
And this is the one
I'm talking about.
-
And you are right, it is normal.
-
Principal unit normal.
-
Remember this very
well for your exam,
-
because it's a very
important notion.
-
How do I get to that?
-
I take t, I differentiate
it, and I divide
-
by the lengths of t prime.
-
Now, can you prove to me
that indeed this fellow
-
is perpendicular to t?
-
Can you do that?
-
STUDENT: That n is
perpendicular to t?
-
PROFESSOR: Mm-hmm.
-
So a little exercise.
-
-
Prove that-- Prove that I don't
have a good marker anymore.
-
Prove that n, the unit
principal vector field,
-
is perpendicular-- you
see, I'm a mathematician.
-
I swear, I hate to write down
the whole word perpendicular.
-
I would love to
say, perpendicular.
-
That's how I write perpendicular
really fast-- to t fore
-
every value of t.
-
For every value of t.
-
OK.
-
How in the world can I do that?
-
I have to think about it.
-
This is hard.
-
Wish me luck.
-
So do I know
anything about Mr. t?
-
What do I know about Mr. t?
-
I'll take it and I'll
differentiate it later.
-
It Mr. t is magic in the
sense that he's a unit vector.
-
I'm going to write that down.
-
t in absolute value equals 1.
-
It's beautiful.
-
If I squared that-- and
you're going to say,
-
why would you want
to square that?
-
You're going to see in a minute.
-
If I squared that,
then I'm going
-
to have the dot product
between t and itself equals 1.
-
-
Can somebody tell me why the
dot product between t and itself
-
is the square of a length of t?
-
What's the definition
of the dot product?
-
Magnitude of the first
vector, times the magnitude
-
of the second vector--
there i am already--
-
times the cosine of the
angle between the two vectors
-
Duh, that's 0.
-
So cosine of 0 is 1, I'm done.
-
Right?
-
Now, I have a vector function
times a vector function--
-
this is crazy, right-- equals 1.
-
I'm going to go ahead
and differentiate.
-
Keep in mind that
this is a product.
-
What's the product?
-
One of my professors,
colleagues,
-
was telling me, now,
let's be serious.
-
In five years, how many
of your engineering majors
-
will remember the product?
-
I really was
thinking about this.
-
I hope everybody, if
they were my students,
-
because we are going to
have enough practice.
-
So the prime rule in
Calc 1 said that if you
-
have f of t times g of
t, you have a product.
-
You prime that product,
and never write
-
f prime times g prime unless you
want me to call you around 2 AM
-
to say you should never do that.
-
-
So how does the
product rule work?
-
The first one prime
times the second unprime
-
plus the first one unprime
times the second prime.
-
My students know
the product rule.
-
I don't care if the rest
of the world doesn't.
-
I don't care about any
community college who
-
would say, I don't want the
product rule to be known,
-
you can differentiate
with a calculator.
-
That's a no, no, no.
-
You don't know calculus if you
don't know the product rule.
-
So the product rule is
a blessing from God.
-
It helps everywhere in physics,
in mechanics, in engineering.
-
It really helps in
differential geometry
-
with the directional
derivative, the Lie derivative.
-
It helps you understand all
the upper level mathematics.
-
Now here you have t prime,
the first prime times
-
the second unprime, plus the
first unprime times the second
-
prime.
-
It's the same as for
regular scalar functions.
-
What's the derivative of 1?
-
STUDENT: 0.
-
PROFESSOR: 0.
-
Look at this guy!
-
Doesn't he look funny?
-
It is the dot product community.
-
Yes it is, by definition.
-
So you have twice T
times T prime equals 0.
-
This 2 is-- stinking
guy, let's divide by 2.
-
Forget about that.
-
What does this say?
-
The dot product of T times--
I mean by T prime is 0.
-
When are two vectors
giving you dot product 0?
-
STUDENT: When they're
perpendicular.
-
-
PROFESSOR: So if both
of them are non-zero,
-
they have to be like that.
-
They have to be like this,
perpendicular, right?
-
So it follows that t has to
be perpendicular to T prime.
-
And now, that's why n
is perpendicular to t.
-
But, because n is
collinear to t prime.
-
Hello.
-
n is collinear to t prime.
-
So this is t prime.
-
Is t prime unitary?
-
I'm going to measure it.
-
No it's not.
-
t prime.
-
So if I want to
make it unitary, I'm
-
going to chop my-- no,
I'm not going to chop.
-
I just take it, t prime,
and divide by its magnitude.
-
Then I'm going to get that
vector n, which is unitary.
-
So from here it follows that t
and n are indeed perpendicular,
-
and your colleague over there
said, hey, it has to be normal.
-
That's perpendicular
to t, but which one?
-
A special one, because
I have many normals.
-
Now, this special one is
easy to find like that.
-
Where shall I put here--
I'll draw him very nicely.
-
-
I'll draw him.
-
Now you guys have to
imagine-- am I drawing
-
well enough for you?
-
I don't even know.
-
t and n should be perpendicular.
-
Can you imagine them having that
90 degree angle between them?
-
OK.
-
Now there is a magic one that
you don't even have to define.
-
And yes sir?
-
STUDENT: In this
thing, can [INAUDIBLE]
-
this T vector [INAUDIBLE]
written by the definition
-
thing?
-
PROFESSOR: No.
-
STUDENT: N vector
times the magnitude
-
of t vector derivative?
-
PROFESSOR: So
technically you have
-
t prime would be the
magnitude of t prime times n.
-
STUDENT: Yes.
-
PROFESSOR: But keep in mind
that sometimes is tricky,
-
because this is, in
general, not a constant.
-
Always keep it in mind,
it's not a constant.
-
We'll have some examples later.
-
There is a magic
guy called binormal.
-
That binormal is the
normal to both t and n.
-
And he's defined as
t plus n because it's
-
normal to both of them.
-
So I'm going to write this
b vector is t cross n.
-
Now I'm asking you to draw it.
-
Can anybody come to
the board and draw it
-
for 0.01 extra credit?
-
Yes, sir?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Draw that on the
picture like t and n, t and n,
-
t is the-- who the heck
is t? t is the red one,
-
and blue is the n.
-
So does it go down or up?
-
We should be perpendicular
to both of them.
-
Is b unitary or not?
-
If you have two unit vectors,
will the cross product
-
be a unit vector?
-
-
Only if the two vectors
are perpendicular,
-
it is going to be, right?
-
So you have-- well, I
think it goes that--
-
in which direction does it go?
-
Because
-
STUDENT: It should
not be how we have it.
-
PROFESSOR: No, no, no.
-
Because this is--
-
STUDENT: Yeah.
-
I'm using--
-
PROFESSOR: So t
goes over n, so I'm
-
going to try-- it is
like that, sort of.
-
STUDENT: Into the chord?
-
PROFESSOR: So again, it's
not very clear because
-
of my stinking art, here.
-
It's really not nice art.
-
t, and this is n.
-
And if I go t going over n.
-
T going over n goes up or down?
-
STUDENT: Down.
-
PROFESSOR: Goes down.
-
So it's going to look
more like this, feet.
-
Now guys, when we--
thank you so much.
-
So you've like a
0.01 extra credit.
-
OK.
-
Tangent, normal, and
binormal form a corner.
-
Yes, sir?
-
STUDENT: Is rt-- rt is
the function at the--
-
for the flag that's flying?
-
PROFESSOR: The r of t
is the position vector
-
of the flag that was
flying that he was drunk.
-
STUDENT: Why wasn't the
derivative of it perpendicular?
-
Why isn't t perpendicular to rt?
-
PROFESSOR: If--
well, good question.
-
-
We'll talk about it.
-
If the length of r
would be a constant,
-
can we prove that r and r
prime are perpendicular?
-
Let's do that as
another exercise.
-
All right?
-
So tnb looks like a corner.
-
Look at the corner that the
video cannot see over there.
-
TN and B are mutually octagonal.
-
-
I'm going to draw them.
-
This is an arbitrary
point on a curve,
-
and this is t, which is
always tangent to the curve,
-
and this is n.
-
Let's say that's the
unit principle normal.
-
And t cross n will
go, again, down.
-
I don't know.
-
I have an obsession
with me going down.
-
This is called the
Frenet Trihedron.
-
-
And I have a proposal
for a problem
-
that maybe I should give
my students in the future.
-
Show that for a circle,
playing in space, I don't know.
-
The position vector and the
velocity vector are always how?
-
Friends.
-
Let's say friends.
-
No, come on, I'm kidding.
-
How are they?
-
STUDENT: Perpendicular.
-
PROFESSOR: How do you do that?
-
Is it hard?
-
We should be smart
enough to do that, right?
-
I have a circle.
-
That circle has what-- what
is the property of a circle?
-
Euclid defined that-- this is
one of the axioms of Euclid.
-
Does anybody know which axiom?
-
That there exists
such a set of points
-
that are all at the same
distance from a given point
-
called center.
-
So that is a circle, right?
-
That's what Mr. Euclid said.
-
He was a genius.
-
So no matter where I put that
circle, I can take r of t
-
in magnitude measured
from the origin
-
from the center of the circle.
-
Keep in mind, always the
center of the circle.
-
I put it at the origin of the
space-- origin of the universe.
-
No, origin of the
space, actually.
-
R of T magnitude
would be a constant.
-
Give me a constant, guys.
-
OK?
-
It doesn't matter.
-
Let me draw.
-
I want to draw in plane, OK?
-
Because I'm getting tired.
x y, and this is r of t,
-
and the magnitude of this r of
t is the radius of the circle.
-
Right?
-
So let's say, this is
the radius of the circle.
-
-
How in the world do I
prove the same idea?
-
Who helps me prove
that r is always
-
perpendicular to r prime?
-
Which way do you want to move,
counterclockwise or clockwise?
-
STUDENT: Counterclockwise.
-
PROFESSOR: Counterclockwise.
-
Because if you are
a real scientist,
-
I'm proud of you guys.
-
It's clear from the
picture that r prime
-
would be perpendicular to r.
-
Why is that?
-
How am I going to do that?
-
Now, mimic everything I--
don't look at your notes,
-
and try to tell me how
I show that quickly.
-
What am I going to do?
-
So all I know, all
that gave me was r of t
-
equals k in magnitude constant.
-
For every t, this same constant.
-
What's next?
-
What do I want to do next?
-
STUDENT: Square it?
-
PROFESSOR: Square
it, differentiate it.
-
I can also go ahead
and differentiate it
-
without squaring
it, but that's going
-
to be a little bit of more pain.
-
So square it, differentiate it.
-
I'm too lazy.
-
When I differentiate,
what am I going to get?
-
From the product rule, twice
r dot r primed of t equals 0.
-
Well, I'm done.
-
Because it means that for
every t that radius-- not
-
the radius, guys, I'm sorry.
-
The position vector will be
perpendicular to the velocity
-
vector.
-
Now, if I draw the
trajectory of my drunken flag
-
this [INAUDIBLE]
is not true, right?
-
This is crazy.
-
Of course this is r,
and this is r prime,
-
and there is an arbitrary
angle between r and r prime.
-
The good thing is that
the arbitrary angle always
-
exists, and is
continuous as a function.
-
I never have that
angle disappear.
-
That's way I want that
prime never to become 0.
-
Because if the bag was
stopping its motion,
-
goodbye angle, goodbye
analysis, right?
-
OK.
-
Very nice.
-
So don't give me more ideas.
-
You smart people, if
you give me more ideas,
-
I'm going to come up with
all sorts of problems.
-
And this is actually one
of the first problems
-
you learn in a graduate
level geometry class.
-
-
Let me give you another
piece of information
-
that you're going
to love, which could
-
be one of those
types of combined
-
problems on a final
exam or midterm,
-
A, B, C, D, E. The
curvature of a curve
-
is a measure of how
the curve will bend.
-
Say what?
-
The curvature of a
curve is a measure
-
of the bending of that curve.
-
-
By definition, you have
to take it like that.
-
If the curve is parameterized
in arc length-- somebody
-
remind me what that is.
-
What does it mean?
-
That is r of s such
that-- what does it mean,
-
parameterizing arc length--
-
STUDENT: r prime of s.
-
PROFESSOR: r primed of
s in magnitude is 1.
-
The speed 1.
-
It's a speed 1 curve.
-
-
Then, the curvature of this
curve is defined as k of s
-
equals the magnitude of
the acceleration vector
-
will respect the S.
Say what, Magdalena?
-
I can also write
it magnitude of d--
-
oh my gosh, second derivative
with respect s of r.
-
I'll do it right now. d2r ds2.
-
And I know you get
a headache when
-
I solve, when I write that,
because you are not used to it.
-
A quick and beautiful example
that can be on the homework,
-
and would also be on the
exam, maybe on all the exams,
-
I don't know.
-
Compute the curvature of a
circle of radius a Say what?
-
Compute the curvature of a
circle of radius a And you say,
-
wait a minute.
-
For a circle of
radius a in plane--
-
why can I assume it's in plane?
-
Because if the circle
is a planar curve,
-
I can always assume
it to be in plane.
-
And it has radius a I
can find infinitely many
-
parameterizations.
-
So what, am I crazy?
-
Well, yes, I am, but
that's another story.
-
Now, if I want to
parameterize, I
-
have to parameterize
in arc length.
-
If I do anything else,
that means I'm stupid.
-
So, r of s will be what?
-
Can somebody tell me
how I parameterize
-
a curve in arc length
for a-- what is this guy?
-
A circle of radius a.
-
Yeah, I cannot do it.
-
I'm not smart enough.
-
So I'll say R of T will be
a cosine t, a sine t and 0.
-
And here I stop, because
I had a headache.
-
t is from 0 to 2 pi, and
I think this a is making
-
my life miserable,
because it's telling me,
-
you don't have
speed 1, Magdalena.
-
Drive to Amarillo
and back, you're
-
not going to get speed 1.
-
Why don't I have speed 1?
-
Think about it.
-
Bear with me.
-
Minus a sine t equals sine t, 0.
-
Bad.
-
What is the speed?
-
a.
-
If you do the math,
the speed will be a.
-
So length of our
prime of t will be a.
-
Somebody help me.
-
Get me out of trouble.
-
Who is this?
-
I want to do it in arc length.
-
Otherwise, how can
I do the curvature?
-
So somebody tell
me how to get to s.
-
What the heck is that?
-
s of t is integral from
0 to t of-- who tells me?
-
The speed, right?
-
Was it not the displacement,
the arc length traveled along,
-
and the curve is integral
in time of the speed.
-
-
OK?
-
So I have-- what is that?
-
Speed is?
-
STUDENT: Um--
-
PROFESSOR: a.
-
So a time t, am I right,
guys? s is a times t.
-
So what do I have to do?
-
Take Mr. t, shake his hand,
and replace him with s over a.
-
OK.
-
So instead of r of t, I'll say--
what other letters do I have?
-
Not r.
-
Rho of s.
-
I love rho.
-
Rho is the Greek [INAUDIBLE].
-
Is this finally an arc length?
-
Cosine of-- what
is t, guys, again?
-
s over a.
-
s over a, a sine
s over a, and 0.
-
This is the parameterization
in arc length.
-
This is an arc length
parameterization of the circle.
-
And then what is this
definition of curvature?
-
It's here.
-
Do that rho once, twice.
-
Prime it twice,
and do the length.
-
So rho prime.
-
Oh my God is it hard.
-
a times minus sine of s over a.
-
Am I done, though?
-
Chain rule.
-
Pay attention, Magdalena.
-
Don't screwed up with this one.
-
1 over a.
-
Good.
-
Next.
-
a cosine of s over a.
-
Chain rule.
-
Don't forget,
multiply by 1 over a.
-
OK, that makes my life easier.
-
We simplify.
-
Thank God a simplifies
here, a simplifies there,
-
so that is that derivative.
-
What's the second derivative?
-
Rho double prime of s will
be-- somebody help me, OK?
-
Because this is a
lot of derivation.
-
STUDENT: --cosine--
-
PROFESSOR: Thank you, sir.
-
Minus cosine of s over a.
-
STUDENT: Times 1 over a.
-
PROFESSOR: Times 1 over a,
comma, minus sine of s over a.
-
That's all I have left
in my life, right?
-
Minus sine of s over a times
1 over a from the chain rule.
-
I have to pay attention and see.
-
What's the magnitude of this?
-
The magnitude of this of this
animal will be the curvature.
-
Oh, my God.
-
So what is k?
-
k of s will be--
could somebody tell me
-
what magnitude I get after I
square all these individuals,
-
sum them up, and take
the square root of them?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Square root
of 1 over 1 squared.
-
And I get 1 over a.
-
You are too fast for
me, you teach me that.
-
No, I'm just kidding.
-
I knew it was 1 over a.
-
Now, how did
engineers know that?
-
Actually, for hundreds of years,
mathematicians, engineers,
-
and physicists knew that.
-
And that's the last thing
I want to teach you today.
-
We have two circles.
-
This is of, let's say, radius
1/2, and this is radius 2.
-
The engineer, mathematician,
physicist, whoever they are,
-
they knew that the curvature
is inverse proportional
-
to the radius.
-
That radius is 1/2.
-
The curvature will
be 2 in this case.
-
The radius is 2, the
curvature will be 1/2.
-
Does that make sense, this
inverse proportionality?
-
The bigger the radius,
the lesser the curvature,
-
that less bent you are.
-
The more fat-- well, OK.
-
I'm not going to say anything
politically incorrect.
-
So this is really curved because
the radius is really small.
-
This less curved,
almost-- at infinity,
-
this curvature
becomes 0, because
-
at infinity, that radius
explodes to plus infinity bag
-
theory.
-
Then you have 1 over
infinity will be 0,
-
and that will be the curvature
of a circle of infinite radius.
-
Right?
-
So we learned something today.
-
We learned about the
curvature of a circle, which
-
is something.
-
But this is the same
way for any curve.
-
You reparameterize.
-
Now you understand why you need
to reparameterize in arc length
-
s.
-
You take the acceleration
in arc length.
-
You get the magnitude.
-
That measures how
bent the curve is.
-
Next time, you're going to
do how bent the helix is.
-
OK?
-
At every point.
-
Enjoy your WeBWorK homework.
-
Ask me anytime, and
ask me also Thursday.
-
Do not have a block about
your homework questions.
-
You can ask me anytime
by email, or in person.
-