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.