PROFESSOR: --everybody.
Do we have any questions or concerns from last time?
I received a few concerns from-- thank you so much-- from three
people about quadrics.
And I'll try to do my best, my very best today
lecturing in quadrics.
The people who expressed the concerns
are right to be concerned, actually.
Thinking back as a freshman, I myself
had some problems with how I identified
a certain type of quadric.
And I think my professor at the time
didn't take his time really explaining
the conic sections you can get, the standard conic sections you
get from cutting off a quadric with the standard planes,
the xy plane, the yz plane, the xz plane.
That would have been very useful to me.
Later on, I had discovered the benefits of reading things
on my own.
That's why I'm always suggesting that you should read the book.
If not the entire book section, at least the examples.
Start example one, example two, example three.
Some people try to solve those on their own.
I would not waste the time if I'm in a time crunch.
I would just go ahead and read the solutions with no problem.
And then when I get homework-- and you will get some WeBWorK
homework over the weekend-- you shouldn't start over
the weekend.
I don't want to ruin your weekend.
But maybe Sunday you'll get some WeBWorK homework,
your first assignment.
And then you will see what this is about.
And then you will start working on it next week.
OK?
When you are going to identify those problems,
we'll say, oh, yeah, she said that in class.
So everything we do in class and at home is connected.
Last time I [INAUDIBLE].
STUDENT: Oh, when would you like us to turn in our extra credit?
PROFESSOR: Oh.
I will collect it at the end.
So at the end of the class, please,
everybody bring the extra credit and leave it on my desk.
And I will take it all at home, and hopefully, this weekend
I will grade it.
Coming back to section 9.7, which
is the section that gives you most headaches in this review
Chapter 9.
I say "review" in quotes, because many of you
did not have Chapter 9 at all at different colleges.
People here in Texas Tech in Calculus 2 had 9.1 through 9.5,
most of them.
Or at least, the instructors were supposed to cover those.
In 9.7, what is challenging?
Not the first things that we covered last time.
So the most beautiful things that you saw
were the model of the ellipsoid and a particular case,
which was the sphere.
Could any of you remind me what the general standard equation
of the ellipsoid was?
STUDENT: x squared over a squared plus y
squared over b squared plus z squared over c squared
equals 1.
PROFESSOR: Very good.
Now, do we have a name for those a, b, c's?
We don't call them a, b, c anymore, right?
We are in college.
STUDENT: Major axis, minor axis.
I don't know the zeros.
PROFESSOR: Major semi-axis, because it's just the length
of half of that axis.
So I'm going to go ahead and draw the favorite shape
of the Texas Tech stadium.
I'm going to write the equation.
I noticed on the videos that it's hard to see the red.
But I will do my best.
x squared over a squared plus y squared
over b squared plus z squared over c squared equals 1.
And last time, we were just talking
that if you take z to be 0, you're
going to get an ellipse of major and minor semi-axis
a and b, respectively.
And I'm going to draw x here, y here, z z here for the axes.
So the a, what is the a?
It's the distance from the origin to this point.
So I'm going to call that little a.
I'm going to call that little b.
And the little c will be this guy, from here to here.
If I would want, for example, to draw
the ellipse as a cross-section which corresponds to what?
To y equals 0, I would have what?
I would have to look at that wall, considering the corner
that you have here in the left-hand side would be just
like x-axis.
[INAUDIBLE] the video doesn't see those z here
and the y-axis along this edge.
So if the y-axis comes towards me, like that, along this edge,
then I have y equals 0 corresponding to this ellipse.
And indeed, I should be able to draw better.
And I apologize that I couldn't do a better job.
Now, what is my technical mistake in this picture?
Could anybody tell me?
Besides the imperfection of the lines,
I should draw a dotted line.
The one that you don't see in perspective is behind.
Right?
OK.
If a and b and c are the same, we have a sphere.
a equals b equals c will be what?
Let's go back.
Big R. And then we are going to have
the sphere x squared plus y squared [? plus z ?] squared
equals R squared.
So far, so good.
Everybody happy with the ellipsoid.
It's something we've played since we were small,
whether we played football or played soccer,
we were happy with both of those.
And it's time to say goodbye to them.
And I'm going to move on to something else, which
we may have, in part, discussed last time.
But not so much.
[? II. ?] Hyperboloid.
And you say, oh, my god, hyperboloid, that
sounds like a mouthful.
It does sound like a mouthful.
And you have two very important standard types of hyperboloids.
I'm going to write the equation of one of them.
x squared over a squared plus y squared
over b squared minus z squared over c squared minus 1
equals 0.
But I would like you to observe--
because it's going to serve you a good purpose later--
is that we have plus, plus, minus, minus,
an alternation in signs.
We have two pluses and two minuses
when it comes to moving all the terms to the left-hand side.
So if I am to move all the terms in one side,
observe that I have this plus, plus, minus,
minus thing, why does that matter?
We will see later.
Now, I'm going to-- yes, sir?
STUDENT: Could it be x squared over a squared minus
y squared over b squared plus z squared over c squared minus 1?
PROFESSOR: OK, that's exactly what
I was trying to tell you-- that if you have plus,
that would be an ellipsoid.
And we already know that.
STUDENT: But if the y squared was minus and the z squared.
PROFESSOR: If this were plus, you
have a different type of hyperboloid.
This is the-- does anybody know?
One-sheeted, respectively, two-sheeted.
Which one is a one-sheeted?
I was getting there?
Which one is a one-sheeted hyperboloid,
and which one is a two-sheeted hyperboloid?
So my next equation was, for comparison, x
squared over a squared plus y squared
over b squared minus z squared over c squared
and plus 1 equals 0.
Let's see what kind of animal that is.
And it's a could it be?
Absolutely.
He's right.
There is some magic being that happens here
of plus, plus, minus, plus.
And you'll say, OK, wait a minute, Dr. [? Tora. ?]
If I multiply by a negative 1, the whole equation,
how does it count that I have three pluses and a minus?
I can three minuses and a plus.
Yes.
So you will have an uneven number of pluses and minuses.
And that should ring a bell.
Or an even number of pluses and minuses like you have here.
And that should ring a different bell.
What are the two bells that I am talking about?
One represents a certain type of surface.
The other one represents another type of surface.
I have to learn by discovery.
When I was taught these things, I was a freshman.
Very naive freshman.
And I was trying to memorize, because I was told to memorize.
Take the equation, memorize the picture.
That's the wrong way to learn, in my opinion,
after 20-something years of teaching.
We have to understand why a certain picture corresponds
to a certain type of equation.
If you don't know which one is which,
then you're going to be confused.
And for the rest of the course, you're
not going to know much about quadrics [INAUDIBLE]
and this type of quadrics.
Let's see what we are going to have.
The magic thing is look at cross-sections.
Does the video see me?
Cross-sections, magic ones.
Z equals 0 is one of the most important ones.
And then it's x equals 0.
And then y equals 0.
And let's see what we have, what those cross-sections will be.
The first one will be a conic, one of our old friends.
What is this, guys?
z squared over a squared plus b squared.
y squared over b squared equals 1.
It's an ellipse.
Right?
OK.
So along the z equals 0, I'm going
to have an ellipse, something like that.
Imagine the surface would have this type of equator.
And then for x equals 0, x-- where is x?
OK. x equals 0 plane is this one.
Right?
So in x equals 0, I'm going to have what type of conic?
Oh, my god.
That's why I had [INAUDIBLE].
See, that's why I had to review those things with you guys
last time.
It was not that I wanted it so badly.
But it was that we needed it.
It's a standard hyperbola in the yz plane.
So if this is the yz plane, it would look like that.
Right?
So I should start drawing.
Do you see me, video?
Yes.
So I'm going to have x, y, and z.
And I'm going to have an ellipse over here as an equator.
And in the yz plane, I should follow my preaching
from last time.
Practically, this is a, this is b.
And [INAUDIBLE].
And then I should draw that magic one.
Rectangle.
And after I draw that magic rectangle in the yz plane,
I should draw the asymptotes.
And I know the first branch has to do what?
Come from paradise, this is the asymptote infinitely
close, right?
Come from paradise, kiss this point
here, kiss the vertical line here.
And go back to-- I'm not going to say to where.
Asymptotically, to the oblique asymptote.
OK?
All right?
Yes, sir?
STUDENT: So that's two-dimensional?
PROFESSOR: That is a two-dimensional object.
It's only one branch.
I'm going to go ahead and draw the other branch, if I can.
Guys, you have to forgive me.
Forgiveness is important in life and also in mathematics.
I don't want to do it.
I cannot draw perfectly well.
These two guys should be perfectly symmetric,
but I feel bad, so I'll do a better job.
Hopefully.
OK?
And I really appreciate all the technology
that's out there on the web, like the Khan Academy
and so on.
And I'm going to send you some videos from Khan Academy.
I'm going to also send you some interactive gallery of quadrics
that was done at University of Minnesota.
Very beautiful, with Java applets, every such quadric.
You [INAUDIBLE].
Some of this interactive art is available in the textbook's
e-book.
Actually, there is a section.
If you have an access code, you have an access code
to your book.
Through that access code, you can
get to an interactive gallery of pictures.
But we don't have all the quadrics there.
So rather than sending you there,
I can send you to a web link from University of Minnesota,
where they have an interactive gallery of quadrics.
You can click on any of these quadrics,
rotate them, look at their cross-sections.
And they will show, with different colors, the ellipse.
The hyperbola is the section in the red things.
And they are in different colors.
And then the other one, the other ellipse
will be also like that.
So instead of this, you will have
one like that with different semi-axis.
What exactly do you have when you put y equals 0?
Yes, sir?
STUDENT: So by setting, like, the individual terms-- x,
y, and z, and 0-- you can see what
it looks like on a two-dimensional plane,
so you can form a three-dimensional image
in your head?
PROFESSOR: Right.
Because y equals 0 would represent what?
The intersection of your surface with the plane y equals 0.
And the plane y equals 0 would be this one.
So I want to see where my surface intersects this wall.
And where does it intersect that wall, I'm going to have,
in the conic, x squared over a squared minus z squared
over c squared equals 1, which is yet another hyperbola that
looks like that.
Are you guys with me?
On that wall.
So I can project it to that wall, but I am here.
Right?
So this is the one that you would have over here.
I should have used a different color.
One of the other authors of the textbook was saying to me,
you draw well, I cannot draw.
That's why I write books.
I don't know about that.
But I'm not drawing well.
I'm just trying to give you a sketch, an idea of what
this water tower looks like.
And what is magic about it, there's
something you don't see in the picture.
I may come and bring you a model.
Somehow either virtual model by email or a real model
to see that this surface, called one-sheeted hyperboloid,
is actually a [? ruled ?] surface.
It contains lines.
And you will say, how in the world does this contain line?
Well, if you look at infinity, these almost
look like lines, the branches of the hyperbola.
Why?
Because they come infinitely close to lines.
They almost look like lines.
But that's the reason why.
So you're actually having families
of surfaces, the families of lines that, in motion, describe
the surface.
I'm trying pretty good at this dance.
I'm not very good.
But anyway, you have two families
of lines, which, in motion, describe this surface.
And I should be able to move the elbow
in a sort of elliptic motion.
But I cannot.
In such a way to describe this one-sheeted hyperboloid.
The thing is one of our professors--
this is a funny story.
I hope he never finds out.
Or maybe he should.
He sold a house to a friend of mine, a little house
by [INAUDIBLE] Canyon.
Or was it-- it's the other one.
What's the other one?
STUDENT: [? Paladero? ?]
PROFESSOR: Buffalo Springs Lake.
And he had this stool that-- the learning
tool, which is a stool.
What?
It's a little stool made of bamboo.
There are these long, straight sticks
made of bamboo that are all put together.
And it sort of looks like-- I cannot draw it.
But practically--
STUDENT: I've seen those before.
PROFESSOR: Yeah.
So it looks like that.
I don't know if you've ever seen it.
It's perfectly symmetric.
And this stool is so nice.
And I offered my friend.
I knew exactly who got it and where it was coming from.
And I offered him $50.
And he said, take it for free.
And I'm really happy, because I was
ready to offer $100 for that.
It's a one-sheeted hyperboloid.
So my friend, who is a car mechanic,
asked me-- he's also Italian, so we speak Italian.
He's many things.
He's Australian, Italian, South African, American.
So I asked him.
And he said, oh, by the way, what the heck is that?
And I said, in mathematics, this is a one-sheeted hyperboloid.
And he said something bad.
And I said, OK, don't, just stop it, OK?
Mathematics deserves respect.
If you don't know what that is, you just keep it to yourself.
But it's really beautiful, this kind of-- it's also light.
It's made of bamboo.
And these sticks are-- together, you can even
imagine them in motion.
One after the other, they are so beautifully put together.
A [? half ?] extra credit homework.
Yes, sir?
STUDENT: What do you mean by a one-sheeted
or a two-sheeted hyperboloid?
PROFESSOR: So you will see next.
Looks like, more or less, like a tube, right?
But it's only one piece.
It's not disconnected.
The other one will be disconnected.
It will be consisting of two different sheets.
Mm-hm.
And I'm going to show you.
So the sheet we were talking about is this one.
The other one is practically one sheet and another sheet,
both of them infinite.
But completely disconnected.
STUDENT: And then, so for example, on this one,
it's the y squared-- if that term was negative
and the z squared term was positive,
would it still be considered a one-sheet hyperboloid.
PROFESSOR: Yes, sir.
And it will be you're just changing--
STUDENT: It'll just rotating?
PROFESSOR: Exactly.
Somebody can give you any combination.
Guys, look at that.
x squared over 4 plus z squared over 9
minus-- very good question-- minus y squared over 7.
And plus or minus 1?
I'm talking one-sheeted.
STUDENT: Minus.
PROFESSOR: Minus 1.
OK, this is still a one-sheeted hyperboloid.
What is different?
Can you tell me what's different?
STUDENT: It's rotated.
PROFESSOR: This is-- the y-axis is different.
The y is different compared to these two.
STUDENT: Is it rotating the [INAUDIBLE].
PROFESSOR: So instead of z-axis in the middle as a rotation
axis, you have the y-axis.
Very good.
All right.
So very good question.
You are ahead of me.
I will try to get a little bit faster in that case.
In this two-sheeted hyperboloid, it's a little bit harder
to imagine what it looks like.
But I'll try to do a good job drawing.
One thing you see when you try z equals 0,
you get a headache immediately.
Well, you shouldn't.
But what happens when you try to put z equals 0?
You see y?
How is that possible?
That's not possible-- a square plus a square plus 1 equals 0.
That's complete nonsense.
It has absolutely no solution.
So you have no intersection at the level of z equals 0.
And actually, if you move a little bit up and a little bit
down from the floor, you're going
to have no intersection for what?
And you may want to think what that y element may be.
So then you're thinking, OK, OK, I
know no intersection empty set.
But then I hope to get some cross-sections in other cases.
Like, y equals 0 should give me something beautiful.
And it does give me something beautiful,
which is x squared over-- let me take a black one.
x squared over a squared minus z squared over c squared plus 1
equals 0.
And you say, oh, wait a minute, I don't like this.
Hm.
If I shift this guy to the right--
you have to be a little bit creative in mathematics-- then
I'm going to have the same thing as z
squared over c squared minus x squared
over a squared minus a-- or equal to 1.
Equal to 1.
OK?
So you say, OK, so this must be some sort of hyperbola as well.
And how about the other one?
I'm going to leave it up to you to go home and experiment,
and draw these hyperbolas.
They will be-- if you look at the xz plane,
what type of hyperbola would be that?
If xz is like x and z, x and z-- look at my arms--
cannot be like that.
Your hyperbola has to be just the conjugate.
Oh, wow.
So instead of these branches in the actual plane,
vertical plane you're looking at,
you are having these branches.
Right?
OK.
You can go ahead and think about this at home and experiment.
You can also take x equals 0.
Who tells me what I have when x equals 0?
I also have the same kind of stuff that drive me crazy.
y squared over b squared minus z squared
over c squared equals minus 1.
What is it that I hate about it?
It's not the standard hyperbola.
I have to multiply again by a minus 1.
So when that drives me crazy, I'm
going to multiply by minus 1 by putting
a plus, a plus, and a minus.
And what is it that I notice again?
That I'm getting z squared over c squared minus y squared
over b squared equals 1.
Is that being the standard orientation of the plane yx?
Who the heck is the plane yz?
This plane.
y is on the bottom, z is going up.
Would I have it like this?
STUDENT: No.
PROFESSOR: Like this?
No.
Again--
STUDENT: Why not?
PROFESSOR: --I would have-- if it were y here
and z here, I would have a standard hyperbola in the yz
plane oriented like that.
But unfortunately, it's not the case.
They are swapped.
So I'm going to have the conjugate one.
So in both cases, the two hyperbolas
are going to look different.
I'm going to go ahead and erase here.
And I'm going to let you go home and-- yes?
Go ahead.
STUDENT: Question on the hyperbola.
How do you know if they're vertical
or if they're horizontal based on looking at the equation?
PROFESSOR: OK.
We said that last time.
It's OK.
So assume that this is xy plane, right?
If you have x squared over a squared minus y squared
over b squared equals 1, the vertical asymptotes
will look like y equals plus/minus b/ax.
Are you guys with me?
OK.
That takes a little bit of work.
That would be what our vertical asymptotes will be.
[? Put ?] oblique asymptotes will be.
For these oblique asymptotes, I'm
going to have a standard hyperbola that looks like that.
OK?
What if I put a plus here and a minus here?
Say it again, Magdalena.
Put a plus here and minus here.
And keep equal to 1.
Then I'm going to have the conjugate.
Right?
So in my case, to make a long story short,
because I really don't have that much time,
I would like you to continue that at home.
You are going to have two separate sheets that
continue to infinity.
What are these branches?
The black branch, I don't like it black.
Let me make it red again.
What is the equation of the red double branch?
Tell me again.
It's the one you obtain by making x equal 0.
And you get y squared over b squared minus z squared
over c squared equals minus 1.
Or if you wanted it in standard form,
you write it, z squared over c squared minus y squared
over b squared equals plus 1.
So if you have a little bit of time,
go home and try this by hand.
What if you don't want to do this by hand?
You hate to draw.
You cannot draw whatsoever, not even as bad as me.
Then I'll just send you that link
for the gallery of quadrics from the University of Minnesota.
And you're going to see them in action.
Rotate them, play with them, see their cross-sections.
There is another cross-section for x
equals 0 that I mentioned today, which was this one.
And I were to draw that, then you have to wish me luck.
I mean, for x equals 0, I did it.
But for y equals 0, I didn't do it.
It would be this one.
And I would have to draw in a different color.
I need to look like this branch that you see.
This part you don't see.
This branch you see, and this part you don't see.
It would still be OK.
It's very hard to mimic this with my hands.
But it would be one branch here and one branch here.
And the whole thing rotated.
And the semi-axis will change.
So really looking weird things.
Now, I want one thing from you.
And maybe you should-- should you do it now?
I think you should think about it now.
How much space is there from the vertex
of this sheet to the vertex of the other sheet?
Exactly what is the dimension from the origin to this peak?
And what is the dimension from the origin to this peak?
STUDENT: It's c/2.
PROFESSOR: Mm.
Why over 2?
STUDENT: Or is it just c?
PROFESSOR: It's c.
Why is it c?
STUDENT: Because if--
PROFESSOR: If x and y are 0, right?
Are you guys with me?
I'm looking at this line. x and y should be 0.
I'm going along z.
Where do I have an intersection?
When z is plus/minus c.
And then when z is plus c, I have it here.
0, 0 plus c.
And 0, 0 minus c.
And actually, you can rigorously prove that there
is nothing in between.
You can actually take any plane that is between
z equals minus c and z equals c.
You're not going to intersect a surface.
All right?
And this is what we call-- tell me again.
I told you there is no stool for that.
Two-sheeted hyperboloid.
It's a disconnected surface.
It consists of two infinite pieces.
So again, if somebody asks you in the exam-- and it
happened before we had problems like that in WeBWorK.
We still have them.
You are going to get one.
And other examples in the book, [? exercises. ?]
How do you recognize a hyperboloid
from just looking at it?
It has to have x squared, y squared, z
squared over some numbers, and the 1, with plus or minus.
The signs matter.
If you have two pluses, two minuses,
when you move everything to the left-hand side,
then it's a what?
[INTERPOSING VOICES]
It's a one sheet.
And if you have three pluses and minus
or three minuses and a plus, then it's
a different kind of animal.
It's a two sheet, OK?
All right.
The thing is that for extra credit--
you interrupted when I said "extra credit."
And that's fine but I want to come back to it.
Maybe you're up to the challenge.
Prove that the one-sheeted hyperboloid is a ruled surface,
is actually--
STUDENT: Sorry, what surface?
PROFESSOR: Ruled, ruled, ruled.
The one-sheeted hyperboloid is a ruled surface
that is a surface generated that is-- i.e., [INAUDIBLE]
in Latin-- a surface generated by lines in motion.
Actually, you have generated by two families,
two separate families of lines, of straight lines.
And-- or but-- I don't know.
Versus.
Right?
How shall I say?
Versus the two-sheeted hyperboloid
that is not a ruled surface.
I once had a genius in my class.
And every now and then, I have a bunch of geniuses in my class.
And after thinking for, like, five minutes,
he said, I think I know why that is.
I think it has to do with those pluses and minuses.
And I said, why do you think that is?
And he said, wait a minute.
Plus, minus, plus, minus.
It's like a pattern that my high school teacher taught us.
I said, your high school teacher must have been good.
Where did you go to high school?
Lubbock High.
I said, good.
And the pattern that he saw from his teacher was very funny.
Actually, he was right.
His teacher showed him, if you have
x squared minus y squared plus z squared minus 1 equals
0-- do you remember this kind of little exercise?
Can you split into two groups of terms
and write the sum of the squares as a product, sum,
and difference.
So he played around with those a lot.
And he said, you know?
The fact that you have plus, minus, plus, minus reminds
me of high school.
And I used to be very good at that.
And then he went ahead and said, what kind of
hyperboloid that could be?
That would be a one-sheeted, because it's two minuses, two
pluses when you move to that.
So he went away and said, x plus y, x minus y plus z plus 1,
z minus 1-- the guy was smart.
Really smart.
And then he said, what if I split--
I don't want to give away the clue.
But I'm always very, very good at giving away the clue.
When I buy gifts for my friends even here in the department,
I sort of give them a clue that I'm
going to buy a gift of a certain sort.
So I spoilt-- completely spoil the surprise.
I don't want to spoil your surprise.
So the guy, based on the idea that he had-- that
was the idea.
Very simple but the idea of a genius.
He said, I think at this point, I can prove to you
that we have lines inside.
And I said, what the heck?
Yes, ma'am, because such a proportion,
like a multiplication equal multiplication, maybe
you can write it as x plus y, x minus y, 1 minus z, 1 plus z.
And I said, stop.
At this point, I said, stop.
You are solving the problem for everybody else.
So he said, oh, [INAUDIBLE] I know how to get the planes.
Intersection of a planes is a line.
Your ruled surface is a ruled surface.
That one-sheeted must be a ruled surface
must contain a family of lines-- I know how to get it.
So he said, stop.
So I don't want to tell you more,
because you have to find those families of lines yourself.
OK?
There are two ways to arrange that.
And you get to do those two families of lines
that generate the surface.
You cannot do that for the two-sheeted hyperboloid,
because if you put a plus here, it's goodbye.
You cannot factor out in real numbers the z squared plus 1.
It's bye-bye.
Right?
OK.
Now, coming back to other surfaces
that are important to us.
You've seen Part 1.
Let's see what happens in Part 1.
We've seen ellipsoid with sphere as the most common and typical
example.
We talked about last time center and radius of the sphere.
In Part 2, we saw hyperboloid of one sheet
and hyperboloid with two sheets.
We saw the difference between them.
Now, Part 3, this is something that
is a little bit easier, hopefully,
to draw and to understand.
And you've seen that before many times.
Something that looks like a single z isolated.
So it's going to be a graph of the form f of xy, where f of xy
is of the following shape.
x squared over a squared plus y squared over b squared.
a and b are positive.
What the heck is that?
Well, when I was asked what that is, I was 18.
First time I saw that.
And I just replaced, mentally, a with a 1 and b with a 1.
And I said, z equals x squared plus y squared?
I don't know.
But it looks familiar.
OK?
So I started thinking.
And then somebody told me there is a different one
that you have.
But if you have z squared equals x
squared over a squared plus y squared over b squared,
which looks a little bit similar, but it's different.
It's different in nature.
And I thought, OK, let me try and draw.
Because if I draw, maybe I find all the answers by drawing.
And sometimes in life, you find lots of your answers
by trying to imagine things, draw a diagram,
visualize them somehow.
So if z would be 0, you only have one solution, which
would be x equals y equals 0.
So you have the origin.
And that's it.
Now, do I say that z equals positive?
No but it's implied.
Why?
Because this whole quantity must be either 0 or positive.
It's greater than or equal to 0.
So I'm only looking at the upper part above the floor.
Everything is above the floor.
What if I take other nice values in case my a and b
would be equal or equal to 1, it doesn't make much difference.
z equals x squared plus y squared
is going to be something nice, in the sense that at level
z equal to 1, I'm going to have a circle,
x squared plus y squared equal 1.
Somebody picked z equals 1 for me.
And I'm going to have that here.
At z equals 4, I'm going to have x squared
plus y squared equals 4.
I have a lot of play in the [? sun. ?]
So how big will the radius be?
2, potentially.
So I'm trying to respect the proportions
and not say anything too deformed.
And if I am to draw many circles, one
on top of the other, by continuity,
I'm going to get this beautiful--
it looks like a cone, but it's not a cone.
STUDENT: It's a paraboloid.
PROFESSOR: It's a paraboloid, right?
It's a paraboloid.
And it's a what?
How do we call this type of vase?
In this case, it's an elliptic paraboloid.
But if a equals b, it's going to be a circular paraboloid.
Because you will have cross-sections
at the horizontal planes being ellipses
if you would deal with this equation-- general [INAUDIBLE].
Or if a equals to b, you are going
to have circle after circle after circle after circle.
Have you seen one of those lamps that
are made with circles of different dimensions?
And you put threads between the circles and hang them?
Yeah.
They are mostly made in Asia.
They're extremely beautiful.
And then if you use white fabric or something, you hang them.
They give you a very nice, calm atmosphere
in-- not like the neon lights-- in the room.
So you can imagine that bunch of circles
for the circular paraboloid, they are called level curves.
So what the heck is a level curve?
A level curve you will learn in Chapter 11.
But I can anticipate a little bit.
Would be the set of all x, y values with the property
that f of xy equals a constant.
OK?
If you were to draw these, they would be circles in plane.
And they would be just the projections
of these circles from the surface to the plane, the ones
I talked about, the circle of radius 1
corresponding to z equals 1.
Projected down, you have the unit circle.
For z equals 4, you get-- what did we say you get?
x squared plus y squared equals 4, with radius 2, and so on.
So if you were to be-- this is the eye of God,
or whatever is here. [INAUDIBLE] external observer.
You see these concentric circles on the floor.
These concentric circles on the floor that
are the projections of the circles in your lamp-- OK,
this is the source of light.
And your thing projects the shadows.
Those concentric circles are called the level curves.
And we will see those again as an obsession in Chapter 11.
All right.
Now, this one looks similar, but it's sharpened.
And I don't know.
The waffle cone, the ice cream cone
you have is more paraboloid, because if you
look at those waffles, the ones at the mall
especially are not sharp.
They don't have perfect straight lines.
They're not in a point, a vertex.
It's more like a paraboloid.
But this should be a perfect cone,
a cone that looks like that.
That's the vertex.
And it would be a double cone, moreover.
And tell me why you have a double cone.
Why do you have the upper part and the lower part?
STUDENT: [INAUDIBLE] plus or minus.
PROFESSOR: Because z could be positive or negative.
And for the negative part, you get the exact opposite,
the symmetric of that.
So we call that a cone in practice, but it's not a cone.
It's a paraboloid.
If you go to an ice cream shop and say, give me an ice cream
cone, waffle, whatever, you don't say,
give me a paraboloid although it looks like a paraboloid.
Right?
All right.
So we have some very important surfaces that we talked about.
But there is yet another type of paraboloid
that is very important in our lives.
And living in Texas, you cannot just neglect this.
You cannot say, I don't want to know about this surface,
it doesn't interest me.
I'm giving you some hints.
So I'll assume that instead of those beautiful z equals--
what was it guys?
x squared over a squared plus y squared over b squared,
you erase this plus and you put a minus.
And you call that [INAUDIBLE], like a scientific term
of a plant, Latin name of a plant.
It sounds sophisticated.
But in reality, it has such a nice, funny name,
such a suggestive name.
Later on.
This should be a what?
A paraboloid.
What kind of paraboloid?
Oh, my god.
It sounds like a monster.
STUDENT: Hyperbolic.
PROFESSOR: Hyper-- hyperbolic paraboloid.
Say it again, Magdalena.
Hyperbolic paraboloid.
That looks like a monster, but it's not a monster.
I'm going to try to make a and b to be 1 for you to enjoy them.
And so I'll erase this picture, and I'll try.
And I cannot promise anything, OK?
But I'll try to draw z equals x squared
minus y squared from scratch.
And this is the picture, to start with.
That's the origin.
That's the x-axis, that's the y-axis, that's the z-axis.
And if I take z-- OK, first of all, let me take x to be 0.
Let me be a disciplined girl, like I haven't been before.
But if I were to take the plane x equals 0 first
and take the intersection, this is sine intersection
with my surface, sigma.
Sigma is my surface.
The Greeks love the [? vocab ?] to use letters as symbols
for mathematical objects.
OK?
You saw that I prefer pi for a plane.
And S is the sigma in Greek.
And then I say surface is sigma.
OK.
What would the intersection be?
Um, a conic.
What kind of conic? z equals minus y squared.
What the heck is z equals minus y squared?
STUDENT: It's a parabola.
PROFESSOR: It's a parabola that opens--
STUDENT: Downwards.
PROFESSOR: Down.
And I should be able to draw that.
If I'm not able to draw that, I don't deserve
to be in this classroom.
And then let's see.
For y equals 1, I would get z equals minus 1.
If it's not a square, you'll forgive me.
But it should look nice enough drawn on this wall.
Many of you are engineers.
Of course, you take technical drawing or stuff,
or you just use a lot of MATLAB.
Don't judge me too harshly.
OK?
So this is what you would have. z equals minus y squared.
I'm going to write on it.
And how about I take y equals 0?
I say, I want y to be equal to 0 to make my life easier?
No.
To get a cross-intersection between the plane y
equals 0 and the surface, sigma.
What do I get?
Another conic.
What conic?
z equals x squared.
STUDENT: Yeah.
PROFESSOR: Now, wish me luck, OK?
All right.
Oh.
My hand was shaking.
But it doesn't matter.
So what I'm trying to draw-- if I
were to draw the other parabolas,
these are not parabolas.
They look like horrible things.
But if I were to draw other parabolas, this you see.
This-- you see the other rim.
This you see.
This you don't see, because it's hidden.
And then I'm going to cut.
How did I cut, actually?
How do you think I tried to cut in my imagination
to get those parabolas?
I tried to say-- what if you take
z to be 7 or something like that?
It's like having 49 minus y squared.
And that's a parabola that opens down.
But it's shifted 49 units up.
And it's going to look exactly like that.
So these guys, the blue ones are along the x-axis
that are coming towards you.
I'm coming along the x-axis, and I say, I fix x to be 7.
And I do a cross-intersection, and I get this parabola
that looks like that.
And I go this way, the other direction, and so on.
And different x values that I set
fixed at different values of x, I tried to see what I have.
I have parabolas opening down, opening down, opening down.
7 minus y squared.
9 minus y squared.
21 minus y squared.
700 minus y squared. [INAUDIBLE].
They become bigger and bigger.
And I should be able to tell you what the rest of the picture
should look like.
In the end, what is this going to look like?
I erase the scientific term of that,
and I will give you a better feel
of what you're going to have.
So guys, you're going to see this edge.
But you have to have an imagination.
Otherwise, you don't understand my pictures.
This is like abstract Picassos.
This is the edge.
This is an edge.
I caught a patch.
And that's another edge.
And this is something you see.
This is something you see.
This is something you don't see.
And I just drew it like that.
You see this part, and you don't see that part.
But you see this part here.
Maybe I can do a better job.
I can round it up a little bit.
Cut this patch with different scissors.
What is this called?
[? STUDENT: Horsey ?] saddle.
PROFESSOR: It's a saddle.
It's a saddle surface, thank god.
Saddle surface is the same thing as a hyperbolic paraboloid.
And for one extra credit point that you
can turn in on a half of a piece of paper or something,
show that a surface z equals xy is of the same type
as the surface z equals x squared minus y squared.
What kind of transformation should you consider?
What type of transformation, parentheses, coordinate
transformation, should you consider?
Now, there are little graphing calculators,
like a TI-92, that can do about just as what MATLAB is doing,
be able to graph such a surface.
And if you graph z equals x squared minus y squared,
you're going to get-- what was the orientation?
Something like-- this is the x-axis going towards you.
The y-axis going in this direction.
The z-axis is going up.
How is the horse standing?
The horse standing either in this direction
or in that direction for the previous one-- z
equals x squared minus y squared.
Am I right?
I forgot how we drew it.
So this is the y-axis.
OK.
So suppose that I'm in y-axis.
I'm on top of the horse.
The saddle point is the point where the Red
Rider sits on the saddle.
So the saddle is shaped like that.
Longitudinally, you have it like that.
Latitudinally, you have the other cross intersection
going down.
So my legs are hanging left and right.
We are in Texas.
Now, if I have z equals xy, what's different?
Is the Red Raider looking straightforward
like that and along the axis like I did before
with this riding attitude or what?
It's gonna look different.
You have to find that kind of transformation.
What do you think it is?
How do you think the surface-- rotation.
Very good.
It's actually a rotation and a rescaling.
OK?
It's a rotation and a rescaling.
Maybe just to give you one idea, we still
have a little bit of time.
I know I shouldn't do trig in this class.
But god, how many of you took trigonometry?
That was a long time ago, right?
Wasn't it?
So if I have-- recall this type of transformation.
This is just a hint, OK?
[INAUDIBLE] in plane xy.
And it was trig in [? plane ?] long time ago.
And I'm changing coordinates.
And I'm saying, x prime, y prime will be the matrix A.
And we didn't know that.
But if you came here to the [? Emmy ?] the other day,
you would know, because I did that in a [INAUDIBLE]
high school day.
I explained how to multiply two matrices.
You have vector multiplied to the left
by a matrix of rotation.
Now, matrix of rotation by 45-degree angles
would be like that.
Cosine of the angle-- cosine of 45 minus sine of 45,
whatever that angle of rotation, [? phi, ?] is.
Sine of 25, cosine of 45.
Let's see what the heck this change of coordinates is.
Right?
Do you guys remember what cosine of 45 was?
STUDENT: Square root of 2/2.
PROFESSOR: Square root of 2/2.
One of my friends and colleagues was telling me in Calc 1
that her students don't know that.
You know that.
I know that we don't remember everything.
But every now and then, we need a little bit of refresher.
Square root of 2/2 minus square root of 2/2.
Square root of 2/2.
Oh, my god, that was a lot of work. x, y.
Did I write it like that for the high school days?
I did because although they don't
know how to multiply two matrices,
I wanted to show them how a system of equations
is actually-- a linear system would
be equivalent to this matrix multiplication.
So what does this mean?
If you take an introduction to C or some programming,
when I took introduction to C++, this was the first thing they
asked me to do-- multiply with a rotation matrix.
And it was fun to program something like that.
So how is this going?
X prime will be.
You go multiply one row by a column.
So row-column multiplication means first times first
plus second times the second.
Root 2/2 x minus root 2/2 y.
Now you don't have to pay tuition for the first two
classes of linear algebra.
Are you taking-- is anybody taking linear algebra
at the same time?
So you guys already knew that.
But anyway, let's do that.
Plus square root 2/2 y.
Now, interestingly enough, there were some high schools
where they teach matrix multiplication
and some high schools where they don't teach matrix
multiplication in algebra.
OK.
Now, what if I multiply x prime and y prime?
What do I get?
What if I z equals x prime, y prime?
What kind of surface is that?
z equals-- you are smart people.
You should know how to do that.
I am running out of gas.
STUDENT: x/2-- x squared over 2 minus y squared over 2.
PROFESSOR: Right.
You are too fast for me.
You are good.
You're really good.
This is a minus b.
This is a plus b.
So it's like he says, product of difference and sum
is the difference of squares.
So it's like this a squared minus b squared.
But he's also smart, and he said, come on,
Magdalena-- he didn't say that, but that's what he meant.
Square root 2/2 is much simpler than you say it.
It's 1 over root 2.
So this guy is x over square root of root--
oh. x over square root of 2.
So when you square that-- that was a lot of explanation.
When you square that, it's x over square root
of 2 squared minus y over square root of 2 squared.
So z equals x squared over 2 minus y squared over 2.
Oh, my god, that was long.
All right.
You've seen I'm almost doing the extra credit homework for you.
I wanted to brush up the details.
How would you get first from such a surface, where you
have x prime, y prime, to xy?
You just rotate the axis of coordinates.
The problem is I'm still getting this annoying and spiteful 2!
And instead of getting z equals x squared minus y squared,
I get z equals x squared minus y squared all over 2.
What the heck does it mean?
Can I do something about it?
STUDENT: Yeah, you can just divide that out-- or multiply.
PROFESSOR: Well, yeah.
What is this called?
You can arrange that.
What is this transformation called?
STUDENT: A rescale.
STUDENT: Rescaling.
PROFESSOR: Rescaling.
How do you know these things?
STUDENT: 'Cause you just said it.
PROFESSOR: I just said it?
Wow. [INAUDIBLE].
So because rescaling is something
that people in my area use a lot.
In differential geometry, we talk
about rescaling coordinates, rescaling matrices.
But most mathematicians don't know that term.
So you are a good recorder.
OK.
STUDENT: I just wrote it down in my notes.
You said that's a rotation and a rescaling.
I wrote it down.
PROFESSOR: Rotation and rescaling will do.
So practically, when you multiply
x and a y in such an equation by the same number,
it's like what I'm doing now.
Look at me, look at me.
So on a whole picture, assume you have z equals
x squared plus y squared.
What if I have z equals 9x squared plus 9y squared,
but I did a rescaling.
What kind of rescaling?
3 times 6 and 3 times y.
And I changed the coordinates.
What's going to happen to my lamp, to the valley?
It's gonna stretch like that from here to here.
But the shape is the same.
The overall shape, the topology of the lamp, is the same.
Very good.
Is there anything I wanted to teach you
and I didn't teach you?
Last time I taught you about circular [INAUDIBLE]
some circular cylinder.
I taught you about other kinds of cylinders
based on other kinds of curves.
Parabolic cylinders or other cylinders.
What is a cylinder?
I didn't tell you what a cylinder is.
And I didn't tell you what a cone is.
And we don't teach you in the book.
[? Beh. ?] I'm one of the authors.
But this is something that I would
like to talk to you a little bit about,
because ruled surfaces are important.
And we don't do them.
We don't cover them in this course.
And it's somewhere between vector calculus
and analytic geometry.
And I would like to know what a cylindrical surface is
and what a conic surface is, because you're an honors class
and you should know a little bit more about quadrics
than anybody else.
So nice quadrics like z equals-- what did we do last time?
It was like x squared plus y squared equals 9.
z equals x squared-- were discussed last time.
And we decided that if they are cylinders
because-- how did we decide?
One variable was missing.
And that variable can be considered to be a parameter.
So what we said is that let's embrace the circle x squared
plus y squared equals 9.
But z could be 0, 1, 2.
That would be a discrete set of values.
But it could be a continuous, real parameter.
So what I'm doing, I'm creating a cylindrical surfaces
with the motion coming from the one family
of a one-parameter family.
So I'm describing a cylinder.
The same way, y is missing here.
Along the y, I can describe a cylinder.
But how would I describe a cylinder
in general outside of the chapter in the book?
Could somebody tell me how-- what's a cylinder?
A cylinder is not a can.
It's not always a round cylinder.
Yes, sir?
STUDENT: Is it a prism with a--
PROFESSOR: It's not a prism.
Any surface that-- OK.
STUDENT: Never mind.
PROFESSOR: Let me show you how I, in general-- [? Nateesh, ?]
can I steal that from you?
I want to generate-- you are going to catch it in a moment.
I have a generating line.
But I say, I want this line to stay.
It's going to move along a curve.
But it has to stay parallel to itself.
Say what, Magdalena?
Say it again.
It has to move along the line, along the contour.
Line doesn't mean shade line.
It could be any curve.
But it could be at an angle, but it
has to stay parallel to itself while moving.
So I'm going to go and start moving it.
I have described a cylindrical surface.
You see how it stays parallel to itself?
OK.
So a cylindrical surface-- I hate this marker.
Cylindrical surface is a ruled surface
generated by the motion of a line,
of a straight line along a curve, which remains parallel
to itself.
And I'll try and draw it.
I did not like my handwriting here.
OK, you will excuse me.
I think sometimes pictures-- that's why I like to draw.
Picture is worth a million words.
So this is the plane.
This is a regular curve.
It could even have self-intersections.
It doesn't matter.
And I'm going to have a continuous motion of a line
that stays parallel to itself.
And it describes a cylindrical surface.
And you say, hey, Magdalena, but-- excuse me,
but the surface presses itself.
And so what?
Sometimes surfaces have cross-intersections.
So that surface would look like that.
Right?
You see?
It's the surface described by my arm.
It could be a curve that's much nicer
with no self-intersection.
It's still a cylindrical surface.
What's a conic surface?
And that is the last thing I want to do.
And before I say that-- you know what you want to say.
Keep your thought.
Before you go home, what do you promise me to do tonight?
Not tonight.
You have whole weekend, thank god, to do that.
You have tonight-- tonight you can
think of it-- Friday, Saturday, Sunday.
A little bit every day.
One hour, two hours every day.
I'm also a student.
I'm taking some classes on life sciences.
For the first time in my life-- I am 48.
But I decided that it's time for me
to learn some anatomy, physiology, chemistry,
biophysics, protein biology, stuff that I never studied.
And it's a little bit related to the mathematics and geometry
I am doing research on.
And then I got into this a little bit more.
So now I'm taking a class on stress management, which
is very interesting, because I realized
that I have no idea how to manage my own stress.
And all my life, I've made mistakes.
And now I'm taking this class.
And we have homework twice a week-- Tuesday, Thursday.
It's so hard!
I said, I promise, I'm not going to put you in such a--
it keeps me on my toes.
I want you to stay on your toes, but I'm not
going to give you homework that's due that often,
because it really doesn't let you do anything else.
All right?
So you have Friday, Saturday, Sunday
to go over those examples in the session 9.7.
Read them.
No homework yet on WeBWorK.
Sunday you're going to get your first WeBWorK homework.
I don't want to overload you.
One of my classes is about research,
medical research based on mathematics and statistics,
also.
But the other class is stress management.
And I was thinking, this class is about stress management,
but the class in itself may stress me out a little bit more
than anything else, because the homework comes so fast.
I mean, having homework twice a week in every class,
how do you manage to have a job and do your job well?
I don't know how to do that.
It's very-- it's practically impossible.
But I go to bed at 1:00.
So it's not [INAUDIBLE].
Wake up early.
And I hope to survive.
About the conic section, what do we have?
The conic surface.
Sorry.
Conic surface.
Could anybody tell me my analogy?
And I think Alexander is ready to tell me what
the conic surface would be.
It's a surface.
Shall I write down? [? I feel something. ?]
Well, I should write down although [INAUDIBLE].
What do you think that is?
I'll take it slowly.
Is it a ruled surface or not, in your opinion?
STUDENT: Yes.
PROFESSOR: Yes, it is.
It's a ruled surface.
Why do I put ruled in parentheses?
Because it's a little bit like an oxymoron.
When you say, what's a ruled surface,
it's a surface generated by the motion of lines.
So since I've already said that it's
generated by the motion of a straight line,
it's saying the same thing twice.
OK?
So it's a ruled surface generated by the motion of--
STUDENT: A line at a fixed point?
PROFESSOR: Very good.
Of a straight line along a curving plane.
which passes through a fixed point-- through a fixed point.
OK.
So have you ever heard the name pencil of lines?
Pencil of lines.
I have discovered-- I was teaching 3350 last semester.
And I came up with this equation.
Well, [INAUDIBLE].
Differential equations you don't know it yet.
You will learn it next.
The family of solutions of that equation was of the type y
equals kx squared.
k was a real parameter.
Real numbers.
Non-zero.
OK?
What is this if you draw that in a plane?
y equals kx will be a pencil of lines.
I didn't know that 15 years ago.
It was called pencil of lines.
But now I know.
So different slopes.
The slope is k.
All the lines pass through the origin.
So it's a family.
Could contain all of them.
Except for-- well, if you put k equals 0,
then you also have y equals 0, which is this.
OK.
What is a pencil of lines in three dimensions?
It's a family of lines that passes through a fixed point.
Of all these lines that are like the radius-- OK.
So you have like a sphere.
And you have like all the radii coming
from the center of the sphere.
OK?
You know, all the directions.
From all of them, you only take those lines
that intersect the given curve.
L is the curve.
And you have here two conditions.
You straight line, give me a name.
Little l.
l intersects big L different from [INAUDIBLE].
What does this mean?
That little l has to touch big L.
And little l passes through P, which is fixed.
And what you get is a conic surface.
It's a cone.
OK?
It's also a ruled surface.
Is there anything else I wanted to tell you?
Not for the time being.
I think I have exhausted everything I wanted
to teach you about conics.
I told you about conics and quadrics.
I taught you a little bit about conics last time.
I showed you a few quadrics.
Showed you a lot of quadrics today.
This is not over.
Do you allow me to disclose our secret to everybody?
OK.
We have that secret website.
The University of Minnesota has the gallery of quadric.
Did you find it entertaining and useful?
And once you go over those pictures and play with them,
it sticks.
You remember those names for the surfaces
and what they look like.
And it's going to be a good start for the next chapters.
STUDENT: Say that again?
PROFESSOR: All right?
I'm going to send it to you.
I promise.
No, no, no.
It's just I'm going to make it public.
By email I'll send it.
I'll send it to you either today or tomorrow.
OK?
And I'll see you Tuesday.
And on Tuesday, we'll start Chapter 10.
And that's about it.
You were going to have some homework on WeBWorK.
Any questions?
Yes, sir?
STUDENT: So the homework that we get on Sunday is due
Tuesday? [INAUDIBLE].
PROFESSOR: No.
No.
After they put me through this, I
promised I would never put you through this.
The homework in general will be due in minimum seven days,
maximum two weeks.
STUDENT: OK.
PROFESSOR: So depending how long it is.
You can go ahead and turn in the assignment.