TTU Math2450 Calculus3 Sec 9.7

Quadric Surfaces

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.

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.

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.