TTU Math2450 Calculus3 Secs 9.1-9.6

Review of vectors, dot and cross product.
Parametric equations and equation of the plane.

PROFESSOR: You have learned
a lot in Calculus 2.

Whether you took Calculus
recently or long time ago,

Chapter 9 is about vectors
in r3 and eventually

[? a plane ?] and
operations with such vectors

and the implications
of the vectors

in the equations of a
line in space or plane

in space-- stuff like that.

Now, 9.1 to 9.5 was considered
to be covered completely

in Calc 2 here at Tech.

However, lots of students
come from South Plains College

and [? Rio ?] College,
lots of colleges

where by the nature of
the course Calculus 2,

vectors in r3 are not covered.

Therefore, I'd like to make an
attempt to review 9.1 and 9.5

quickly with the knowledge
you have now as grown-ups

in the area of vectors in r2.

So again, what are vectors?

They are oriented segments.

Not only that they
are oriented segments,

but we make the distinction
between a vector that is fixed

in the sense that his origin
is fixed-- we cannot move him--

and a free vector who is
not married to the origin.

He can shift by parallelism
anywhere in space.

And we call that a free vector.

The distinction between those
vectors would be vr of v bar.

As you remember, v bar was
the free guy, free vector,

which is the-- actually,
it's an equivalence class

of all vectors that can be
obtained from the generic v

bounded.

So I'm going to have to
point by translation.

So you have this kind
of-- same magnitude

for all vectors, same
magnitude, same orientation,

and parallel directions,
parallel lines.

What have we done
to such a vector?

As you remember very
well, we decomposed him,

being on the standard canonical
basis, which for most of you

engineers and engineering
majors is denoted as ijk where

ijk is an orthonormal
frame with respect

to the Cartesian coordinates.

So i, j, and k will
be their unit vectors

on the x, y, z axes of
coordinates, Cartesian axes

of coordinates.

So remember always that ijk
are orthogonal to one another.

Since this is review, I'd
like to attract your attention

to the fact that k is plus j.

Think about it-- what
happens you bring i over j.

And you get k
because you move up.

Because it's like you
are turning [INAUDIBLE]

connection and the screw
or whatever from the faucet

is pointing upwards.

It's like the right hand rule.

If you would do the other way
around, if you do j cross i,

what are you going to have?

Minus k-- so the properties
of the cross product being

antisymmetric are
supposed to be,

no, pay attention to the signs
in all the exams that you have.

What do we know about
their respective products

for vectors in
space or in plain?

If you have two vectors
in their standard basis,

you want i plus
u2j plus u3k where

ui is a real number and
e1i plus v2j plus v3k where

vi are [INAUDIBLE] real numbers
the dot product or the scalar

product-- now, I saw that in
all your engineering and physics

classes, you will
use this notation.

Mathematicians
sometimes say, no, I'm

going to use angular
brackets because it's

a scalar product in r3
or the scalar product

and the dot product
is the same thing,

being that's the
standard one here.

You want v1 plus u2v2 plus u3v3.

So what do you to
remember what you do?

First component
plus first component

times second component
times second component

plus third component
times third component, OK?

If you are in
computer science, I

saw that you use this notation.

I was very happy to see that.

the summation notation.

But you don't have to
use that in our class.

Now, above the [? fresh ?]
product of two vectors,

you have the definition
ijk the first row.

So what you get is
going to be a vector.

Here, what you get is
a scalar as a result.

Here's what you get as
a vector, as the result

of the first
product is a vector.

So you have u1,
u2, u3, v1, v2, v3.

These are all friends of yours.

I'm just reminding you
the lucrative definitions.

Now, some people
said, yes, but I'd

like to see the lucrative
definitions that

have to do with trig as well.

OK, let's see.

For those of you who asked me
to remind you what they were,

I will remind you
what they were.

For u.v, you get the same
thing as writing magnitude u

magnitude v and cosine
of the angle between them

no matter in which
direction you take it

because the cosine is the same.

Cosine of pi is equal to
cosine of negative phi

or theta [INAUDIBLE].

How about the other one?

Here's where one of
you had a little bit

of a misunderstanding.

And I saw that happen in
two finals, unfortunately.

This is not the scalar
vector that I'm right here.

It's a vector.

So what's missing?

This is the scalar part.

And then you have times e
where e is the unit vector

of the direction of the
vector, the direction of u.v.

Why I cannot use another notion?

Because u is already taken.

But e in itself
should suggest to you

that you have a unit vector,
[? length of ?] one vector, OK?

All right, what is the--
let's review a little

bit the absolute value.

Well, the absolute
value is a scalar.

So that scalar will be
magnitude of your magnitude

of [INAUDIBLE]
sine of the angle.

And do you guys remember
the geometric interpretation

of that?

STUDENT: [INAUDIBLE]

PROFESSOR: The area
of the parallelogram

based on the two vectors--
very good [INAUDIBLE].

U plus b is the area
of the parallelogram

that you would draw based
on those two vectors.

All right, good.

Now, say goodbye, vectors.

We've seen-- you've seen
them through 9.4, 9.5.

What was important
to remember was

that these vectors
were the building

blocks, the foundations,
of the equations

of the lines in space.

That's your work [INAUDIBLE].

So what did we work with?

Lines in space-- lines in space
can be given in many ways.

But now that you
remember them, I'm

going to give you the symmetric
equation of a line in space.

OK, [INAUDIBLE] this
can see [INAUDIBLE]

included on the final.

You are expected to know it.

So that is the
symmetric equation,

meaning the equation of a what?

Of a line in space passing
through or containing

the point of p, not of
coordinates x0, y0, [? z0, ?]

and of direction [INAUDIBLE]
in the sense of a vector.

Now, if I were to
draw such a line,

I'm going to have
the line over here.

Going to have a vector for
the point p0 on the line.

I can put this free
vector because he's free.

He says, I'm a free guy.

I can slide any way I want.

So I'm going to have
li plus mj plus mk.

This is the blue vector.

Now, you don't have blue
markers or blue pens,

but you can still do a
good job taking notes.

Now somebody asked
me just a week ago,

saying that I've started
doing review already

and I don't understand
what the difference is

between the symmetric
equation of a line in space

and the parametric equations
of a line in space.

This is no essential difference.

So what do we do?

We denote this whole
animal by t, a real number.

And then we erase the board.

And then we write
the three equations

that govern-- I'm going
to put if and only

if xyz satisfy the following.

So I'm going to have, what?

X equals lt plus x0, y equals
mt plus y0, n equals nt plus z0.

Well, of course, we
understand-- we know the meaning

that lmn are like what
[INAUDIBLE] physics direction

cosines were
telling me about it.

And then x0, y0,
z0 is a fixed point

that belongs to that line.

Now, since you know a
little bit more than

you knew in Calculus
2 when you saw

that for the first time, what
is the typical notation that we

use all through Calc 3,
all through the chapters?

The position vector-- the
position vector of the point

on the line that is
related to, what?

So practically you
have the origin here.

[? Op0 ?] represent the
vector x0i plus y0j plus e0k.

So now you have a little bit
of a different understanding

of what's going on.

And then after, let's say, t
equals 1 hour, what do you do?

You are adding the
blue vector here.

Let's say at t equals
1, you are here.

You are here at p1.

So to get to p1, you have to
add two vectors, right guys?

This is the addition between the
blue vector and the red vector.

So what you get is your result.

So if I am smart enough to
understand my concepts are all

connected, the
position in this case

will be r of t, which is--
I hate angular brackets,

but just because
you like them, I'm

going to use them--
x of ty tz of tm

to be consistent with the book.

This is the same as
xi plux yj plus zk.

And what is this by
the actual notations

from the parametric equation?

This is nothing
but a certain lmn

vector that is the vector
li plus mj plus nk written

with angular brackets
because I know

you like that times the time
t plus the fixed vector x0,

y0, z0.

You can say, yeah, I
thought it was a point.

It is a point and a vector.

You identified the point p0 with
the position of the point p0

starting with respect
to the origin.

So whether you're
talking about mister p0

being a point in
space-- x0, y0, z0.

Or you're talking about
the [INAUDIBLE] position

vector that [INAUDIBLE]
is practically

the same after identification.

So you have something very nice.

And if I asked you with
the mind and the knowledge

you have now what that
does is mean-- r prime of t

equals what?

It's the velocity vector.

And what is that as a vector?

Do the differentiation.

What do we get in terms
of velocity vector?

Prime with respect
to t-- what do I get?

STUDENT: Lmn.

PROFESSOR: Lmn as a vector.

But of course, as I
hate angular notations,

I will rewrite it--
li plus mj plus nk.

So this is your velocity.

What can you say about
this type of motion?

This is a--

STUDENT: [INAUDIBLE]
constant velocity.

PROFESSOR: Yeah, you
have a constant velocity

for this motion.

If somebody would ask you
you have-- 10 years from now,

you have a boy who
said, dad-- or a girl.

let's not be biased.

So he learns, math, good at
math or physics, and says,

what is the difference
between velocity and speed?

Well, most parents will
say it's the same thing.

Well, you're not most parents.

You are educated parents.

So this is-- don't tell
your kid about vectors,

but you can show them you
have an oriented segment.

So make your child run around
around in circles and say,

this is the velocity that's
always tangent to the circle

that you are running on.

That's a velocity.

And if they ask,
well, they will catch

the notions of acceleration
and force faster than you

because they see
all these cartoons.

And my son was telling me
the other thing-- he's 10

and I asked him, what
the heck is that?

It looked like an
electromagnetic field

surrounding some hero.

And he said, mom, that's
the force field of course.

And I was thinking, force field?

This is what I taught
the other day when

I was talking about [? crux. ?]
Double integral of f.n

[INAUDIBLE] f was
the force field.

So he was, like, talking
about something very normal

that you see every day.

So do not underestimate your
nephews, nieces, children.

They will catch
up on these things

faster than you, which is good.

Now, the speed in this
case will be, what?

What is the speed
of this-- the speed

of this motion, linear motion?

STUDENT: Square
root of l squared.

PROFESSOR: Square root of l
squared plus m squared plus n

squared, which again is
different from velocity.

Velocity is a vector,
speed is a scalar.

Velocity is a vector,
speed is a scalar.

In general, doesn't
have to be constant,

but this is the
blessing because lmn

are given constants.
[INAUDIBLE] in this case,

you are on cruise control.

You are moving on
a line directly

in your motion on
cruise control driving

to Amarillo at 60 miles
an hour because you

are afraid of the cops.

And you are doing
the right thing

because don't mess with Texas.

I have friends who came here
to visit-- Texas, New Mexico,

go to Santa Fe, go
to Carlsbad Caverns.

Many of them got caught.

Many of them got tickets.

So it's really serious.

OK, that's go further
and see what we

remember about planes in space.

Because planes in
space are magic?

No.

Planes in space
are very important.

Planes in space are
two dimensional objects

embedded three dimensional
[? area ?] spaces.

This is what we're
talking about.

But even if you lived
in a four dimensional

space, five dimensional
space, n dimensional space,

in the space of
your imagination,

if you have this two
dimensional object,

it would still be
called a plane.

All right, so how about planes?

What is their equation?

In your case ax plus by plus cz
plus d is the general equation.

We now have a plane in r3.

You should not forget about it.

It's going to haunt
you in the final

and in other exams in your
life through at least two

or three different exercises.

Now I'm going to ask you
to do a simple exercise.

What is the equation of the
plane normal to the given line?

And this is the given line.

Look at it, how
beautiful [INAUDIBLE].

And passing through-- that
passes through the point

another point-- x1, y1,
z1-- that I give you.

How do you solve solution?

How do you solve this quickly?

You should just remember
what you learned

and write that as
soon as possible.

Because, OK, this
may be a little piece

of a bigger problem in my exam.

STUDENT: [INAUDIBLE]

[? PROFESSOR: Who is ?] a?

If this is normal to the line--

STUDENT: A is 1.

PROFESSOR: You
pick up abc exactly

from the lmn of the line.

Remember this was an essential
piece of information.

So the relationship between
a line and its normal plane

is that the direction
of that line lmn

gives the coefficients abc
of the plane, all right?

Don't forget that because you're
going to stumble right into it

in the exams [? lx ?] in the
coming up-- in the one that's

coming up.

And c, is this good?

No, I cannot say d
and then look for d.

I could-- I could [INAUDIBLE].

Whatever you want.

But then it's more work for me.

Look, I don't know-- suppose
I don't know who d is.

I have to make the
plane satisfy--

make the point
x1, y1, z1 satisfy

the equation of the plane.

And that is more work.

I can do that if I forget.

If I forget the theory,
I can always do that.

Subtract the two lines, subtract
the second out of the first.

I get something
magic that I should

have known from my
previous knowledge,

from a previous life-- no.

L times x minus x1 plus m
times y minus y1 plus z times

z minus 1.

And I notice that most of
you-- you prove me on exams,

you prove me on homework--
know that if you have

the coefficients and you
also have the point that

is containing the plane,
you can go ahead and write

this equation from the start.

So you know very well
that x1, y1, z1 satisfies

your [INAUDIBLE] the plane Then
you can go ahead and write it.

Save time on that
exam Don't waste time.

It's like a star test
that's a four hour test.

No, ours is only two
hours and a half.

But still, the pressure
is about the same.

So we have to remember
these notions.

We cannot survive without them.

Let's move on.

And one of you asked me.

Do I need to know by
heart the formula that

give-- a formula that will give
the distance between a point

in space and a line in space?

No, that is not assumed.

You can build up to that one.

It's not so immediate.

It takes about 15 minutes.

That's not a problem.

What you are
supposed to remember,

though, is that the formula
for distance between a given

point in plane and a
point in space and a given

plane in space-- that was a long
time ago that you knew that,

but I said you should
never for get it

because it's similar
to the formula

for the distance between a point
in plane and a line in plane.

I'm not testing you, but I
will-- I hope-- maybe I do.

I hope that you remember how
to write this as a fraction.

I'm already giving you hits.

What is--

STUDENT: [INAUDIBLE]

PROFESSOR: Absolute value
because it's a distance.

STUDENT: [INAUDIBLE]

PROFESSOR: Of what?

STUDENT: Ax.

PROFESSOR: Ax0
plus by0 plus cz0--

STUDENT: Plus b.

PROFESSOR: Plus b, O. Good.

STUDENT: [INAUDIBLE]

PROFESSOR: Square root of--

STUDENT: A squared.

PROFESSOR: A squared plus
b squared plus c squared.

Right, so it's a generalization
of the formula of the--

in plane if you have a
point and a line that

doesn't contain the point, you
have a similar type of formula.

Good, let's remember
the basics of conics.

Because I'm afraid that
you forgot them from Calc 2

and from analytic or
trigonometry class.

What were the standard conics
that were used in this class

and I would like
you to never forget?

Well, when you are
in an exam, you

may be asked the [INAUDIBLE]
inside of an ellipse.

But if you don't know
the standard equation

of an ellipse, that's bad.

So you should.

What is that?

Ab are semi-axis.

STUDENT: X squared
over a squared.

PROFESSOR: X squared over
a squared plus y squared

over b squared equals 1.

Excellent, and what
if I have-- I'm

going to draw a
rectangle with these kind

of semi axes a and b.

And I'm going to draw the
diagonals-- the diagonals.

And I'm going to draw
a [INAUDIBLE] something

that is touching, kissing
at this point tangent to it.

And it's asymptotic to
the blue asymptotes.

What is this animal?

STUDENT: Hyperbola.

PROFESSOR: The
standard hyperbola?

Tell me what-- it
has these branches.

The equation is what?

STUDENT: [INAUDIBLE]

PROFESSOR: X squared over
a squared minus y squared

over b squared equals 1.

If I were to draw
its brother-- oh--

that brother would
be the conjugate, OK?

And you would have to swap
the sides of plus minus.

And you'll get the conjugate.

Quadrics-- OK, the parabola, I
don't remind you the parabola

because you see it everywhere.

I'm going to review it when
I work with some quadrics.

So the [INAUDIBLE]
quadrics-- and I really

would like you to, if you feel
the need to remind yourself

when quadrics are, go to the
so-called gallery of quadrics.

Type these magic words
as keywords in Google.

And it's going to send
you to a beautiful website

from University of Minnesota
that has a gallery of quadrics

where not only do you see
the most important quadrics

in standard forms, but you also
see the cross sections that you

have when you curve those
quardics with horizontal planes

or other planes parallel to
the planes of coordinates.

So I don't know in which
order to present them to you.

But how about I present
them to you in the order

that they were mostly
frequently used

rather than starting with--
so ellipsoid and respectively

a sphere.

Depends if you like football--
American football or soccer.

Well, let's see what
the equations were.

X squared over a
squared plus y squared

over b squared plus z
squared over c squared

equals 1 for the ellipsoids.

If abc are equal and
equal to r, what is that?

That's a sphere of center
origin-- standard sphere--

in radius .

R These are your friends.

Don't forget about them.

When you draw the
ellipsoid, remember

that the first line,
the dotted one,

is an ellipse on the
other behind the board.

And that is obtained
as x squared

over a squared plus y squared
over b squared equals 1.

So it's going to be an
intersection with z equals 0

And similarly, you can take
the plain that's x equals 0.

And you get this ellipse,
the plane that is y equals 0.

And you get this ellipse.

So those are all
friends of yours.

Remember that all
the cross sections

you have cutting with planes,
the football, you have, what?

Ellipses.

That is easy and
beautiful and it's not

something you need a
lot of thinking about.

But let's move on some other
guys that I'm afraid you forgot

and you should not
forget in any case.

And the hyperboloids--
hyperboloids,

the most standard ones,
the classification

that we had in the classroom
was based on putting everybody

to the left hand side.

How many pluses, how many
minuses you have had?

If you have plus, plus, plus,
minus or minus, minus, minus,

plus, you have an uneven
number of pluses and minus.

That was the
two-sheeted hyperbola.

If you had an even number
of pluses and minuses,

that's the one sheet hyperbola.

So let us remember
how that went.

Assuming that I
love this one, this

is the first one--
the first kind which

is the one-sheeted hyperboloid.

What is the symmetry axis?

The surface of
revolution-- What axis?

Of axis 0x.

So I'm going to go
ahead and draw that.

I'm going to draw
as well as I can.

I cannot draw very well today.

Although I had three cups
of coffee, doesn't matter.

I'm still shaking when
it comes to drawing.

So in order to get the cross
section, the first cross

section, the red one,
what do you guys do?

STUDENT: [INAUDIBLE]

PROFESSOR: It's a-- what?

It's an ellipse
because you said z

equal to 0 just as you said now.

So I get the ellipse
of semi axis a and b.

This is the x-axis.

This is a.

This is b.

Well, it looks
like horrible in b.

And that's the
[INAUDIBLE] we have.

But now you say,
but wait a minute.

I would like to draw the cross
section that corresponds to x

equals 0.

And that should be in
the plane of the board.

So if you set x to be 0, then
you have the standard hyperbola

based on semi axes b and c.

Now, b, you believe me.

But c, you don't believe me
at all because you cannot see.

So if I were to be proactive--
which right now I'm

not very proactive,
but I'll try--

I'm going to have
to draw-- look,

I'm not done even if I
didn't have enough coffee.

So the rectangle--
you see b and c here?

OK, you see the asymptote?

It was not a bad guess
of the asymptote.

This branch of the cross
section looks like, really,

a good branch for the asymptote.

Good, and the other
one in a similar way,

you can find the
other cross section,

which is also a hyperbola.

So your old friend which
is one-sheeted hyperboloid,

hyperboloid-- it sounds
like a monster-- what

was special about him?

You have some extra credit.

STUDENT: [INAUDIBLE]

PROFESSOR: It's a
[? ruled ?] surface generated

by two families of lines.

And thanks again for the model.

I will keep it for
the rest of my life.

You got five bonus
points because of that.

I'm just-- well,
this is something

I will always remember.

Number two, how do I write
that two-sheeted hyperboloid

if I wanted me to have
the same axis of symmetry?

It should be a
surface of revolution

consisting of two parts, two.

They are disconnected, right?

You have two sheets,
two somethings,

two connected components.

It's not hard at all.

What do I need to do?

The same thing as here-- just
change the minus to a plus.

All righty, x squared over
a squared plus y squared

over b squared minus z squared
over c squared plus 1 equals 0.

Great, so I can go ahead and
reminds you what that was.

You didn't like
it when you first,

but maybe now you
like it better.

This is always yz.

And I'm going to
draw the two sheets.

And I'm going to
ask you eventually,

because I am mean, how
far apart they are.

It's the surface of revolution.

These two guys
should be symmetric.

Well, so when I were-- if
I were to take z equals 0,

I would get no solution
because this is impossible.

I have a sum of
squares equal 0, right?

It's impossible
to get 0 this way.

When would I get 0 on
the axis of rotation?

Well, axis of rotation
means forget about x and y.

X is 0, y is 0.

Z would be how much?

STUDENT: C.

PROFESSOR: Plus minus c.

Plus minus-- very good.

C, practically c, if c is
positive, and minus c here.

So I know how far apart they
are, these two-- [INAUDIBLE]

this is not [? x ?] [INAUDIBLE]
minimum and the maximum

over here.

Now, one last question.

Well-- OK, no.

More questions-- when I were
to intersect with, let's

say, a z that is bigger
than c, a z plane that

is bigger than c over here,
what am I going to get?

No--

STUDENT: An ellipse.

PROFESSOR: An
elipse-- excellent.

An ellipse here, an
ellipse there everything

is symmetrical.

And finally, what
if I take x to be 0?

I'm in the plane of the board.

I hide the x.

I get this.

What is this?

A hyperbola in the plane
of the board, which is yz.

Y is going this
way, z is going up.

X doesn't exist anymore.

So what kind of
hyperbola is this?

Do you like it?

So--

STUDENT: [INAUDIBLE]

PROFESSOR: Right, mean smart.

go ahead and multiply by
negative-- who said that?

Zander, you got two extra
points, extra [INAUDIBLE].

Minus y squared over
b squared equals 1.

What did he notice?

What did he-- he gets my mind.

I'm trying to say you have
no hyperbola like that.

So Zander said, I
know what which ones.

She wants these two branches
to be the hyperbola.

But that's a
conjugate hyperbola.

That is a conjugate
hyperbola because you

don't have y and z with minus
between the squares and a y.

So this is the conjugate
hyperbola-- hyperbola--

that I'm going to draw.

In what color?

That's the question.

It's really essential what
color I'm going to use.

So I'm going to use--
I'm going to use green.

And this is the hyperbola
we are talking about.

It's a conjugate one drawn
in the plane of the board.

OK, all right.

So if I wanted to
drop those asymptotes,

they will look very ugly.

And I cannot do better,
but that's [INAUDIBLE].

So we have reviewed the
most awful quadrics.

A friend of yours that
by now all of you love

is mister paraboloid.

You have used that in
all sorts of examples.

I'm going to remind you
what the standard one was

that we used before.

So [INAUDIBLE] paraboloids,
elliptic paraboloid.

Circular paraboloid is
just the particular case.

The elliptic paraboloid
that you're used to

is the following-- z equals
x squared over a squared

plus y squared over b squared.

They may be positive
if you want.

They don't-- in general,
they are not equal.

The circular paraboloid--
well, you simply

assume that a and b are equal.

And then you put-- you want
a c squared or an r squared.

Let's put an r squared on top.

It really doesn't matter
what you're putting there.

Can I draw?

Hopefully, hopefully,
hopefully I can draw.

It looks like a
valley whose minimum

is at the origin
I'm going to draw

so that the intersection
with the horizontal plane

will be visible to you.

And I take this
z greater than 0.

And then I'm going to
have some sort of ellipse.

Under that, there is nothing.

Under the origin,
there is nothing

because z is going to be
positive at x equals 0,

y equals 0, and passing
through the origin-- very

nice and [? sassy ?] Quadric.

There is one that occurred in
many examples like a nightmare.

And it was based on that one.

And I'm going to
draw-- no, no, no.

I'm going to write
it and you draw it

with the eyes of
your imagination

and see what that is.

Because you are, again, going
to bump into it into the exam.

We had all sorts
of patches of that.

Look at the areas of the patch.

And you cannot get rid of that.

It's haunting your dreams.

What is this?

STUDENT: [INAUDIBLE]

PROFESSOR: Upside
down paraboloid--

what is the vertex?

Where is the vertex at?

STUDENT: 0, 0, 1.

PROFESSOR: 0, 0, 1-- very good.

What's special about it?

So assume that I
would draw the-- I

would draw it to compute
the normal to the surface.

How would I do that?

STUDENT: [INAUDIBLE]

PROFESSOR: Uh, yeah.

Well, it's a little
bit more complicated.

I would have to shift
everybody to once side,

the side that I have a certain
increase in form [? than to ?]

the gradient
[? to stuff ?] like that.

So don't forget about this type
of project is an essential one.

Am I missing anybody important?

Yes.

We live in tests.

We cannot say goodbye to the
last section of the chapter

nine, which is 9.7, without
meeting again our friend

the saddle, right?

The saddle is-- this
is elliptic paraboloid.

And the last very
important quadric

that I wanted to talk
about today is the--

STUDENT: What about a cone?

PROFESSOR: Huh?

STUDENT: How about a cone?

PROFESSOR: Oh, a
cone is too easy.

But yeah, let's talk
about the cone as well.

Give me an example
of the standard cone.

Thank you, [INAUDIBLE].

X squared-- well--

STUDENT: T squared
equals x squared.

PROFESSOR: I'm going
to draw it first

so that you know what I want.

Unless I draw it, how
would you know what

to invent or to come up with?

It's not an ice cream
cone-- it's a double cone.

So I can have a positive
z and a negative z-- two

different sheets that are
symmetric with one another.

So how do I write that?

STUDENT: T squared.

PROFESSOR: Yes, equals?

STUDENT: [INAUDIBLE]

PROFESSOR: Well, would you like
it to be like most of those

that we see in the examples
in the book, right?

But it doesn't have
to be like that.

Of course, if it's
like that, of course

you realize z-- set the z.

Set the plane and altitude.

Then you're going to have
circle, circle, circle,

circle-- circle after
circle of different radii

as cross sections.

Z could also be negative.

Except for the case of the
origin, where you have 0, 0, 0.

Now, if you were to set
x equals 0, of course

you would get y
equals plus minus

z, which are exactly these
lines, the red lines that I'm

drawing in this picture.

So practically, this is, what?

Called a what?

A circular cone.

If I wanted to make
it more interesting,

I would put a squared
and b squared.

And it would be
an elliptic cone.

And we stayed away from
that as much as we could.

We brought it up now because
Zander asked about it.

So how about the number four,
number five, whatever it

is-- number four?

The [INAUDIBLE] what
was the typical equation

of the hyperbolic paraboloid
that I had in mind?

STUDENT: [INAUDIBLE]

PROFESSOR: Z equals x squared
minus y squared, very good.

So I will try again and draw it.

It's not so easy to draw.

If I were to choose x
to be 0 and draw exactly

in the plane of the board,
z equals minus y squared

would be some
coordinate line, right?

This is what we
call such a thing.

If we fix the x to be x0,
we get a coordinate line.

If we fix the y to be y0, we
get another coordinate line.

There are two families of lines.

Why is z equals minus y
squared drawn in this board,

on the board, in this plane?

It's a parabola that
opens upside down.

OK, so you have something like
this which you are drawing,

right?

And then what if y would be 0?

Then you get z equals x0.

So it's going to a
parabola that opens up.

Then I have to locate myself
and draw it on that wall.

But I can't.

Because if I do that, I'm
going to get in trouble.

So I better draw it like
this in perspective.

And you guys should
imagine what we have.

So if we were to cut
down with a knife,

we would get-- we will
still get these parabolas

that all point down.

And in those directions, these
are just the highest parts

of the saddle.

And let's say this would be
the lowest part of the saddle.

Where-- where is the
part of the rider?

A guy's butt is here.

And his leg is following the
shape of the saddle going down.

That's the cowboy boot, OK?

And he is-- hold on.

I don't know how-- what's the
attitude of the [INAUDIBLE]?

Well, it doesn't look
like a cowboy hat.

But anyway, I'm sorry.

He looks a little
bit Vietnamese.

That was not the intention.

STUDENT: [INAUDIBLE]

PROFESSOR: Then
let him be Mexican.

Half of the population
in this town are Mexican.

So this is his leg
that goes down.

OK, very good.

Look-- he even has a--
what do you call that?

That's so beautiful.

In the Mexican culture, they
make those embroidered by hand

with many colors belts.

But there are some
special belts.

OK-- depends on the area
of Mexico You visit.

I liked several of them.

They're so beautiful.

But my favorite
one is, of course,

the Rivera Maya, which is where
you go to the Chichen Itza,

to the mystic areas, to the
sea, and eat the good food

and go to Cozumel and forget
about school for a week.

That is paradise for me.

But [INAUDIBLE]
is not bad either.

If I were to choose where
to live and I had money,

I would live in Cozumel
for the rest of my life.

OK, so this is the
saddle that is oriented

so that you have a parabola
going in this direction,

going up, a parabola going
down in this direction.

What is magic about
a saddle point?

Do you remember?

STUDENT: It was [INAUDIBLE]

PROFESSOR: It's-- one direction
is like a max and one direction

is like a min.

So when you compute that
discriminant, you get negative.

You get like the product
between the curvatures.

One curvature in one
direction would be positive.

The other one would be negative.

So it's like getting the product
plus minus equals y, [? yes. ?]

But again, we will talk
about this sometimes later.

You don't even
have to know that.

Shall we say goodbye to This?

I guess it's time.

And chapter nine is now
fresh in your memory.

You would be really
to start chapter 10.

What I want to do, I want
to-- without being recorded.