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.
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.