PROFESSOR: So let's forget about this example
and review what we learned in 11.3, chapter 11.
Chapter 11, again, was functions of several variables.
In our case, I'll say functions of two variables.
11.3 taught you, what?
Taught you some beautiful things.
Practically, if you understand this picture,
you will remember everything.
This picture is going to try and [INAUDIBLE] a graph
that's sitting above here somewhere
in Euclidean free space, dimensional space.
You have the origin.
And you say I want markers.
No, you don't say I want markers.
I say I want markers.
We want to fix a point x0, y0 on the surface,
assuming the surface is smooth.
That x0 of mine should be projected.
I'm going to try to draw better than I did last time.
X0, y0 corresponds to a certain altitude
z0 that is projected like that.
And this is my [INAUDIBLE] 0 here.
But I don't care much about that right now.
I care about the fact that locally, I
represent the function as a graph-- z of f-- f
of x and y defined over a domain.
I have a domain that is an open set.
And you connect to-- that's more than you need to know.
Could be anything.
Could be a square, could be a-- this could be something,
a nice patch of them like.
So the projection of my point here is x0, y0.
I'm going to draw these parallels as well as I can.
But I cannot draw very well.
But I'm trying.
X0 and y0-- and remember from last time.
What did we say?
I'm going to draw a plane of equation x equals x0.
All right, I'll try.
I'll try and do a good job-- x equals x0 is this plane.
STUDENT: Don't you have the x amount
and the y amounts backward?
Or [INAUDIBLE]
PROFESSOR: No.
STUDENT: [INAUDIBLE]
PROFESSOR: x is this 1 coming towards you like that.
And I also think about that always, Ryan.
Do I have them backward?
This time, I was lucky.
I didn't have them backward.
So y goes this way.
For y0, let me pick another color, a more beautiful color.
For y0, my video is not going to see the y0.
But hopefully, it's going to see it, this beautiful line.
Spring is coming.
So this is going to be the plane.
Label it [INAUDIBLE] y equals 0y.
Now, the green plane cuts the surface into a plane curve,
of course, because teasing the plane
that I drew with the line.
And in the plane that I drew with red-- was it red or pink?
It's Valentine's Day.
It's pink.
OK, so I have it like that.
So what is the pink curve?
The pink curve is the intersection between z
equals-- x equals x0 plane with my surface.
My surface is black.
I'm going to say s on surface.
And then I have a pink curve.
Let's call it c1.
Because you cannot see pink on your notes.
You only can imagine that it's not the same thing.
C2 is y equals y0 plane intersected with s.
And what have we learned last time?
Last time, we learned that we introduce some derivatives
at the point at 0, y is 0, so that they represent
those partial derivatives of the function z with respect
to x and y.
So we have the partial z sub x at x0, y0 and the partial and z
sub y at x0, y0.
Do we have a more elegant definition?
That's elegant enough for me, thank you very much.
But if I wanted to give the original definition, what
was that?
That is d of bx at x0, y0, which is a limit of the difference
quotient.
And this time, we're going to-- not going to do the x of y.
I I'm different today.
So I do h goes to 0.
H is my smallest displacement of [INAUDIBLE].
Here, I have f of-- now, who is the variable?
X.
So who is going to say fixed?
Y. So I'm going to say I'm displacing mister x0 with an h.
And y0 will be fixed minus f of x0, y0, all over h.
So again, instead of-- instead of a delta x, I call the h.
And the derivative with respect to y
will assume that x0 is a constant.
I saw how well [INAUDIBLE] explained that
and I'm ambitious.
I want to do an even better job than [INAUDIBLE].
Hopefully, I might manage.
D of ty equals [INAUDIBLE] h going to 0 of that CF of-- now,
who's telling me what we have?
Of course, mister x, y and y, yy.
F of x0, y0 is their constant waiting for his turn.
H is your parameter.
And then you'll have, what?
H0 is fixed, right?
STUDENT: So h0 is--
PROFESSOR: --fixed.
Y is the variable.
So I go into the direction of y starting from y0.
And I displace that with a small quantity, right?
So these are my partial velocity--
my partial derivatives, I'm sorry, not partial velocities.
Forget what I said.
I said something that you will learn later.
What are those?
Those are the slopes at x0, y0 of the tangents at the point
here, OK?
The tangents to the two curves, the pink one-- the pink one
and the green one, all right?
For the pink one, for the pink curve, what is the variable?
The variable is the y, right?
So this is c1 is a curve that depends on y.
And c2 is a curve that depends on x.
So this comes with x0 fixed.
I better write it like that.
F of x is 0.
Y, instead of c2 of x, I'll say f of y-- f of x and yz.
So, which slope is which?
The d of dy at this point is the slope to this one.
Are you guys with me?
The slope of that tangent.
Considered in the plane where it is.
How about the other one?
S of x will be the slope of this line in the green plane, OK?
That is considered as a plane of axis of coordinates.
Good, good-- so we know what they are.
A quick example to review-- I've given you some really ugly,
nasty functions today.
The last time, you did a good job.
So today, I'm not challenging you anymore.
I'm just going to give you one simple example.
And I'm asked you, what does this guy look like
and what will the meanings of z sub x and z sub y be?
What will they be at?
Let's say I think I know what I want to take at the point 0, 0.
And maybe you're going to tell me what else it will be.
And eventually at another point like z
sub a, so coordinates, 1 over square root of 2 and 1
over square root of 2.
And v sub y is same-- 1 over square root of 2,
1 over square root of 2.
Can one draw them and have a geometric explanation
of what's going on?
Well, I don't want you to forget the definitions,
but since you absorbed them with your mind hopefully
and with your eyes, you're not going to need them anymore.
We should be able to draw this quadric that you love.
I'm sure you love it.
When it's-- what does it look like?
STUDENT: [INAUDIBLE]
PROFESSOR: Wait a minute, you're not awake or I'm not awake.
So if you do x squared plus y squared,
don't write it down please.
It would be that.
And what is this?
STUDENT: That's a [INAUDIBLE].
PROFESSOR: A circular paraboloid-- you are correct.
We've done that before.
I'd say it looks like an egg shell,
but it's actually-- this is a parabola
if it's going to infinity.
And you said a bunch of circles.
Yes, sir.
STUDENT: So is it an upside down graph?
PROFESSOR: It's an upside down paraboloid.
STUDENT: [INAUDIBLE]
PROFESSOR: So, very good-- how do we do that?
We make this guy look in the mirror.
This is the lake.
The lake is xy plane.
So this guy is looking in the mirror.
Take his image and shift it just like he
said-- shift it one unit up.
This is one.
You're going to have another paraboloid.
So from this construction, I'm going to draw.
And he's going to look like you took a cup
and you put it upside down.
But it's more like an eggshell, right?
It's not a cup because a cup is supposed
to have a flat bottom, right?
But this is like an eggshell.
And I'll draw.
And for this fellow, we have a beautiful picture
that looks like this hopefully But I'm going to try and draw.
STUDENT: Are you looking from a top to bottom?
PROFESSOR: We can look it whatever you want to look.
That's a very good thing.
You're getting too close to what I wanted to go.
We'll discuss in one minute.
So you can imagine this is a hill full of snow.
Although in two days, we have Valentine's Day
and there is no snow.
But assume that we go to New Mexico
and we find a hill that more or less looks
like a perfect hill like that.
And we start thinking of skiing down the hill.
Where am I at 0, 0?
I am on top of the hill.
I'm on top of the hill and I decide
to analyze the slope of the tangents
to the surface in the direction of-- who is this?
Like now, and you make me nervous.
So in the direction of y, I have one slope.
In the direction of x, I have another slope in general.
Only in this case, they are the same slope.
And what is that same slope if I'm here on top of the hill?
This is me-- well, I don't know, one of you guys.
That looks horrible.
What's going to happen?
We don't want to think about it.
But it definitely is too steep.
So this will be the slope of the line in the direction
with respect to y.
So I'm going to think f sub y and f sub
x if I change my skis go this direction and I go down.
So I could go down this way and break my neck.
Or I could go down this way and break my neck as well.
OK, it has to go like-- right?
Can you tell me what these guys will be?
I'm going to put them in pink because they're beautiful.
STUDENT: 0 [INAUDIBLE].
PROFESSOR: Thank God, they are beautiful.
Larry, what does it mean?
That means that the two tangents, the tangents
to the curves, are horizontal.
And if I were to draw the plane between those two tangents--
one tangent is in pink pen, our is in green.
Today, I'm all about colors.
I'm in a good mood.
And that's going to be the so-called tangent plane--
tangent plane to the surface at x0, y0, which is the origin.
That was a nice point.
That is a nice point.
Not all the points will be [INAUDIBLE]
and nice but beautiful.
[INAUDIBLE] I take the nice-- well, not so nice,
I don't know.
You'll have to figure it out.
How do I get-- well, first of all, where is this point?
If I take x to be 1 over square 2 and y to be 1 over square 2
and I plug them in, what's z going to be?
STUDENT: 0.
PROFESSOR: 0, and I did that on purpose.
Because in that case, I'm going to be on flat line again.
This look like [INAUDIBLE].
Except [INAUDIBLE] is not z equal 0.
What is [INAUDIBLE] like?
z equals--
STUDENT: [INAUDIBLE]
PROFESSOR: Huh?
STUDENT: I don't know.
PROFESSOR: Do you want to go in meters or in feet?
STUDENT: [INAUDIBLE].
It's about a mile.
PROFESSOR: Yes, I don't know.
I thought it's about one kilometer, 1,000 something
meters.
But somebody said it's more so.
It's flat land, and I'd say about a mile above the sea
level.
All right, now, I am going to be in flat land right here,
1 over a root 2, 1 over a root 2, and 0.
What happened here?
Here, I just already broke my neck, you know.
Well, if I came in this direction,
I would need to draw a prospective trajectory that
was hopefully not mine.
And the tangent would-- the tangent, the slope
of the tangent, would be funny.
Let's see what you need to do.
You need to say, OK, prime with respect to x, minus 2x.
And then at the point x equals 1 over [INAUDIBLE] 2 y
equals 1 over 2, you just plug in.
And what do you have?
STUDENT: Square root of [INAUDIBLE].
PROFESSOR: Negative square root of 2-- my god
that is really bad as a slope.
It's a steep slope.
And this one-- how about this one?
Same idea, symmetric function.
And it's going to be exactly the same-- very steep slope.
Why are they negative numbers?
Because the slope is going down, right?
That's the kind of slope I have in both directions--
one and-- all right.
If I were to draw this thing continuing,
how would I represent those slopes?
This circle-- this circle is just making my life harder.
But I would need to imagine those slopes as being
like I'm here, all right?
Are you guys with me?
And I will need to draw x0-- well, what is that?
1 over root 2 and 1 over root 2 And I would draw two planes.
And I would have two curves.
And when you slice up, imagine this
would be a piece of cheese.
STUDENT: [INAUDIBLE]
PROFESSOR: And you cut--
STUDENT: [INAUDIBLE]
PROFESSOR: Right?
STUDENT: Yeah.
PROFESSOR: And you cut in this other side.
Well, this is the one that's facing you.
You cut like that.
And when you cut like this, it's facing--
STUDENT: [INAUDIBLE]
PROFESSOR: Hm?
But anyway, let's not draw the other one.
It's hard, right?
STUDENT: [INAUDIBLE] angle like this-- just the piece
of the corner of the cheese.
PROFESSOR: Right.
STUDENT: The corner is facing you.
PROFESSOR: So yeah, so it's-- the corner is facing you.
STUDENT: So basically, you [INAUDIBLE] this.
PROFESSOR: But-- exactly, but--
STUDENT: Like this.
PROFESSOR: Yeah, well-- yeah, it's hard to draw.
So practically, this is what you're
looking at it is slope that's negative in both directions.
So you're going to go this way or this way.
And it's much steeper than you imagine [INAUDIBLE].
OK, they are equal.
I'm trying to draw them equal.
I don't know how equal they can be.
One belongs to one plane just like you said.
This belongs to this plane.
And the green one belongs to the plane that's facing you.
So the slope goes this way.
But the two slopes are equal.
You have to have a little bit of imagination.
We would need some cheese to make a mountain of cheese
and cut them and slice them.
We'll eat everything after, yeah.
All right, let's move on to something more challenging
now that we got to the tangent plane.
So if somebody would say, wait a minute,
you said this is the tangent plane to the surface.
You just introduced a new notion.
You were fooling us.
I'm fooling you guys.
It's not April 1, but this kind of a not a neat thing.
I just tried to introduce you into the section 11.4.
So if you have a piece of a curve that's smooth
and you have a point x0, y0, can you
find out the equation of the tangent plane?
Pi, and this is s form surface.
How can I find the equation of the tangent plane?
That x0, y0-- 12 is going to be also z0.
But what I mean that x0, y0 is in on the floor
as a projection.
So I'm always looking at the graph.
And that's why.
The moment I stop looking at the graph,
things will be different.
But I'm looking at the graph of independent variables x, y.
And that's why those guys are always on the floor.
A and z would be a function to keep in the variable.
Now, does anybody know?
Because I know you guys are reading in advance
and you have better teachers than me.
You have the internet.
You have the links.
You have YouTube.
You have Khan Academy.
I know from a bunch of you that you have already gone
over half of the chapter 11.
I just hope that now you can compare what you learned
with what I'm teaching you, And I'm not
expecting you to go in advance, but several of you
already know this formula.
We talked about it in office hours on yesterday.
Because Tuesday, I didn't have office hours.
I had a coordinator meeting.
So what equation corresponds to the tangent plate?
STUDENT: [INAUDIBLE]
PROFESSOR: Several of you know it.
You know what I hated?
It's fine that you know it.
I'm proud of you guys and I'll write it.
But when I was a freshman-- or what the heck was I?
A sophomore I think-- no, I was a freshman when they fed me.
They spoon-fed me this equation.
And I didn't understand anything at the time.
I hated the fact that the Professor painted it
on the board just like that out of the blue.
I want to see a proof.
And he was able to-- I think he could have done a good job.
But he didn't.
He showed us a bunch of justifications
like if you generally have a surface in implicit form,
I told you that the gradient of F
represents the normal connection, right?
And he prepared us pretty good for what could
have been the proof of that.
He said, OK, guys.
You know the duration of the normal
as even as the gradient over the next of the gradient,
if you want unit normal.
How did he do that?
Well, he had a bunch of examples.
He had the sphere.
He showed us that for the sphere,
you have the normal, which is the continuation
of the position vector.
Then he said, OK, you can have approximations
of a surface that is smooth and round with oscillating spheres
just the way you have for a curve, a resonating circle,
a resonating circle-- that's called oscillating circle.
Resonating circle-- in that case, what will the normal be?
Well, the normal will have to depend
on the radius of the circle.
So you have a principal normal or a normal if it's a plane
curve.
And it's easy to understand that's the same
as the gradient.
So we have enough justification for the direction
of the gradient of such a function is always
normal-- normal to the surface, normal to all
the curves on the surface.
If we want to find that without swallowing
this like I had to when I was a student, it's not hard.
And let me show you how we do it.
We start from the graph, right?
Z equals f of x and y.
And we say, well, Magdalena, but this is a graph.
It's not an implicit equation.
And I'll say, yes it is.
Let me show you how I make it an implicit equation.
I move z to the other side.
I put 0 equals f of xy minus z.
Now it is an implicit equation.
So you say you cheated.
Yes, I did.
I have cheated.
It's funny that whenever somebody gives you a graph,
you can rewrite that graph immediately
as an implicit equation.
So that implicit equation is of the form big F of xyz
now equals a constant, which is 0.
F of xy is your old friend and minus z.
Now, can you tell me what is the normal to this surface?
Yeah, give me a splash in a minute like that.
So what is the gradient of f?
Gradient of f will be the normal.
I don't care if it's unit or not.
To heck with the unit or normal.
I'm going to say I wanted prime with respect to x, y,
and z respectively.
And what is the gradient?
Is the vector.
Big F sub x comma big F sub y comma big F sub z.
We see that last time.
So the gradient of a function is the vector
whose coordinates are the partial velocity--
your friends form last time.
Can we represent this again?
I don't know.
You need to help me.
Who is big F prime with respect to x?
There is no x here.
Thank God that's like a constant.
I just have to take this little one, f, and prime it
with respect to x.
And that's exactly what that's going to be-- little f sub x.
What is big F with respect to y?
STUDENT: [INAUDIBLE]
PROFESSOR: Little f sub y prime with respect
to y-- differentiated with respect to y.
And finally, if I differentiated with respect to z,
there is no z here, right?
There is no z.
So that's like a constant.
Prime [INAUDIBLE] 0 and minus 1.
So I know the gradient.
I know the normal.
This is the normal.
Now, if somebody gives you the normal, there you are.
You have the normal to the surface-- normal to surface.
What does it mean?
Equals normal to the tangent plane to the surface.
Normal or perpendicular to the tangent plane-
to the plane-- of the surface.
At that point-- point is the point p.
All right, so if you were to study a surface that's-- do you
have a [INAUDIBLE]?
STUDENT: Uh, no.
Do you?
PROFESSOR: OK.
OK, I want to study the tangent plane
at this point to the surface.
Well, that's flat, Magdalena.
You have no imagination.
The tangent plane is this plane, is the same as the surface.
So, no fun-- no fun.
How about I pick my favorite plane here
and I take-- what is-- OK.
I have-- this is Children Internationals.
I have a little girl abroad that I'm sponsoring.
So you have a point here and a plane
that passes through that point.
This is the tangent plane.
And my finger is the normal.
And the normal, we call that normal to the surface
when it's normal to the tangent plane.
At every point, this is what the normal is.
All right, can we write that based on chapter nine?
Now I will see what you remember from chapter nine if anything
at all.
All right, how do we write the tangent plane
if we know the normal?
OK, review-- if the normal vector is ai plus bj plus ck,
that means the plane that is perpendicular to it
is of what form?
Ax plus by plus cz plus d equals 0, right?
You've learned that in chapter nine.
Most of you learned that last semester in Calculus 2
at the end.
Now, if my normal is f sub x, f sub y, and minus 1,
those are ABC for God's sake.
Well, good.
Big A, big B, big C at the given point.
So I'm going to have f sub x at the given point d times
x plus f sub y at any given point d times y.
Who is c?
C is minus 1.
Minus 1 times z is-- say you're being silly.
Magdalena, why do you write minus 1?
Just because I'm having fun.
And plus, d equals 0.
And you say, well, wait, wait, wait.
This starts looking like that but it's not the same thing.
All right, what?
How do you get to d?
Now, actually, the plane perpendicular
to n that passes through a given point
can be written much faster, right?
So if a plane is perpendicular to a certain line,
how do we write if we know a point?
If we know a point in the normal ABC--
I have to go backwards to read it backwards-- then
the plane is going to be x minus x0
plus b times y times y0 plus c times z minus c0 equals 0.
So who is the d?
The d is all the constant that gets out of here.
So the point x0, y0, z0 has to verify the plane.
And that's why when you plug in x0, y0, z0,
you get 0 plus 0 plus 0 equals 0.
That's what it means for a point to verify the plane.
When you take the x0, y0, z0 and you plug it into the equation,
you have to have an identity 0 equals 0.
So this can be rewritten zx plus by plus cz
just like we did there plus a d.
And who in the world is the d?
The d will be exactly minus ax0 minus by0 minus cz0.
If that makes you uncomfortable, this is in chapter nine.
Look at the equation of a plane and the normal to it.
Now I know that I can do better than that if I'm smart.
So again, I collect the ABC.
Now I know my ABC.
I put them in here.
So I have f sub x at the point in time.
Oh, OK, x minus x0 plus, who is my b?
F sub y computed at the point p times y minus y0.
And, what?
Minus, right?
Minus-- minus 1.
I'm not going to write minus 1.
You're going to make fun of me.
Minus z minus cz.
And my proof is done.
QED-- what does it mean, QED?
In Latin.
QED means I proved what I wanted to prove.
Do you know what it stands for?
Did you take Latin, any of you?
You took Latin?
Quod erat demonstrandum.
So this was to be proved.
That's exactly what it was to be proved.
That, what, that c minus z0, which
was my fellow over here pretty in pink, is going to be f sub x
times x minus x0 plus yf sub y times y minus y0.
So now you know why the equation of the tangent plane is that.
I proved it more or less, making some assumptions,
some axioms as assumption.
But you don't know how to use it.
So let's use it.
So for the same valley-- not valley, hill--
it was full of snow.
Z equals 1 minus x squared-- what was you
guys have forgotten?
OK, 1 minus x squared minus y squared.
Find the tangent plane at the following points.
Ah, x0, y0 to be origin.
And you say, did you say that that's trivial?
Yes, it is trivial.
But I'm going to do it one more time.
And what was my [INAUDIBLE] point before?
STUDENT: [INAUDIBLE]
PROFESSOR: 1 over 2 and 1 over 2.
OK, and what will be the corresponding point in 3D?
1 over 2, 1 over 2, I plug in.
Ah, yes.
And with this, I hope to finish the day so we
can go to our other businesses.
Is this hard?
Now, I was not able-- I have to be honest with you.
I was not able to memorize the equation of a tangent plane
when I was-- when I was young, like a freshman and sophomore.
I wasn't ready to understand that this
is a linear approximation of a curved something.
This practically like the Taylor equation
for functions of two variables when
you neglect the quadratic third term and so on.
You just take the-- I'll teach you
next time when this is, a first order linear approximation.
All right, can we do this really quickly?
It's going to be a piece of cake.
Let's see.
Again, how do we do that?
This is f of x and y.
We computed that again.
F of 0, 0 was this 0.
Guys, if I say something silly, will you stop me?
F of f sub x-- f of y at 0, 0 is 0.
So I have two slopes.
Those are my hands.
The slopes of my hands are 0.
So the tangent plane will be z minus z0 equals 0.
What is the 0?
STUDENT: 1
PROFESSOR: 1, excellent.
STUDENT: [INAUDIBLE]
PROFESSOR: Why is that 1?
0 and 0 give me 1.
So that was the picture that I had z equals 1
as the tangent plane at the point corresponding
to the origin.
That look like the north pole, 0, 0, 1.
OK, no.
It's the top of a hill.
And finally, one last thing [INAUDIBLE].
Maybe you can do this by yourselves,
but I will shut up if I can.
I can't in general, but I'll shut up.
Let's see-- f sub x at 1 over root 2, 1 over root 2.
Why was that?
What is f sub x?
STUDENT: The square root of-- negative square root of 2.
PROFESSOR: Right, we've done that before.
And you got exactly what you said-- [INAUDIBLE]
2 f sub y at the same point.
I am too lazy to write it down again-- minus root 2.
And how do we actually express the final answer
so we can go home and whatever-- to the next class?
Is it hard?
No.
What's the answer?
Z minus-- now, attention.
What is z0?
STUDENT: 0.
PROFESSOR: 0, right.
Why is that?
Because when I plug 1 over a 2, 1 over a 2, I got 0.
0-- do I have to write it down?
No, not unless I want to be silly.
But if you do write down everything
and you don't simplify the equation of the plane,
we don't penalize you in any way in the final, OK?
So if you show your work like that, you're going to be fine.
What is that 1 over 2?
Plus minus root 2 times y minus 1 over root 2.
Is it elegant?
No, it's not elegant at all.
So as the last row for today, one final line.
Can we make it look more elegant?
Do we care to make it more elegant?
Definitely some of you care.
Z will be minus root 2x.
I want to be consistent and keep the same style in y.
And yet the constant goes wherever
it wants to go at the end.
What's that constant?
STUDENT: 2 [INAUDIBLE].
PROFESSOR: So you see what you have.
You have this times that.
It's a 1, this then that is a 1.
1 plus 1 is 2.
All right, are you happy with this?
I'm not.
I'm happy.
You-- if this were a multiple choice,
you would be able to recognize it right away.
What's the standardized general equation of a plane, though?
Something x plus something y plus something z plus something
equals 0.
So if you wanted to make me very happy,
you would still move everybody to the left hand side.
Do you want equal to or minus 3?
Yes, it does.
STUDENT: [INAUDIBLE]
PROFESSOR: Huh?
Negative 2-- is that OK?
Is that fine?
Are you guys done?
Is this hard?
Mm-mm.
It's hard?
No.
Who said it's hard?
So-- so I would work more tangent planes next time.
But I think it's something that we can practice on.
And do expect one exercise like that from one
of those, God knows, 15, 16 on the final.
I'm not sure about the midterm.
I like this type of problem.
So you might even see something with tangent planes
on the midterm-- normal to a surface tangent plane.
It's a good topic.
It's really pretty.
For people who like to draw, it's also nice to draw them.
But do you have to?
No.
Some of you don't like to.
OK, so now I say thank you for the attendance
and I'll see you next time on Thursday-- on Tuesday.
Happy Valentine's Day.