TTU Math2450 Calculus3 Sec 11.4 and 11.5

Tangent Plane, Differentiability and Approximations and intro to Chain Rule

PROFESSOR TODA: Any
questions so far?

I mean, conceptual,
theoretical questions first,

and then we will
do the second part

of [INAUDIBLE] applications.

Then you can ask
for more questions.

No questions so far?

I have not finished 11-4.

I still owe you a long
explanation about 11-4.

Hopefully it's going to
make more sense today

than it made last time.

I was just saying
that I'm doing 11-4.

This is a lot of chapter.

So second part of 11-4 today--
tangent plane and applications.

Now, we don't say what
those applications are

from the start, but these are
some very important concepts

called the total differential.

And the linear
approximation number

is going under the [INAUDIBLE].

Thank you, sir.

Linear approximation for
functions of the type z

equals f of xy, which means
graphs of two variables.

At the end of the chapter, I'll
take the notes copy from you.

So don't give me
anything until it's over.

When is that going to be over?

We have four more
sections to go.

So I guess right before
spring break you give me

the notes for chapter 11.

All right, and then
I'm thinking of making

copies of both chapters.

You get the-- I'm
distributing them to you.

I haven't started
and yet go ahead.

Could anybody tell
me what the equation

that we used last time--
we proved it, actually.

What is the equation
of the tangent plane

to a smooth surface or a patch
of a surface at the point

m of coordinates x0, y0,
z0, where the graph is

given by z equals f of x and y.

I'm going to label it on
the patch of a surface.

OK, imagine it
labeled brown there.

And can somebody tell me the
equation of the other plane?

But because you
have better memory,

being much younger, about 25
years younger than me or so.

So could you-- could anybody
tell me what the tangent

planes equation-- I'll start.

And it's going to come to you.

z minus z0 equals.

And now let's see.

I'll pick a nice color.

I'll wait.

STUDENT: fx of x.

PROFESSOR TODA: f sub x, the
partial derivative measured

at f0 i0 times the
quantity x minus x0 plus--

STUDENT: f sub y.

PROFESSOR TODA: f
sub y, excellent.

f sub y.

STUDENT: x0, y0.

PROFESSOR TODA: x0,
y0 times y minus y0.

OK.

All right.

Now thinking of what those
quantities mean, x minus x0, y

minus y0, z minus
z0, what are they?

They are small
displacements, aren't they?

I mean, what does it
mean small displacement?

Imagine that you are near
the point on both surfaces.

So what is a small
neighborhood--

what's a typical small
neighborhood [INAUDIBLE]?

It's a disk, right?

There are many kinds of
neighborhoods, but one of them,

I'd say, would be
this open disk, OK?

I'll draw that.

Now, if I have a
red point-- I don't

know how to do that pink point--
somewhere nearby in planes--

this is the plane.

In plane, I have this
point that is close.

And that point is xyz.

And you think, OK, can
I visualize that better?

Well, guys, it's hard to
visualize that better.

But I'll draw a triangle
[? doing ?] a better job.

That's the frame.

This is a surface.

Imagine it's a surface, OK?

That's the point of x0, y0.

[? It's ?] the 0 and that.

Where is the point xyz again?

The point xyz is not
on the pink stuff.

This is a pink surface.

It looks like Pepto
Bismol or something.

You shaded it.

No.

That's not what I want.

I want the close enough
point on the blue plane.

It's actually in the blue plane
pie and this guy would be xyz.

So now say, OK, how
far I x be from x0?

Well, I don't know.

We would have to check
the points, the set 0,

check the blue point.

This is x.

So between x and x0, I
have this difference,

which is delta x displacement,
displacement along the x-axis,

away from the
point, fixed point.

This is the fixed
point, this point.

This point is p.

OK.

y minus y0, let's call
that delta y, which

is the displacement
along the y-axis.

And then the z minus z0 can be.

Just because I'm a mathematician
and I don't like writing down

a lot, I would use
s batch as I can,

compact symbols, to
speed up my computation.

So I can rewrite
this whole thing

as a delta z equals f sub
x, x0 y0, which is a number.

It's a slope.

We discussed about
that last time.

We even went skiing
last time, when

we said that's like the slope
in-- what's the x direction?

Slope in the x direction
and slope in the y direction

on the graph that was the
white covered with snow hill.

That was what we had last time.

Delta x plus f sub
0, another slope

in the y direction, delta y.

And fortunately-- OK, the book
is a very good book, obviously,

right?

But I wish we could've done
certain things better in terms

of comparisons between
this notion in Calc III

and some corresponding
notion in Calc I.

So you're probably
thinking, what the heck

is this witch thinking about?

Well, I'm thinking
of something that you

may want to remember
from Calc I.

And that's going to come
into place beautifully

right now because you have the
Calc I, Calc III comparison.

And that's why it would be
great-- the books don't even

talk about this comparison.

In Calc I, I reminded
you about Mr. Leibniz.

He was a very nice guy.

I have no idea, right?

Never met him.

One of the fathers of calculus.

And he introduced the
so-called Leibniz notation.

And one of you in office
hours last Wednesday

told me, so the
Leibnitz notation

for a function g of
x-- I'm intentionally

changing notation-- is what?

Well, this is just
the derivative

which is the limit of
the different quotients

of your delta g over
delta x-- as done by some

blutches-- 0, right, which
would be the same as lim

of g of x minus g of x0 over
x minus x0 as x approaches x0,

right?

Right.

So we've done that in Calc I.
But it was a long time ago.

My mission is to teach
you all Calc III,

but I feel that
my mission is also

to teach you what you may not
remember very well from Calc I,

because everything is related.

So what was the way we
could have written this,

not delta g over delta
x equals g prime.

No.

But it's an approximation of
g prime around a very small

[INAUDIBLE], very close to x0.

So if you wanted to
rewrite this approximation,

how would you have rewritten it?

Delta g--

STUDENT: g prime sub x.

PROFESSOR TODA: g prime
of x0 times delta x.

OK?

Now, why this approximation?

What if I had put equal?

If I had put equal, it
would be all nonsense.

Why?

Well, say, Magdalena, if you
put equal, it's another object.

What object?

OK.

Let's look at the objects.

Let's draw a picture.

This is g.

This is x0.

This is g of x.

What's g prime?

g prime-- thank god-- is the
slope of g prime x0 over here.

So if I want to write the
line, the line is exactly this.

The red object is the line.

So what is the red object again?

It's y minus y over x
minus x0 equals m, which

is g prime number 0.

m is the slope.

That's the point slope
formula, thank you very much.

So the red object is this.

This is the line.

Attention is not the same.

The blue thing is my
curve, more precisely

a tiny portion of my curve.

This neighborhood around the
point is what I have here.

What I'm actually-- what?

I'm trying to
approximate my curve

function with a little line.

And I say, I would rather
approximate with a red line

because this is the
best approximation

to the blue arc of a curve
which is on the curve, right?

So this is what it is
is just an approximation

of a curve, approximation of
a curve of an arc of a curve.

But Magdalena's lazy
today-- approximation

of an arc of a curve
with a segment of a line,

with a segment of
the tangent line

of the tangent [INAUDIBLE].

How do we call
such a phenomenon?

An approximation of
an arc of a circle

with a little segment
of a tangent line

is like a discretization, right?

But we call it
linear approximation.

It's called a linear
approximation.

A-P-P, approx.

Have you ever seen a
linear approximation

before coming from Calc II?

Well, in Calc II you've
seen the Taylor's formula.

What is the Taylor's formula?

It's a beautiful
thing that said what?

I don't know.

Let's remember together.

So relationship
with Calc II, I'm

going to go and make an arrow--
relationship with Calc II,

because everything
is actually related.

In Calc II-- how did we
introduce Taylor's formula?

Well, instead of little a that
you're so used to in Calc II,

we are going to put x0
is the same thing, right?

So what was Taylor's
formula saying?

You have this kind of
smooth, beautiful curve.

But being smooth is not enough.

You have that real analytic.

Real analytic means
that the function can be

expanded in Taylor's formula.

So what does it mean?

It means that we have f of x
prime is f of x0 equals-- or g.

You want-- it doesn't matter.

f prime of x0 times
x minus x0 plus

dot, dot, dot, dot something
that I'm going to put.

This is [? O. ?] It's a small
quantity that's maybe not

so small, but I declare
it to be negligible.

And so they're going
to be negligible.

I have to make a face,
a smiley face and eyes,

meaning that it's OK to
neglect the second order

term, the third order term.

So what happens, that
little h, when I square it,

say the heck with it.

It's going to be very small.

Like if h is 0.1 and then
h squared will be 0.0001.

And I have a certain range
of error that I allow,

a threshold.

I say that's negligible.

If h squared and h cubed and h
to the fourth are negligible,

then I'm fine.

If I take all the
other spot, that's

the linear approximation.

And that's exactly
what I wrote here

with little g instead of f.

The only difference is this is
little f and this is little g.

But it's the same exact
formula, linear approximation.

Do you guys remember then next
terms of the Taylor's formula?

STUDENT: fw--

PROFESSOR TODA: fw--

STUDENT: w over--

PROFESSOR TODA: So
fw prime at x0 over--

STUDENT: 1 factorial.

PROFESSOR TODA: 2 factorial.

This was 1 factorial.

This was over 1 factorial.

But I don't write
it because it's one.

STUDENT: Right.

PROFESSOR TODA: Here I would
have f double prime of blah,

blah, blah over-- what did
you say-- 2 factorial times x

minus x0 squared plus, plus,
plus, the cubic [INAUDIBLE]

of the-- this is the quadratic
term that I neglect, right?

So that was Taylor's formula.

Do I mention anything
about it now?

We should.

But practically, the
authors of the book

thought, well, everything
is in the book.

You can go back and forth.

It's not like that unless
somebody opens your eyes.

For example, I didn't
see that when I was 21.

I couldn't make any connection
between these Calc I,

Calc II, Calc III notions.

Because nobody told me, hey,
Magdalena, open your eyes

and look at that in
perspective and make

a comparison between what you
learned in different chapters.

I had to grow.

After 20 years, I
said, oh, I finally

see the picture of linearization
of a function of, let's say,

n variables.

So all these total
differentials will come in place

when time comes.

You have a so-called
differential in Calc I.

And that's not delta g.

Some people say, OK,
no, that's delta g.

No, no, no, no.

The delta x is a displacement.

The delta g is the
induced displacement.

If you want this to be
come a differential,

then you shrink
that displacement

to infinitesimally small.

OK?

So it's like going from
a molecule to an atom

to an electron to subatomic
particles but even more,

something infinitesimally small.

So what do we do?

We shrink delta x into dx
which is infinitesimally small.

It's like the notion of
God but microscopically

or like microbiology
compared to the universe, OK?

So dx is multiplied
by g prime of x0.

And instead of delta g, I'm
going to have a so-called dg,

and that's a form.

In mathematics, this is
called a form or a one form.

And it's a special
kind of object, OK?

So Mr. Leibniz was very smart.

He said, but I can rewrite this
form like dg dx equals g prime.

So if you ever forget
about this form which

is called differential,
differential form,

you remember Mr.
Leibniz, he taught you

how to write the derivative in
two different ways, dg dx or g

prime.

What you do is just formally
multiply g prime by dx

and you get dg.

Say it again, Magdalena--
multiply g prime by dx

and you get dg.

And that's your
so-called differential.

Now, why do you say total
differential-- total

differential, my god, like
complete differentiation?

In 11.4, we deal with
functions of two variables.

So can we say differentials?

Mmm, it's a little bit
like a differential

with respect to what variable?

If you say with respect
to all the variables,

then you have to be thinking
to be smart and event,

create this new object.

If one would write
Taylor's formula,

there is a Taylor's
formula that we don't give.

OK.

Now, you guys are looking
at me with excitement.

For one point extra
credit, on the internet,

find Taylor's formula for
n variables, functions

of n variables or at
least two variables,

which was going to look
like z minus z0 equals

f sub x at the point x0
at 0 times x minus x0 plus

f sub y at x0 y0 times x minus
x0 plus second order terms

plus third order terms
plus fourth order terms.

And the video cannot see me.

So what do we do?

We just truncate this
part of Taylor's I say,

I already take the Taylor
polynomial of degree one.

And the quadratic terms and
everything else, the heck

with that.

And I call that a
linear approximation,

but it's actually Taylor's
formula being discussed.

We don't tell you in
the book because we

don't want to scare you.

I think we would better
tell you at some point,

so I decided to tell you now.

All right.

So this is Taylor's formula
for functions of two variables.

We have to create
not out of nothing

but out of this the
total differential.

Who tells me?

Shrink the
displacement, Magdalena.

The delta x shrunk to
an infinitesimally small

will be dx.

Delta y will become dy.

The line is a smiley from the
skies, just looking at us.

He loves our notations.

And this is dz.

So I'm going to write dz or df's
the same thing equals f sub x.

At the point, you
could be at any point

you are taking in particular,
dx plus f sub y xy dy.

So this is at any point
at the arbitrary point xy

in the domain where your
function e is at least c1.

What does it mean, c1?

It means the function
is differentiable

and the partial
derivatives are continuous.

I said several times, I
want even more than that.

I want it maybe second
order derivatives

to exist and be continuous
and so on and so forth.

And I will assume
that the function can

be expanded [INAUDIBLE] series.

All right, now example
of a final problem

that was my first problem
on the final many times

and also on the common
final departmental final.

And many students
screwed up, and I

don't want you to ever
make such a mistake.

So this is a mistake not
to make, OK, mistake not

to make because after 20
something years of teaching,

I'm quite familiar with
the mistakes students

make in general and I don't
want you to make them.

You are too good to do this.

So problem 1.

On the final, I said-- we
said-- the only difference was

on some departmental finals,
we gave a more sophisticated

function.

I'm going to give only
some simple function

for this polynomial.

That's beautiful.

And then I said we said
write the differential

of this function at an
arbitrary point x, y.

And done.

And [INAUDIBLE].

Well, let me tell you what
some of my students-- some

of my studentss-- don't do that.

I'm going to cross it with red.

And some of my students
wrote me very beautifully df

equals 2x plus 2y.

And that can send
me to the hospital.

If you want to go to the ER
soon, do this on the exam

because this is nonsense.

Why is this nonsense?

This is not--

STUDENT: [INAUDIBLE] dx or dy.

PROFESSOR TODA: Exactly.

So the most important thing
is that the df is like-- OK,

let me come back to driving.

I'm driving to Amarillo-- and I
give this example to my calc 1

students all the time because
it's a linear motion in terms

of time.

And let's say I'm on
cruise control or not.

It doesn't matter.

When we drive and I'm looking at
the speedometer and I see 60--

I didn't want to say more, but
let's say 80, 80 miles an hour.

That is a miles an hour.

That means the hour is a huge
chunk delta h or delta t.

Let's call it delta
t because it's time.

I'm silly.

Delta t is 1.

Delta s, the space,
the space, is going

to be the chunk of 60 miles.

But then that is the
average speed that I had.

So that's why I said 60.

That's the average
speed I had in my trip,

during my trip [INAUDIBLE].

There were moments when my
speed was 0 or close to 0.

Let's assume it was never 0.

But that means there were many
moments when my speed could've

been 100, and nobody knows
because they didn't catch me.

So I was just lucky.

So in average, if somebody is
asking you what is the average,

that doesn't tell them anything.

That reminds me of that
joke-- overall I'm good,

the statistician joke
who was, are you cold?

Are you warm?

And he was actually sitting
on with one half of him

on a block of ice and the
other half on the stove,

and he says, in
average, I'm fine.

But he was dying.

This is the same kind of thing.

My average was 60 miles
an hour, but I almost

got caught when I was
driving almost 100.

But nobody knows because I'm
not giving you that information.

That's the infinitesimally small
information that I have not

put correctly here
means that what is

what I see on the speedometer?

It's the instantaneous
rate of change

that I see that
fraction of second.

So that means maybe a few feet
per a fraction of a second.

It means how many
feet did I travel

in that fraction of a second?

And if that fraction of a second
is very tiny that I cannot even

express it properly, that's
what I'm going to have--

df equals f prime dx.

So df and dx have to be small
because their ratio will be

a good number, like 60, like
80, but [? them in ?] themselves

delta m delta [? srv, ?]
very tiny things.

It's the ratio that matters
in the end to be 60, or 80,

or whatever.

So I have 2x dx plus 2y dy.

Never say that the
differential, which

is something
infinitesimally small,

is equal to this scalar
function that it doesn't even

make any sense.

Don't do that because
you get 0 points

and then we argue,
and I don't want

you to get 0 points on
this problem, right.

So it's a very simple problem.

All I want to test you on
would be this definition.

Remember, you're going to
see that again on the midterm

and on the final, or
just on the final.

Any questions about that?

All right.

So I want to give you the
following homework out

of section 11.4 on
top of the web work.

Read all the solved
examples of the section.

OK.

So for example,
somebody tells you

I have to apply this
knowing that I have

an error of measurement of
some sort in the s direction

and an error of measurement of
some sort in the y direction.

There are two or three
examples like that.

They will give you all this
data, including the error

measurement.

For delta, it should be 0.1.

Don't confuse the 0.1 with
dx. dx is not a quantity.

dx is something like
micro cosmic thing.

It's like infinitely
[? small ?].

Infinitesimally small.

So saying that dx should be
0.1 doesn't make any sense,

but delta x being
0.1 make sense.

Delta y being 0.3 makes sense.

And they ask you to
plug it in and find

the general difference.

For example, where
could that happen?

And you see examples
in the book.

Somebody measures something--
an area of a rectangle

or a volume of a cube.

But when you measure,
you make mistakes.

You have measurement errors.

In the delta x, you have
an error of plus minus 0.1.

In the y direction, you have
displacement error 0.2 or 0.3,

something like that.

What is the overall
error you are

going to make when you measure
that function of two variables?

That's what you have.

So you plug in all
those displacements

and you come up with the
computational problem.

Several of you Wednesday we
discussed in my office already

solved those problems through
web work and came to me,

and I said, how did you know
to plug in those [? numbers ?]?

Well, it's not so hard.

It's sort of common sense.

Plus, I looked in the book
and that gave me the idea

to remind you to
look in the book

for those numerical examples.

You will have to
use your calculator.

So you don't have it with
you, you generally, we

don't use in the classroom,
but it's very easy.

All you have to do is use the
calculator and [INAUDIBLE]

examples and see how it goes.

I wanted to show you
something more interesting

even, more beautiful
regarding something

we don't show in the
book until later on,

and I'm uncomfortable with the
idea of not showing this to you

now.

An alternate way, or
more advanced way,

more advanced way, to
define the tangent plane--

the tangent plane-- to a
surface S at the point p.

And I'll draw again.

Half of my job is drawing
in this class, which I like.

I mean, I was having an argument
with one of my colleagues who

said, I hate when they are
giving me to teach calculus 3

because I cannot draw.

I think that the
most beautiful part

is that we can represent
things visually,

and this is just pi, the
tangent plane I'm after,

and p will be a
coordinate 0 by 0, z0.

And what was the label?

Oh, the label.

The label.

The label was internal
where z equals f of xy.

But more generally, I'll say
this time plus more generally,

what if you have f of xyz
equals c for that surface.

Let's call it [INAUDIBLE].

F of xy is [INAUDIBLE].

And somebody even said, can
you have a parametrization?

And this is where
I wanted to go.

Ryan was the first
one who asked me,

but then there were
three more of you

who have restless
minds plus you--

because that's the essence
of being active here.

We don't lose our connections.

We lose neurons anyway, but
we don't lose our connections

if we think, and
anticipate things,

and try to relate concepts.

So if you don't want to
get Alzheimer's, just

think about the parametrization.

So can I have a
parametrization for a surface?

All righty, what do you mean?

What if somebody says for a
curve, we have r of t, right,

which was what?

It was x of ti plus y of tj plus
z of tk, and we were so happy

and we were happy
because we were traveling

in time with respect
to the origin,

and this was r of t at time t.

[INAUDIBLE]

But somebody asked
me, [INAUDIBLE],

can you have such a position
vector moving on a surface?

Like look, it's a rigid motion.

If you went to the
robotics science

fair, Texas Tech, or something
like that, you know about that.

Yeah, cities.

So how do we introduce
such a parametrization?

We have an origin of course.

An origin is always important.

Everybody has an origin.

And I take that position
vector, and where does it start?

It starts at the origin, and
the tip of it is on the surface,

And it's gliding on the
surface, the tip of it.

And that's going to be r, but
it's not going to be r of t.

It's going to be r of
longitude and latitude.

Like imagine, that would
be the radius coming

from the center of the earth.

And it depends on
two parameters.

One of them would be latitude.

Am I drawing this right?

Latitude--

STUDENT: [INAUDIBLE] longitude.

PROFESSOR TODA:
--from a latitude 0.

I'm at the equator.

Then latitude 90 degrees.

I'm at the North Pole.

In mathematics, we are funny.

We say latitude 0,
latitude 90 North Pole,

latitude negative 90,
which is South Pole.

And longitude from 0 to 2 pi.

Meridian 0 to all around.

So r will be not a function of
t but a function of u and b,

thank god, because u and b
are the latitude and longitude

sort of.

So we have x of uv i plus
y of uv j plus z of uv k.

You can do that.

And you say, but can you give
us an example, because this

looks so abstract for god sake.

If you give me the graph
the way you gave it to me

before z equals f of xy,
please parametrize this for me.

Parametrize it for
me because I'm lost.

You are not lost.

We can do this together.

Now what's the simplest
way to parametrize

a graph of the type
z equals f of xy?

Take the xy to be
u and v. Take x

and y to be your
independent variables

and take z to be the
dependent variable.

I'm again expressing these
things in terms of variables

like I did last time.

Then I say, let's take this kind
of parametrization. [INAUDIBLE]

vu, right.

y would be v. Then I'm
going to write r of x and y

just like that guy will
be [INAUDIBLE] of xn.

[? y ?] will say, wait a minute.

I will have to re-denote
everybody with capitals.

Then my life will become
better because you

don't have to erase.

You just make little
x big, little y bigs,

bigs, big, capitalized XYZ.

And then I'll say OK, XYZ
will be my setting here in 3D.

All right.

So how am I going
to re-parametrize

the whole surface?

Whole surface will be r of
xy equals in this case, well,

let's think about it.

In this case, I'm
going to have xy.

And where's the little f?

I just erased it.

I was smart, right,
that I erased f of xy.

So I have x, y, and
z, which is f of xy.

And this is the generic point
p of coordinates xy f of xy.

So I say, OK, what does it mean?

I will project this point.

And this is the point
when big x becomes little

x, when big y becomes--
where is my y-axis?

Somebody ate my y axis.

[INAUDIBLE]

So when big Y becomes
little y, little y

is just an instance of big Y.
And big Z will take what value?

Well, I need to project that.

How do you project from
a point to the z-axis?

You have to take the
parallel from the point

to the horizontal
plane until you

hit the-- [INAUDIBLE] the whole
plane parallel to the floor

through the point p.

And what do I get here?

STUDENT: [INAUDIBLE].

PROFESSOR TODA: Not
z0, but it's little z

equals f of xy, which is an
instance of the variable xz.

For you programmers, you know
that big z will be a variable

and little z will be
[INAUDIBLE] a variable.

OK.

So I parametrized my graph
in a more general way,

general parametrization
for a graph.

And now, what are-- what's the
meaning of r sub x and r sub y?

What are they?

STUDENT: [INAUDIBLE].

PROFESSOR TODA: Now, we
don't say that in the book.

Shame on us.

Shame on us.

We should have because I was
browsing through the projects

about a year and a half ago.

The senior projects of
a few of my students

who are-- two of them were
in mechanical engineering.

One of them was in
petroleum engineering.

And he actually showed me
that they were doing this.

They were taking vectors
that depend on parameters--

this is a vector [INAUDIBLE]--
and differentiated them with

respect to those parameters.

And I was thinking OK, did we
do the partial derivatives r sub

x, r sub y?

Not so much.

But now I want to do it
because I think that prepares

you better as engineers.

So what is r sub x
and what is r sub y?

And you say, well,
OK. [INAUDIBLE],

I think I know how to do
that in my sleep, right.

If you want me to do
that theoretically

from this formula,
but on the picture,

I really don't know what it is.

So I'm asking you what
I'm going to have in terms

of r sub x and r sub y.

They will be vectors.

This should be a
vector as well, right.

And for me, vector triple
means the identification

between the three coordinates
and the physical vector.

So this is the physical vector.

Go ahead and write x prime
with respect to x is 1.

y prime with respect to x is 0.

The third [INAUDIBLE]
prime with respect

to x is just whatever
this little f is,

it's not any of my business.

It's a [INAUDIBLE]
function f sub x.

Well, what is the second vector?

STUDENT: 0, 1, f sub y.

PROFESSOR TODA: 0, 1, f sub y.

Now, are they slopes?

No.

These are slopes.

That's a slope and
that's a slope.

And we learned
about those in 11.3,

and we understood that those
are ski slopes, they were.

In the direction of x
and the direction of y,

the slopes of the tangents
to the coordinate lines.

But this looks like I have
a direction of a line,

and this would be the lope, and
that's the direction of a line,

and that would be the slope.

What are those lines?

STUDENT: [INAUDIBLE] to
the function [INAUDIBLE].

PROFESSOR TODA: Let me draw.

Then shall I erase
the whole thing?

No.

I'm just going to keep--
I'll erase the tangent.

Don't erase anything
on your notebooks.

So this is the point p.

It's still there.

This is the surface.

It's still there.

So my surface will be x,
slices of x, [? S ?] constant

are coming towards you.

They are these [? walls ?]
like that, like this, yes.

It's like the CT scan.

I think that when they
slice up your body,

tch tch tch tch tch
tch, take pictures

of the slices of your body,
that's the same kind of thing.

So x0, x0, x0, x0.

I'm going to [INAUDIBLE]
planes and I had x equals x0.

And in the other direction, I
cut and I get, what do I get?

Well, I started bad.

Great, Magdalena, this is--

What is this pink?

It's not Valentine's Day
anymore. y equals [INAUDIBLE].

And this is the point.

So, as Alex was
trying to tell you,

our sub x would represent the
vector, the physical vector

in 3D, that is originating
at p and tangent to which

of the two, to the purple
one or to the red one?

STUDENT: Red.

Uh, purple.

PROFESSOR TODA:
Make up your mind.

STUDENT: The purple one.

PROFESSOR TODA: [INAUDIBLE]
constant and [INAUDIBLE]

constant in the red
one, y equals y0, right?

So, this depends on x.

So this has r sub x.

This is the velocity with
respect to the variable x.

And the other one, the
blue one, x equals x0,

means x0 is held fixed
and y is the variable.

So I have to do r sub y,
and what am I gonna get?

I'm gonna get the blue vector.

What's the property
of the blue vector?

It's tangent to the purple line.

So r sub y has to be
tangent to the curve.

x0, y, f of x0 and
y is the curve.

And r sub x is tangent
to which curve?

Who is telling me which curve?

x, y0 sub constant,
f of x and y0.

So that's a curve that
depends only on y,

y is the time in this case.

And that's the curve
that depends only on x.

x is the time in this case.

r sub x and r sub y are
the tangent vectors.

What's magical about them?

If I shape this
triangle between them,

that will be the tangent plane.

And I make a smile because I
discovered the tangent plane

in a different way than
we did it last time.

So the tangent plane represents
the plane of the vector r sub

x and r sub y.

The tangent plane
represents the plane

given by vectors r sub x and
r sub y with what conditions?

It's a conditional.

r sub x and r sub
y shouldn't be 0.

r sub x different from 0,
r sub y different from 0,

and r sub x and r sub
y are not collinear.

What's gonna happen
if they are collinear?

Well, they're gonna
collapse; they are not

gonna determine a plane.

So there will be
no tangent planes.

So they have to be
linearly independent.

For the people who are taking
now linear algebra, I'm saying.

So we have no other
choice, we have

to assume that these vectors,
called partial velocities,

by the way, for the
motion across the surface.

OK?

These are the partial
velocities, or partial velocity

vectors.

Partial velocity vectors
have to determine a plane,

so I have to assume
that they are non-zero,

they never become 0, and
they are not collinear.

If they are collinear,
life is over for you.

OK?

So I have to assume that I
throw away all the points where

the velocities become 0, and
all the points where--those are

singularity points--where
my velocity vectors are 0.

Have you ever studied design?

Any kind of experimental design.

Like, how do you design a car,
the coordinate lines on a car?

I'm just dreaming.

You have a car, a beautiful
car, and then you have-- Well,

I cannot draw really
well, but anyway.

I have these coordinate
lines on this car.

It's a mesh what I have there.

Actually, we do that in
animation all the time.

We have meshes over the
models we have in animation.

Think Avatar.

Now, those are all
coordinate lines.

Those coordinate lines would be,
even your singularities, where?

For example, if you take a body
in a mesh like that, in a net,

in, like, a fishnet, then
you pull from the fishnet,

all the coordinate lines
will come together,

and this would be a singularity.

We avoid this kind
of singularity.

So these are points where
something bad happened.

Either the velocity
vectors become collinear.

You see what I'm talking about?

Or the velocity
vectors shrank to 0.

So that's a bad point;
that's a singularity point.

They have this
problem when meshing.

So when they make
these models that

involve two-dimensional meshing
and three-dimensional ambient

space, like it is in
animation, the mesh

is called regular
if we don't have

this kind of singularity, where
the velocity vectors become 0,

or collinear.

It's very important for a
person who programs in animation

to know mathematics.

If they don't understand
these things, it's over.

Because you write the matrix,
and you will know the vectors

will become collinear when the
two vectors--let's say two rows

of a matrix--

STUDENT: Parallel.

PROFESSOR TODA:
Are proportional.

Or parallel.

Or proportional.

So, everything is numerical
in terms of those matrices,

but it's just a discretization
of a continuous phenomenon,

which is this one.

Do you remember Toy Story?

OK.

The Toy Story people,
the renderers,

the ones who did the rendering
techniques for Toy Story,

both have their
master's in mathematics.

And you realize why
now to do that you

have to know calc I, calc
II, calc III, linear algebra,

be able to deal with matrices.

Have a programming course
or two; that's essential.

They took advanced calculus
because some people

don't cover thi-- I was about to
skip it right now in calc III.

But they teach that in
advanced calculus 4350, 4351.

So that's about as
far as you can get,

and differential equation's
also very important.

So, if you master those and
you go into something else,

like programming,
electrical engineering,

you're ready for animation.

[INAUDIBLE] If you went
I want to be a rendering

guy for the next movie,
then they'll say no,

we won't take you.

I have a friend who
works for Disney.

She wanted to get a PhD.

At some point, she
changed her mind

and ended up just with a
master's in mathematics

while I was in Kansas,
University of Kansas,

and she said, "You know what?

Disney's just giving me
$65,000 as an intern."

And I was like OK and probably
asked [INAUDIBLE] $40,000 as

a postdoc.

And she said,
"Good luck to you."

Good luck to you, too.

But we stayed in touch,
and right now she's

making twice as much as
I'm making, for Disney.

Is she happy?

Yeah.

Would I be happy?

No.

Because she works
for 11 hours a day.

11 hours a day, on a chair.

That would kill me.

I mean, I spend about six hours
sitting on a chair every day

of the week, but
it's still too much.

She's a hard worker, though.

She loves what she's doing.

The problem is your eyes.

After a while, your
eyes are going bad.

So, what is the normal for
the plane in this case?

I'll try my best
ability to draw normal.

The normal has to
be perpendicular

to the tangent space, right?

Tangent plane.

So, n has to be
perpendicular to our sub

x and has to be
perpendicular to our sub y.

So, can you have any
guess how in the world

I'm gonna get n vector?

STUDENT: [INAUDIBLE]

PROFESSOR TODA:
[INAUDIBLE] That's

why you need to
know linear algebra

sort of at the same time, but
you guys are making it fine.

It's not a big deal.

You have a matrix, i, j, k
in the front row vectors,

and then you have r sub x that
you gave me, and I erased it.

1, 0, f sub x.

0, 1, f sub y.

And you have exactly 18
seconds to compute this vector.

STUDENT: [INAUDIBLE]

PROFESSOR TODA: You want k, but
I want to leave k at the end

because I always
order my vectors.

Something i plus something
j plus something k.

[INTERPOSING VOICES]

PROFESSOR TODA: Am I right?

Minus f sub x--

STUDENT: Minus f of x plus k.

PROFESSOR TODA: --times i.

For j, do I have to change sign?

Yeah, because 1 plus 2 is odd.

So I go minus 1.

And do it slowly.

You're not gonna make fun of
me; I gotta make fun of you, OK?

And minus 1 times--

STUDENT: Did you forget f y?

PROFESSOR TODA: --f sub y--I go
like that--sub y times j plus

k.

As you said very well
in the most elegant way

without being like yours,
but I say it like this.

So you have minus f
sub x, minus f sub y,

and 1 as a triple with angular
brackets--You love that.

I don't; I like it parentheses
[INAUDIBLE]--equals n.

But n is non-unitary,
but I don't care.

Why don't I care?

I can write the
tangent plane very well

without that n being
unitary, right?

It doesn't matter in the end.

These would be my a, b, c.

Now I know my ABC.

I know my ABC.

So, the tangent plane
is your next guess.

The tangent plane would
be perpendicular to n.

So this is n.

The tangent plane passes
through the point p

and is perpendicular to n.

So, what is the equation
of the tangent plane?

STUDENT: Do you want
scalar equations?

PROFESSOR TODA: A by x minus 0.

Very good.

That's exactly what I
wanted you to write.

All right, so, does
it look familiar?

Not yet.

[STUDENT SNEEZES]

STUDENT: Bless you.

STUDENT: Bless you.

PROFESSOR TODA: Bless you.

Who sneezed?

OK.

Am I almost done?

Well, I am almost done.

I have to go backwards,
and whatever I get

I'll put it big here in
a big formula on top.

I'm gonna say oh, my God.

No, that's not
what I'm gonna say.

I'm gonna say minus f sub x at
my point p--that is a, right?

Times x minus x0.

Plus minus f sub y at
the point p; that's b.

y minus y0 plus--c is 1, right?

c is 1.

I'm not gonna write
it because if I write

it you'll want to make fun
of me. z minus z0 equals 0.

Now it starts looking like
something familiar, finally.

Now we discovered
that the tangent plane

can be written as z minus z0.

I'm keeping the guys z minus
z0 on the left-hand side.

And these guys are gonna
move to the right-hand side.

So, I'm gonna have
again, my friend,

the equation of the tangent
plane for the graph z equals f

of x,y.

But you will say
OK, I think by now

we've learned these
by heart, we know

the equation of the tangent
plane, and now we're asleep.

But what if your surface
would be implicit the way

you gave it to us at first.

Maybe you remember the sphere
that was an implicit equation,

x squared plus x squared
plus x squared equals--

What do you want it to be?

STUDENT: 16.

PROFESSOR TODA: Huh?

STUDENT: 16.

PROFESSOR TODA: 16.

So, radius should be 4.

And in such a case, the equation
is of the type f of x, y, z

equals constant.

Can we write again the
equation [INAUDIBLE]?

Well, you say well,
you just taught

us some theory that says I have
to think of u and v, but not x

and y.

Because if I think of x
and y, what would they be?

I think the sphere
as being an apple.

Not an apple, something
you can cut easily.

Well, an apple, an
orange, something.

A round piece of soft cheese.

I started being hungry,
and I'm dreaming.

So, this is a huge something
you're gonna slice up.

If you are gonna
do it with x and y,

the slices would be like this.

Like that and like this, right?

And in that case,
your coordinate curves

are sort of weird.

If you want to do it in
different coordinates,

so we want to
change coordinates,

and those coordinates should
be plotted to the longitude,

then we cannot use x and y.

Am I right?

We cannot use x and y.

So those u and v will be
different coordinates,

and then we can do it
like that, latitude.

[INAUDIBLE] minus [INAUDIBLE].

And longitude.

We are gonna talk about
spherical coordinates

later, not today.

Latitude and longitude.

1 point extra credit,
because eventually we

are gonna get
there, chapter 12.7.

12.7 comes way
after spring break.

But before we get there, who
is in mechanical engineering

again?

You know about Euler's
angles, and stuff like that.

OK.

Can you write me
the equations of x

and y and z of the sphere
with respect to u and v,

u being latitude and
v being longitude?

These have to be
trigonometric functions.

In terms of u and v, when u is
latitude and v is longitude.

1 point extra credit
until a week from today.

How about that?

U and v are latitude
and longitude.

And express the xyz point in
the ambient space on the sphere.

x squared plus x squared
plus x squared would be 16.

So you'll have lots of
cosines and sines [INAUDIBLE]

of those angles, the latitude
angle and the longitude angle.

And I would suggest to you that
you take--for the extra credit

thing--you take the longitude
angle to be from 0 to 2pi,

from the Greenwich 0 meridian
going back to himself,

and--well, there are two ways
we do this in mathematics

because mathematicians
are so diverse.

Some of us, say, for me,
I measure the latitude

starting from the North Pole.

I think that's because we all
believe in Santa or something.

So, we start measuring
always from the North Pole

because that's the most
important place on Earth.

They go 0, pi over 2, and then--
what is our lat--shame on me.

STUDENT: It's 33.

PROFESSOR TODA: 33?

OK.

Then pi would be the
equator, and then pi

would be the South Pole.

But some other mathematicians,
especially biologists

and differential geometry
people, I'm one of them,

we go like that.

Minus pi over 2, South Pole
0, pi over 2 North Pole.

So we shift that
kind of interval.

Then for us, the trigonometric
functions of these angles

would be a little
bit different when we

do the spherical coordinates.

OK, that's just extra credit.

It has nothing to do with
what I'm gonna do right now.

What I'm gonna do right now
is to pick a point on Earth.

We have to find Lubbock.

STUDENT: It's on the left.

PROFESSOR TODA: Here?

Is that a good point?

This is LBB.

That's Lubbock
International Airport.

So, for Lubbock--let's call it
p as well--draw the r sub u,

r sub v. So, u was latitude.

So if I fix the latitude,
that means I fix

the 33 point whatever you said.

u equals u0.

It is fixed, so I have u
fixed, and v equals v0 is that.

I fixed the meridian
where we are.

What is this tangent vector?

To the pink parallel,
the tangent vector

would be r sub what?

STUDENT: v.

PROFESSOR TODA: r
sub v. You are right.

You've got the idea.

And the blue vector would
be the partial velocity.

That's the tangent vector
to the blue meridian,

which is r sub u.

And what is n gonna be? n's
gonna be r sub u [INAUDIBLE].

But is there any other way
to do it in a simpler way

without you guys going oh, man.

Suppose some of you don't
wanna do the extra credit

and then say the
heck with it; I don't

care about her stinking extra
credit until chapter 12,

when I have to study the
spherical coordinates,

and is there another
way to get n.

I told you another way to get n.

Well, we are getting there.

n was the gradient of f
over the length of that.

And if we want it unitary,
the length of f was what?

f sub x, f sub y, f
sub z vector, where

the implicit equation of
the surface was f of x, y, z

equals c.

So now we've done this before.

You say Magdalena, you're
repeating yourself.

I know I'm repeating myself, but
I want you to learn this twice

so you can remember it.

What is f of x, y, z?

In my case, it's x squared
plus y squared plus z squared

minus 16, or even nothing.

Because the constant
doesn't matter anyway

when I do the gradient.

You guys are doing homework.

You saw how the gradient goes.

So gradient of f would
be 2x times-- and that's

the partial derivative times i
plus 2y times j plus 2z times

k-- that's very important.

[? Lovett ?] has some
coordinates we plug in.

Now, can we write-- two things.

I want two things from you.

Write me a total
differential b tangent plane

at the point-- so, a, write
the total differential.

I'm not going to ask you you
to do a linear approximation.

I could.

B, write the tangent plane
to the sphere at the point

that-- I don't know.

I don't want one that's trivial.

Let's take this 0, square root
of 8, and square root of 8.

I just have to make
sure that I don't

come with some
nonsensical point that's

not going to be on the sphere.

This will be because I
plugged it in in my mind.

I get 8 plus 8 is 16 last
time I checked, right?

So after we do this
we take a break.

Suppose that this is a
problem on your midterm,

or on your final or
on your homework,

or on somebody [? YouTubed it ?]
for a lot of money,

you asked them, $25 an hour
for me to work that problem.

That's good.

I mean-- it's-- it's a
class that you're taking

for your general requirement
because your school wants you

to take calc 3.

But it gives you-- and
I know from experience,

some of my students came
back to me and said,

after I took calc
3, I understood it

so well that I was able to
tutor calc 1, calc 2, calc 3,

so I got a double job.

Several hours a week,
the tutoring center,

math department,
and several hours

at the [INAUDIBLE] center.

You know what I'm talking about?

So I've had students who did
well and ended up liking this,

and said I can tutor
this in my sleep.

So-- and also private tutoring
is always a possibility.

OK.

Write total differential.

df equals, and now
I'll say at any point.

So I don't care what
the value will be.

I didn't say at what point.

It means in general.

Why is that?

You tell me, you
know that by now.

2x times what?

Now, you learned
your lesson, you're

never gonna make mistakes.

2y plus 2z dz.

That is very good.

That's the total differential.

Now, what is the equation
of the tangent plane?

It's not gonna be that.

Because I'm not
considering a graph.

I'm considering an
implicitly given surface

by this implicit equation f of
x, y, z, equals c, your friend.

So what was, in that case,
the equation of the plane

written as?

STUDENT: [INAUDIBLE]

PROFESSOR TODA: I'm--
yeah, you guys are smart.

I mean, you are fast.

Let's do it in general.

F sub x-- we did that last
time, [INAUDIBLE] times--

do you guys remember?

x minus x0.

And this is at the point plus
big F sub y at the point times

y minus y0 plus big F sub
z at the point z minus z0.

This is just review.

Equals 0.

Stop.

Where do these guys come from?

From the gradient.

From the gradient.

Which are the a,b,c, now I
know my ABCs, from the normal.

My ABCs from the normal.

So in this case-- I
don't want to erase

this beautiful picture.

The last thing I have to do
before the break is-- you

said 0.

I'm a lazy person by definition.

Can you tell me why
you said 0 times?

STUDENT: Because the
x value is [INAUDIBLE]

PROFESSOR TODA: You said
2x, plug in and x equals 0

from your point,
Magdalena, so you don't

have to write down everything.

But I'm gonna write down 0
times x minus 0 plus-- what's

next for me?

STUDENT: 2 square root 8.

PROFESSOR TODA: 2y, 2 root 8.

Is root 8 beautiful?

It looks like heck.

At the end I'm gonna
brush it up a little bit.

This is the partial-- f sub y of
t times y minus-- who is y, z?

Root 8.

Do I like it?

I hate it, but it
doesn't matter.

Because I'm gonna simplify.

Plus again, 2 root 8, thank you.

This is my c guy.

Times z minus root 8 equals 0.

I picked another example
from the one from the book,

because you are gonna
read the book anyway.

I'm gonna erase that.

And I'm gonna brush
this up because it

looks horrible to me.

Thank God this goes away.

So the plane will
simply be a combination

of my y and z in a constant.

And if I want to
make my life easier,

I'm gonna divide by what?

By this.

So in the end, it
doesn't matter.

Come on.

I'll get y minus root 8 plus
c minus root 8 equals 0.

Do I like it?

I hate it.

No, you know, I don't like it.

Why don't I like it?

It's not simplified.

So in any case, if this
were multiple choice,

it would not be written
like that, right?

So what would be the
simplified claim in this case?

The way I would write
it-- a y plus a z minus--

think, what is root 8?

STUDENT: 2 root 2.

PROFESSOR TODA: And 2 root 2.

And 2 root 2, how
much-- minus 4 root 2.

And this is how you are expected
to leave this answer boxed.

This is that tangent
plane at the point.

To the sphere.

There are programs--
one time I was teaching

advance geometry, 4331, and one
thing I gave my students to do,

which was a lot of fun--
using a parametrization,

plot the entire
sphere with MathLab.

We did it with MathLab.

Some people said they know
[INAUDIBLE] I didn't care.

So MathLab for me
was easier, so we

plotted the sphere in MathLab.

We picked a point,
and we drew-- well,

we drew-- with MathLab we
drew the tangent plane that

was tangent to the
sphere at that point.

And they liked it.

It was-- you know
what this class is,

is-- if you're math
majors you take it.

It's called advanced geometries.

Mainly it's theoretical.

It teaches you Euclidian
axioms and stuff,

and then some
non-Euclidian geometries.

But I thought that I would
do it into an honors class.

And I put one third of that
last class visualization

with MathLab of geometry.

And I think that was what
they liked the most, not so

much the axiomatic
part and the proofs,

but the hands-on computation
and visualization in the lab.

We have this lab, 113.

We used to have two labs,
but now we are poor,

we only have one.

No, we lost the lab.

The undergraduate
lab-- 009, next to you,

is lost because-- I used
to each calc 3 there.

Not because-- that's
not why we lost it.

We lost it because we-- we
put some 20 graduate students

there.

We have no space.

And we have 130 graduate
students in mathematics.

Where do you put them?

We just cram them into cubicles.

So they made 20 cubicles
here, and they put some,

so we lost the lab.

It's sad.

All right.

So that's it for now.

We are gonna take a
short break, and we

will continue for one more hour,
which is mostly application.

I'm sort of done with 11.4.

I'll jump into 11.5 next.

Take a short break.

Thanks for the attendance.

Oh, and you did the calculus.

Very good.

Did this homework give you
a lot of headaches, troubles

or anything, or not?

Not too much?

It's a long homework.

49 problems-- 42 problems.

It wasn't bad?

OK, questions from the-- what
was it, the first part-- mainly

the first part of chapter 11.

This is where we are.

Right now we hit the
half point because 11.8

is the last section.

And we will do that, that's
Lagrange multipliers.

So, let's do a little
bit of a review.

Questions about homework.

Do you have them?

Imagine this would
be office hour.

What would you ask?

STUDENT: I know it's
a stupid question,

but my visualization [INAUDIBLE]
coming along, and question

three about the sphere passing
the plane and passing the line.

So you have a 3, 5,
and 4 x, y, and z,

and you have a radius of 5.

Is it passing the x, y plane?

Is it passing [INAUDIBLE]
x plane and [INAUDIBLE]

passing the other plane.

PROFESSOR TODA: So-- say again.

So you have 3 and 4 and 5--

STUDENT: x minus-- yes.

PROFESSOR TODA: What
are the coordinates?

STUDENT: 3, 4, and 5.

PROFESSOR TODA: 3, 4, and
5, just as you said them.

You can--

STUDENT: And the radius is 5.

PROFESSOR TODA: Radius of?

STUDENT: 5.

Radius is equal to 5.

[INAUDIBLE]

PROFESSOR TODA: Yeah, well, OK.

So assume you have a
sphere of radius 5, which

means you have 25.

If you do the 3 squared plus
4 squared plus 5 squared,

what is that?

For this point.

You have two separate points.

For this point you
have 25 plus 25.

Are you guys with me?

So you have the
specific x0, y0, z0.

You do the sum of the
squares, and you get 50.

My question is, is this point
outside, inside the sphere

or on the sphere?

On the sphere,
obviously, it's not,

because it does not verify the
equation of the sphere, right?

STUDENT: [INAUDIBLE] those the
location of the center point.

STUDENT: Where's the
center of the sphere?

STUDENT: [INAUDIBLE]

PROFESSOR TODA: The center
of the sphere would be at 0.

STUDENT: [INAUDIBLE]

PROFESSOR TODA: We are
making up a question.

So, right?

So practically, I am
making up a question.

STUDENT: Oh, OK.

PROFESSOR TODA: So I'm saying if
you have a sphere of radius 5,

and somebody gives you this
point of coordinates 3, 4,

and 5, where is the point?

Is it inside the sphere, outside
the sphere or on the sphere?

On the sphere it cannot be
because it doesn't verify

the sphere.

Ah, it looks like a Mr. Egg.

I don't like it.

I'm sorry, it's a sphere.

So a point on a sphere that
will have-- that's a hint.

A point on a sphere that
will have coordinates 3 and 4

would be exactly 3, 4, and 0.

So it would be where?

STUDENT: 16, 4.

PROFESSOR TODA: 3 squared plus
4 squared is 5 squared, right?

So those are
Pythagorean numbers.

That's the beauty of them.

I'm trying to draw well.

Right.

This is the point a.

You go up how many?

You shift by 5.

So are you inside or outside?

STUDENT: Outside.

PROFESSOR TODA: Yeah.

STUDENT: Are you outside
or are you exactly on-- oh.

Sorry, I thought--

PROFESSOR TODA: You go--

STUDENT: I thought you
were saying point a.

Point a is like
exactly-- [INAUDIBLE]

PROFESSOR TODA: You
are on the equator,

and from the Equator
of the Earth,

you're going parallel to the
z-axis, then you stay outside.

But the question is
more subtle than that.

This is pretty--
you figured it out.

1 point-- 0.5 extra credit.

That we don't have--
I wish we had-- maybe

we'll find some time.

When I-- when we rewrite the
book, maybe we should do that.

So express the points outside
the sphere, inside the sphere,

and on the sphere
using exclusively

equalities and inequalities.

And that's extra credit.

So, of course, the
[INAUDIBLE] is obvious.

The sphere is the set of
the triples x, y, z in R3.

OK, I'm teaching you a little
bit of mathematical language.

x, y, z belongs to R3,
R3 being the free space,

with the property that x squared
plus y squared plus z squared

equals given a squared.

What if you have less than,
what if you have greater than?

Ah, shut up, Magdalena.

This is all up to you.

You will figure
out how the points

on the outside and the points
on the inside are characterized.

And unfortunately we don't
emphasize that in the textbook.

I'll erase.

You figured it out.

And now I want to
move on to something

a little bit challenging,
but not very challenging.

STUDENT: Professor, [INAUDIBLE]

PROFESSOR TODA: The
last requirement

on the extra credit?

So I said the sphere
represents the set of all

triples x, y, z in
R3 with the property

that x squared plus y squared
plus y squared plus z squared

equals a squared.

With the equality sign.

Represent the points on
the inside of the sphere

and the outside of the sphere
using just inequalities.

Mathematics.

No writing, no words,
just mathematics.

In set theory symbols.

Like, the set of points
with braces like that.

OK.

I'll help you review a little
bit of stuff from the chain

rule in-- in chapter--
I don't know, guys,

it was a long time ago.

Shame on me.

Chapter 3, calc 1.

Versus chain rule rules in
calc in-- chapter 5 calc 3.

This is a little
bit of a warmup.

I don't want to
[INAUDIBLE] again

next time when we
meet on Thursday.

Bless you.

The bless you was
out of the context.

What was the chain rule?

We did compositions
of functions,

and we had a diagram that we
don't show you, but we should.

There is practically a function
that comes from a set A

to a set B to a set
C. These are the sets.

And we have g and an f.

And we have g of f of t.

t is your favorite letter here.

How do you do the
derivative with respect

to g composed with f?

I asked the same question to
my Calc 1 and Calc 2 students,

and they really had a hard
time expressing themselves,

expressing the chain rule.

And when I gave them
an example, they

said, oh, I know how to
do it on the example.

I just don't know how to do it
on the-- I like the numbers,

but I don't like them letters.

So how do we do
it in an example?

I chose natural log,
which you find everywhere.

So how do you do d
dt of this animal?

It's an animal.

STUDENT: [INAUDIBLE]

PROFESSOR TODA: So the idea
is you go from the outside

to the inside, one at a time.

My students know that.

You prime the function,
the outer function,

the last one you applied,
to the function inside.

And you prime that with
respect to the argument.

This is called the
argument in that case.

Derivative of natural
log is 1 over what?

The argument.

And you cover up natural
log with your hand,

and you keep going.

And you say, next I go,
times the derivative

of this square, plus 1,
prime with respect to t.

So I go times 2t.

And that's what we have.

And they say, when you explain
it like that, they said to me,

I can understand it.

But I'm having a
problem understanding it

when you express this diagram--
that it throws me off.

So in order to avoid that kind
of theoretical misconception,

I'm saying, let us see
what the heck this is.

d dt of g of f of t, because
this is what you're doing,

has to have some understanding.

The problem is that Mister
f of t, that lives here,

has a different argument.

The letter in B should
be, let's say, u.

That doesn't say
anything practically.

How do you differentiate
with respect to what?

You cannot say d dt here.

So you have to call f
of t something generic.

You have to have a
generic variable for that.

So you have then dg du, at
what specific value of u?

At the specific value of
u that we have as f of t.

Do you understand the
specificity of this?

Times-- that's the chain
rule, the product coming

from the chain rule-- df pt.

You take du dt or d of dt.

It is the same thing.

Say it again, df dt.

I had a student ask me,
what if I put du dt?

Would it be wrong?

No, as long as you understand
that u is a-something,

as the image of this t.

Do you know what he liked?

He said, do you know
what I like about that?

I like that I can imagine
that these are two cowboys-- I

told the same thing to my son.

He was so excited,
not about that,

but about these two cowboys.

Of course, he is 10.

These are the cowboys.

They are across.

One is on top of
the building there,

shooting at this
guy, who is here

across the street on the bottom.

So they are
annihilating each other.

They shoot and they die.

And they die, and
you're left with 1/3.

The same idea is that, actually,
these guys do not simplify.

du and-- [? du, ?] they're not
cowboys who shoot at each other

at the same time and both
die at the same time.

It is not so romantic.

But the idea of remembering
this formula is the same.

Because practically, if you want
to annihilate the two cowboys

and put your hands over them
so you don't see them anymore,

du dt, you would
have to remember, oh,

so that was the
derivative with respect

to t that I initially
have of the guy on top,

which was g of f of
the composed function.

So if you view g of f of t
as the composed function,

who is that?

The composition g
composed with f of t

is the function g of f of t.

This is the function that
you want to differentiate

with respect to time, t.

This is this, prime
with respect to t.

It's like they would be killing
each other, and you would die.

And I liked this
idea, and I said,

I should tell that to my
students and to my son.

And, of course, my son
started jumping around

and said that he understands
multiplication of fractions

better now.

They don't learn about
simplifications-- I don't

know how they teach these kids.

It became so complicated.

It's as if mathematics--
mathematics is the same.

It hasn't changed.

It's the people
who make the rules

on how to teach it that change.

So he simply doesn't see
that this simplifies.

And when I tell him simplify,
he's like, what is simplify?

What is this word simplify?

My teacher doesn't use it.

So I feel like sometimes
I want to shoot myself.

But he went over that and
he understood about the idea

of simplification.

[? He ?] composing
something on top

and the bottom finding the
common factors up and down,

crossing them out, and so on.

And so now he knows
what it means.

But imagine going to
college without having

this early knowledge.

You come to college,
you were good in school,

and you've never learned
enough simplification.

And then somebody like me,
and tells you simplification.

You say, she is a foreigner.

She has a language barrier
that is [INAUDIBLE] she has

that I've never heard before.

So I wish the people who
really re-conceive, re-write

the curriculum for K12
would be a little bit

more respectful of the history.

Imagine that I
would teach calculus

without ever telling you
anything about Leibniz, who

was Leibniz, he doesn't exist.

Or Euler, or one
of these fathers.

They are the ones who
created these notations.

And if we never tell you
about them, that I guess,

wherever they are, it is an
injustice that we are doing.

All right.

Chain rule in
Chapter 5 of Calc 3.

This is a little bit
more complicated,

but I'm going to teach it
to you because I like it.

Imagine that you have z equals
x squared plus y squared.

What is that?

It's an example of a graph.

And I just taught
you what a graph is.

But imagine that
xy follow a curve.

[INAUDIBLE] with
respect to time.

And you will say, Magdalena,
can you draw that?

What in the world do you mean
that x and y follow a curve?

I'll try to draw it.

First of all, you are on a walk.

You are in a beautiful valley.

It's not a vase.

It's a circular
paraboloid, as an example.

It's like an egg shell.

You have a curve on that.

You draw that.

You have nothing better
to do than decorating eggs

for Easter.

Hey, wait.

Easter is far, far away.

But let's say you want to
decorate eggs for Easter.

You take some color of paint
and put paint on the egg.

You are actually describing
an arc of a curve.

And x and y, their
projection on the floor

will be x of t, y of t.

Because you paint in time.

You paint in time.

You describe this in time.

Now, if x of ty of t is
being projected on the floor.

Of course, you have a curve
here as well, which is what?

Which it will be x
of t, y of t, z of t.

Oh, my god.

Yes, because the altitude also
depends on the motion in time.

All right.

So what's missing here?

It's missing the third
coordinate, duh, that's

0 because I'm on the floor.

I'm on the xy plane, which
is the floor z equals z.

But now let's
suppose that I want

to say this is f of x and y,
and I want to differentiate

f with respect to t.

And you go, say what?

Oh, my god.

What is that?

I differentiate f
with respect to t.

By differentiating
f with respect to t,

I mean that I have f of
x and y differentiated

with respect to t.

And you say, wait, Magdalena.

This doesn't make any sense.

And you would be right to say
it doesn't make any sense.

Can somebody tell me why
it doesn't make any sense?

It's not clear where in the
world the variable t is inside.

So I'm going to say, OK,
x are themselves functions

of t, functions of that.

x of t, y of t.

If I don't do that,
it's not clear.

So this is a composed
function just like this one.

Look at the similarity.

It's really beautiful.

This is a function of
a function, g of f.

This is a function
of two functions.

Say it again, f is a function
of two functions, x and y.

This was a function
of a function of t.

This was a function
of two functions of t.

Oh, my God.

How do we compute this?

There is a rule.

It can be proved.

We will look a little bit into
the theoretical justification

of this proof later.

But practically what
you do, you say,

I have to have some
order in my life.

OK.?

So the way we do that,
we differentiate first

with respect to the first
location, which is x.

I go there, but I cannot write
df dx because f is a mother

of two babies.

f is a function of two
variables, x and y.

She has to be a mother
to both of them;

otherwise, they get
jealous of one another.

So I have to say, partial
of f with respect to x,

I cannot use d.

Like Leibniz, I have
to use del, d of dx.

At the point x of dy of t,
this is the location I have.

Times what?

I keep derivation.

I keep derivating, like
don't drink and derive.

What is that?

The chain rule.

Prime again, this guy x
with respect to t, dx dt.

And then you go,
plus because she has

to be a mother to both kids.

The same thing for
the second child.

So you go, the derivative
of f with respect

to y, add x of ty
of t times dy dt.

So you see on the surface, x and
y are moving according to time.

And somehow we want to
measure the derivative

of the resulting function,
or composition function,

with respect to time.

This is a very
important chain rule

that I would like
you to memorize.

A chain rule.

Chain Rule No.

1.

Is it hard?

No, but for me it was.

When I was 21 and I saw
that-- and, of course,

my teacher was good.

And he told me, Magdalena,
imagine that instead of del you

would have d's.

So you have d and d and d and d.

The dx dx here, dy dy here,
they should be in your mind.

They are facing each other.

They are across on a diagonal.

And then, of course, I didn't
tell my teacher my idea

with the cowboys,
but it was funny.

So this is the chain rule
that re-makes, or generalizes

this idea to two variables.

Let's finish the example
because we didn't do it.

What is the derivative
of f in our case?

df dt will be-- oh, my god--
at any point p, how arbitary,

would be what?

First, you write
with respect to x.

2x, right?

2x.

But then you have to compute
this dx, add the pair you give.

And the pair they
gave you has a t.

So 2x is add x of
ty-- if you're going

to write it first
like that, you're

going to find it weird-- times,
I'm done with the first guy.

Then I'm going to take
the second guy in red,

and I'll put it here.

dx dt, but dx dt
everybody knows.

[INAUDIBLE] Let me
write it like this.

Plus [INAUDIBLE] that
guy again with green-- dy

computed at the pair x
of dy of [? t ?] times,

again, in red, dy dt.

So how do we write
the whole thing?

Could I have written it
from the beginning better?

Yeah.

2x of t, dx dt plus 2y of t dy.

Is it hard?

No, this is the idea.

Let's have something
more specific.

I'm going to erase
the whole thing.

I'll give you a problem
that we gave on the final

a few years ago.

And I'll show you how my
students cheated on that.

And I let them cheat, in
a way, because in the end

they were smart.

It didn't matter how they did
the problem, as long as they

got the correct answer.

So the problem was like that.

And my colleague did that many
years ago, several years ago,

did that several times.

So he said, let's do f of
t, dt squared and g of t.

I'll I'll do this
one, dq plus 1.

And then let's
[INAUDIBLE] the w of u

and B, exactly the same thing I
gave you before, [INAUDIBLE] I

remember that.

And he said, compute the
derivative of w of f of t,

and g of t with respect to t.

And you will ask,
wait a minute here.

Why do you put d and not del?

Because this is a composed
function that in the end

is a function of t only.

So if you do it as
a composed function,

because this goes like this.

t goes to two
functions, f of t and u.

And there is a function w
that takes both of them, that

is a function of both of them.

In the end, this composition
that's straight from here

to here, is a function
of one variable only.

So my students then-- it was in
the beginning of the examine,

I remember.

And they said, well,
I forgot, they said.

I stayed up almost all night.

Don't do that.

Don't do what they did.

Many of my students
stay up all night

before the final because
I think I scare people,

and that's not what I mean.

I just want you to study.

But they stay up before
the final and the next day,

I'm a vegetable.

I don't even remember
the chain rule.

So they did not
remember the chain rule

that I've just wrote.

And they said, oh, but I
think I know how to do it.

And I said, shh.

Just don't say anything.

Let me show you how the
course coordinator wanted

that done several years ago.

So he wanted it done
by the chain rule.

He didn't say how you do it.

OK?

He said just get to
the right answer.

It doesn't matter.

He wanted it done like that.

He said, dw of f of tg
of p with respect to t,

would be dw du, instead
of u you have f of t.

f of tg of t times df
dt plus dw with respect

to the second variable.

So this would be u, and
this would be v with respect

to the variable v,
the second variable

where [? measure ?]
that f of dg of t.

Evaluate it there times dg dt.

So it's like dv dt, which is dg
dt. [INAUDIBLE] So he did that,

and he expected
people to do what?

He expected people to take
a u squared the same 2 times

u, just like you
did before, 2 times.

And instead of u, since u is
f of t to [INAUDIBLE] puts

2f of t, this is the first
squiggly thing, times v of dt.

2t is this smiley face.

This is 2t plus--
what is the f dv?

Dw with respect to dv is
going to be 2v 2 time gf t.

When I evaluate add gf
t, this funny fellow

with this funny fellow, times qg
d, which, with your permission

I'm going to erase
and write 3p squared.

And the last row he expected
my students to write

was 2t squared times 2t plus
2pq plus 1, times 3t squared.

Are you guys with me?

So [INAUDIBLE] 2t 2x
2t squared, correct.

I forgot to identify
this as that.

All right.

So in the end, the answer
is a simplified answer.

Can you tell me what it is?

I'm too lazy to write it down.

You compute it.

How much is it simplified?

Find it as a polynomial.

STUDENT: [INAUDIBLE].

PROFESSOR TODA:
So you have 6, 6--

STUDENT: 16 cubed plus 3--

PROFESSOR TODA: T
to the 5th plus--

STUDENT: [INAUDIBLE].

PROFESSOR TODA: In
order, in order.

What's the next guy?

STUDENT: [INAUDIBLE].

PROFESSOR TODA: 4t cubed.

And the last guy--

STUDENT: 6t squared.

PROFESSOR TODA: 6t squared.

Yes?

Did you get the same thing?

OK.

Now, how did my students do it?

[INAUDIBLE]

Did they apply the chain rule?

No.

They said OK, this
is how it goes.

W of U of T and V of T is U is
F. So this guy is T squared,

T squared squared,
plus this guy is T

cubed plus 1 taken and
shaken and squared.

And then when I do the
whole thing, derivative

of this with respect
to T, I get--

I'm too lazy-- T to the
4 prime is 40 cubed.

I'm not going to do on the map.

2 out T cubed plus 1 times
chain rule, 3t squared.

40 cubed plus 16 to the 5 plus--
[INAUDIBLE] 2 and 6t squared.

So you realize that I
have to give them 100%.

Although they were very
honest and said, we blanked.

We don't remember
the chain rule.

We don't remember the formula.

So that's fine.

Do whatever you can.

So I gave them 100% for that.

But realize that the
author of the problem

was a little bit naive.

Because you could have
done this differently.

I mean if you wanted to
actually test the whole thing,

you wouldn't have given-- let's
say you wouldn't have given

the actual-- yeah, you wouldn't
have given the actual functions

and say write the chain
formula symbolically

for this function applied
for F of T and G of T.

So it was-- they
were just lucky.

Remember that you need
to know this chain rule.

It's going to be
one of the problems

to be emphasized in the exams.

Maybe one of the top 15 or
16 most important topics.

Is that OK?

Can I erase the whole thing?

OK.

Let me erase the whole thing.

OK.

Any other questions?

No?

I'm not going to let
you go right away,

we're going to work one
more problem or two more

simple problems.

And then we are going to go.

OK?

So question.

A question.

What do you think the
gradient is good at?

Two reasons, right.

Review number one.

If you have an increasingly
defined function,

then the gradient of F was what?

Equals direction of the
normal to the surface S--

let's say S is given
increasingly at the point

with [INAUDIBLE].

But any other reason?

Let's take that again.

Z equals x squared
plus y squared.

Let's compute a few
partial derivatives.

Let's compute the gradient.

The gradient is Fs of x, Fs
of y, where this is F of xy

or Fs of xi plus Fs of yj.

[INAUDIBLE]

And we drew it.

I drew this case, and we also
drew another related example,

where we took Z equals 1 minus
x squared minus y squared.

And we went skiing.

And we were so happy last week
to go skiing, because we still

had snow in New
Mexico, and we-- and we

said now we computed the
Z to be minus 2x minus 2y.

And we said, I'm
looking at the slopes.

This is the x duration
and the y duration.

And I'm looking at the slopes of
the lines of these two curves.

So one that goes
down, like that.

So this was for what?

For y equals 0.

And this was for x equals 0.

Curve, x equals
0 curve in plane.

Right?

We just cross-section
our surface,

and we have this [INAUDIBLE].

And then we have the two
tangents, two slopes.

And we computed them everywhere.

At every point.

But realize that to go
up or down these hills,

I can go on a curve
like that, or I

can go-- remember the
train of Mickey Mouse going

on the hilly point on the hill?

We try to take different paths.

We are going hiking.

We are going hiking, and we'll
take hiking through the pass.

OK.

How do we get the maximum
rate of change of the function

Z equals F of x1?

So now I'm
anticipating something.

I'd like to see your intuition,
your inborn sense of I

know what's going to happen.

And you know what
that from Mister--

STUDENT: Heinrich.

PROFESSOR TODA: [? Heinrich ?]
from high school.

So I'm asking-- let me
rephrase the question

like a non-mathematician.

Let's go hiking.

This is [INAUDIBLE] we
go to the lighthouse.

Which path shall I take
on my mountain, my hill,

my god knows what
geography, in order

to obtain the maximum
rate of change?

That means the
highest derivative.

In what direction do I get
the highest derivative?

STUDENT: In what direction you
get the highest derivative--

PROFESSOR TODA: So
in which direction--

in which direction
on this hill do

I get the highest derivative?

The highest rate of change.

Rate of change means I want to
get the fastest possible way

somewhere.

STUDENT: The shortest slope?

Along just the straight line up.

PROFESSOR TODA: Along--

STUDENT: You don't want
to take any [INAUDIBLE].

PROFESSOR TODA: Right.

STUDENT: [INAUDIBLE].

It could be along any axis.

PROFESSOR TODA: So could
you see which direction

those are-- very good.

Actually you were getting
to the same direction.

So [INAUDIBLE] says
Magdalena, don't be silly.

The actual maximum rate of
change for the function Z

is obviously, because
it is common sense,

it's obviously happening if
you take the so-called-- what

are these guys?

[INAUDIBLE], not meridians.

STUDENT: Longtitudes?

PROFESSOR TODA: OK.

That is-- OK.

Suppose that we don't hike,
because it's too tiring.

We go down from the
top of the hill.

Ah, there's also very good idea.

So when you let yourself
go down on a sleigh,

don't think bobsled or
anything-- just a sleigh,

think of a child's sleigh.

No, take a plastic bag
and put your butt in it

and let yourself go.

What is their
direction actually?

Your body will find the
fastest way to get down.

The fastest way to get
down will happen exactly

in the same
directions going down

in the directions
of these meridians.

OK?

And now, [INAUDIBLE].

The maximum rate of
change will always

happen in the direction
of the gradient.

You can get a little
bit ahead of time

by just-- I would like this
to [INAUDIBLE] in your heads

until we get to that section.

In one section we will be there.

We also-- it's also reformulated
as the highest, the steepest,

ascent or descent.

The steepest.

The steepest ascent or
the steepest descent

always happens in the
direction of the gradient.

Ascent is when you hike
to the top of the hill.

Descent is when you let yourself
go in the plastic [INAUDIBLE]

bag in the snow.

Right?

Can you verify this happens
just on this example?

It's true in general,
for any smooth function.

Our smooth function is
a really nice function.

So what is the gradient?

Well again, it was 2x 2y, right?

And that means at a certain
point, x0 y0, whenever you are,

guys you don't
necessarily have to start

from the top of the hill.

You can be-- OK,
this is your cabin.

And here you are with
friends, or with mom and dad,

or whoever, on the hill.

You get out, you take the
sleigh, and you go down.

So no matter where
you are, there you go.

You have 2x0 times
i plus 2y0 times j.

And the direction of the
gradient will be 2x0 2y0.

Do you like this one?

Well in this case,
if you were-- suppose

you were at the
point [INAUDIBLE].

You are at the point
of coordinates--

do you want to be here?

You want to be here, right?

So we've done that before.

I'll take it as 1 over
[? square root of ?]

2-- I'm trying to be creative
today-- [INAUDIBLE] y equals 0,

and Z equals-- what's left?

1/2, right?

Where am I?

Guys, do you realize where I am?

I'll [? take a ?] [INAUDIBLE].

y0.

So I need to be on this
meridian on the red thingy.

And somewhere here.

What's the duration
of the gradient here?

Delta z at this p.

Then you say ah,
well, I don't get it.

I have-- the second guy will
become 0, because y0 is 0.

The first guy will become
1 over square root of 2.

So I have 2 times 1 over square
root of 2 times i plus 0j.

It means in the direction of i--
in the direction of i-- from p,

I have the fastest-- fastest,
Magdalena, fastest-- descent

possible.

But we don't say in
the direction of i

in our everyday life, right?

Let's say geographic points.

We are-- I'm a bug,
and this is north.

This is south.

This is east.

And this is west.

So if I go east, going east
means going in the direction i.

Now suppose-- I'm going
to finish with this one.

Suppose that my house
is not on the prairie

but my house is here.

House, h.

Find me a wood
point to be there.

STUDENT: Northeast.

Or to get further down.

PROFESSOR TODA: Anything, what
would look like why I'm here?

x0, y0, z0.

Hm.

1/2, 1/2, and I
need the minimum.

So I want to be on the
bisecting plane between the two.

You understand?

This is my quarter.

And I want to be in
this bisecting plane.

So I'll take 1/2, 1/2, and
what results from here?

I have to do math.

1 minus 1/4 minus 1/4 is 1/2.

Right?

1/2, 1/2, 1/2.

This is where my house
is [? and so on. ?]

And this is full of smoke.

And what is the
maximum rate of change?

What is the steepest
descent is the trajectory

that my body will take
when I let myself go down

on the sleigh.

How do I compute that?

I will just do the same thing.

Delta z at the point x0
equals 1/2, y0 equals 1/2,

z0 equals 1/2.

Well what do I get as direction?

That will be the
direction of the gradient.

2 times 1/2-- you
guys with me still?

i plus 2 times 1/2 with j.

And there is no Mr.
z0 In the picture.

Why?

Because that will
give me the direction

like on-- in a geographic way.

North, west, east, south.

These are the
direction in plane.

I'm not talking
directions on the hill,

I'm talking
directions on the map.

These are directions on the map.

So what is the direction
i plus j on the map?

If you show this to a
geography major and say,

I'm going in the direction
i plus j on the map,

he will say you are crazy.

He doesn't understand the thing.

But you know what you mean.

East for you is the
direction of i in the x-axis.

[INAUDIBLE]

And this is north.

Are you guys with me?

The y direction is north.

So I'm going perfectly
northeast at a 45-degree angle.

If I tell the
geography major I'm

going northeast perfectly in
the middle, he will say I know.

But you will know that
for you, that is i plus j.

Because you are
the mathematician.

Right?

So you go down.

And this is where you are.

And you're on the meridian.

This is the direction i plus j.

So if I want to project my
trajectory-- I went down

with the sleigh, all the way
down-- project the trajectory,

my trajectory is a
body on the snow.

Projecting it on the
ground is this one.

So it is exactly the
direction i plus j.

Right, guys?

So exactly northeast
perfectly at 45-degree angles.

Now one caveat.

One caveat, because
when we get there,

you should be ready
already, in 11.6 and 11.7.

When we will say direction,
we are also crazy people.

I told you, mathematicians
are not normal.

You have to be a
little bit crazy

to want to do all the stuff
in your head like that.

i plus j for us is not a
direction most of the time.

When we say direction, we mean
we normalize that direction.

We take the unit
vector, which is unique,

for responding to i plus j.

So what is that
unique unit vector?

You learned in Chapter 9
everything is connected.

It's a big circle.

i plus j, very good.

So direction is a unit vector
for most mathematicians,

which means you will be i
plus j over square root of 2.

So in Chapter 5, please
remember, unlike Chapter 9,

direction is a unit vector.

In Chapter 9, Chapter 10,
it said direction lmn,

direction god knows what.

But in Chapter 11, direction
is a vector in plane,

like this one, i
plus [INAUDIBLE]

has to be a unique
normal-- a unique vector.

OK?

And we-- keep that in mind.

Next time, when we
meet on Thursday,

you will understand why
we need to normalize it.

Now can we say goodbye to
the snow and everything?

It's not going to
show up much anymore.

Remember this example.

But we will start with
flowers next time.

OK.

Have a nice day.

Yes, sir?

Let me stop the video.