-
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.
-