-
-
PROFESSOR: I would like to
review just briefly what
-
we discussed last time.
-
We gave very important results,
and that was Green's Theorem.
-
And I would like to
know if you remember
-
when I said about the
settling for this problem.
-
So we'll assume we have
a domain without a hole,
-
D. D is a domain
without a hole inside,
-
without punctures or holes.
-
-
There is a scientific name in
mathematics for such a domain.
-
This is going to be
simply connected.
-
-
And this is a difficult
topological theorem,
-
but this is what we expect, OK?
-
And what does it mean?
-
What does it mean?
-
It means that in the C being
a Jordan curve was what?
-
How?
-
This was continuous,
no self intersections.
-
-
In such a case, we set up
M and N to be C1 functions.
-
-
And then we proceed through
the path integral of C.
-
Do you like this kind of C,
or you prefer a straight C?
-
The path integral of C of M
of xy dx class, N of xy, dy,
-
everything is in plane.
-
I'm sorry that I
cannot repeat that,
-
but we discussed that
time, is in the plane of 2.
-
And then what-- do
you remember in terms
-
of how this path integral,
[INAUDIBLE] inside,
-
is connected to a double
integral over the whole domain.
-
In particular, do you remember--
this is easy to memorize--
-
but do you remember
what's inside?
-
Because for the final, you are
expected to know his result.
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: N sub X.
-
STUDENT: Minus M sub Y.
-
PROFESSOR: Minus M
sub Y. [INAUDIBLE]
-
must M-- M and N-- M sub Y.
Here is the Y. Of course this
-
would be dA in plane,
and in the-- if you
-
want to represent this
in the general format,
-
the MdX minus the MdY.
-
Feel free to do that.
-
One was a correlary
or a consequence.
-
This theorem was that if I
were to take this big M to be
-
the minus Y as a function,
then this function N will
-
be plus X, what will I get?
-
I would get that minus
YdX plus NdY will be what?
-
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Two times, excellent.
-
You are very awake.
-
So I wanted to catch you.
-
I couldn't catch you.
-
I thought you would say
the A of the domain,
-
but you said it right.
-
You said Y is the
area of the domain.
-
You probably
already in your mind
-
did the math saying X sub X
is one, minus Y sub 1 is 1.
-
1 plus 1 is two, so the
two part [INAUDIBLE].
-
OK, so what did we do with it?
-
We just stared at it?
-
No.
-
We didn't just stare at it.
-
We did something nice
with it last time.
-
We proved that, finally,
that the area, this radius R
-
will be pi R
squared, and we also
-
proved that the area
[INAUDIBLE] is what?
-
I'm testing you to
see if you remember.
-
STUDENT: AB pi.
-
PROFESSOR: AB pi.
-
Very good.
-
Or pi AB.
-
It's more, I like it the
way you said it, AB pi,
-
because pi is a transcendental
number, and you go around
-
and it's like partly
variable to put at the end.
-
And the real numbers
that could be anything,
-
so [INAUDIBLE] they are the
semi axes of the ellipse.
-
So we gain new knowledge and
we are ready to move forward.
-
And we're going to move
forward to something
-
called section 13.5, which
is the surface integral.
-
We will come back
to Green's Theorem
-
because there are
generalizations
-
of the Green's Theorem to
more complicate the case.
-
But in order to
see those, we have
-
to learn a little bit more.
-
In mathematics, you need to
know many things, many pieces
-
of the puzzle, and then
you put them together
-
to get the whole picture.
-
All right, so what
is 13.5 about?
-
This is just review.
-
13.5, if should be looking like
a friend, old friend, to you.
-
And I'll show you
in a minute why this
-
is called the surface integral.
-
-
I saw that US natives
don't pronounce integral,
-
they pronounce in-negral.
-
And everybody that I heard
in romance language-speaking
-
countries like Spanish,
Italian, Portuguese,
-
they put the T there
out, very visibly.
-
So it doesn't matter.
-
Even some accent difference
in different parts
-
of the United States
pronounce it differently.
-
So what is the surface
integral about?
-
It's about integrating a smooth
function, not a vector value,
-
but a real value function.
-
Let's say you have G or XY being
a nice interglobal function
-
over some surfaces.
-
And you say, I'm
going to take it,
-
double integral, over S of GDS,
where DS will be area level.
-
-
I had a student one time who
looked at two different books
-
and said, I have a problem
with this, [INAUDIBLE].
-
In one book it shows a
big, fat snake over S.
-
And in another book, a
double integral over it,
-
and I don't know which one it
is because I don't understand.
-
No matter how you denote it,
it's still a double integral.
-
You know why?
-
Because it's an
integral over a surface.
-
The same thing, integral over
a surface or a domain plane,
-
or anything two-dimensional
will be a double integral.
-
So it doesn't matter
how you denote it.
-
In the end, it's going
to be a double integral.
-
Now, what in the world
do we mean by that?
-
DS is an old friend of
yours, and I don't know
-
if you remember him at all.
-
He was infinitesimal element
on some curved or linear patch.
-
Imagine your favorite surface.
-
Let's assume it's a graph.
-
It doesn't have to be a graph,
but let's assume it's a graph.
-
And that's your
favorite surface S.
-
And then you draw
coordinate lines,
-
and you are looking at a patch.
-
-
And this patch looks small,
but it's not small enough.
-
I want this to be
infinitesimally small.
-
Imagine that these curvature
lines become closer and closer
-
to one another.
-
And then we look in the
directions of DX and DY,
-
and then you say, wait a
minute, I'm not in plane.
-
If I were in plane,
DA will be DX, DY.
-
If you work with [INAUDIBLE],
I will be DX with DY.
-
So we've matched
the orientation.
-
If you would change
DY, [INAUDIBLE]
-
put the minus in front.
-
But this happens
because-- thank God this
-
will be a rectangular 1 patch
in plane, in the plane of 2.
-
But what if you
were on the surface?
-
On the surface, you
don't have this animal.
-
You will have-- which animal--
I'm testing your knowledge.
-
I'm doing review with you.
-
For sure, you will
see something that
-
involves the S in the final.
-
Have you started browsing
through those finals
-
I sent you?
-
Just out of curiosity.
-
And do they look awful to you?
-
They look awful to you.
-
Come on.
-
I'm going to work with
you on some of those.
-
I don't want you to
have-- I don't want
-
you to be afraid of this final.
-
Because compared to
other exams that you'll
-
have in other courses,
where a lot of memorization
-
is emphasized, this
should not be a problem.
-
So you could go over
the types of problems
-
that are significant
in this course,
-
you will not have any-- you
shouldn't have any problem.
-
And I sent you three samples.
-
Didn't I send you three
samples with solutions?
-
Those are going to help
you once you read the exam
-
and you can go ahead
and try the exam
-
or go ahead, read the solutions.
-
If I give you more of that, then
you should be doctors in those,
-
and you would be able to
solve them yourselves.
-
What about this one?
-
This is not DA.
-
It's a DA times something.
-
There is some factor in front
of that, and why is that?
-
In case of Z equals
F of X and Y,
-
you should know that by heart,
and I know that some of you
-
know it.
-
You just have to ring
the bell, and I'll
-
start ringing the bell.
-
Look at my first step.
-
And now you know, right?
-
STUDENT: [INAUDIBLE] 1--
-
PROFESSOR: I start with 1.
-
You said it right.
-
1 plus--
-
STUDENT: F of X.
-
PROFESSOR: F of X--
-
STUDENT: F squared.
-
PROFESSOR: Squared plus--
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: --SY squared.
-
So this what you're doing.
-
What are you going to do?
-
You're going to
do wait a minute.
-
This animal of mine,
that looks so scary,
-
this is nothing but what?
-
It's the same thing as, not
the picture, my picture.
-
It's going to be double integral
over a plane or domain D.
-
Well, I just said
goodbye to the picture,
-
but I find you are really smart.
-
I would have drawn the
[INAUDIBLE] of a picture here.
-
This is S and this
is D. What is D?
-
It's the projection,
projects the shadow.
-
The projection of S
on the plane XY when
-
I have to deal with a graph.
-
So when I have to deal with a
graph, my life is really easy.
-
And I said I'd get double
integral over D of G of God
-
knows what in the end will
be a function of X and Y. OK?
-
And here I'm going to have
square root of this animal.
-
Let me change it,
F sub X squared
-
like-- because in this
one it is like that.
-
Plus 1.
-
It doesn't matter
where I put the 1.
-
DXDY.
-
DXDY will be like the area of an
infinitesimally small rectangle
-
based on displacement
DX and displacement DY
-
and disintegration.
-
So this is DA.
-
Make the distinction
between the DA and the DX.
-
Can I draw the two animals?
-
Let me try again.
-
So you have the
direction of X and Y.
-
You have to be imaginative and
see that some coordinate lines
-
are [INAUDIBLE] for fixing Y.
-
When I fix Y, I sliced a
lot like that very nicely.
-
That's the same piece of
cheese that I've been dreaming
-
because I didn't have lunch.
-
I was too busy not
to have any lunch.
-
So you slice it
up like that where
-
Y equals constant to slice
it up like that for X
-
equals constant.
-
What you get are so-called
coordinate lines.
-
So the coordinate
lines are [INAUDIBLE].
-
Y equals my zeros, and
X equals the zeros.
-
And when they get to be
many dense and refined,
-
your curvilinear element is
this-- between two curves
-
like this two curves like that.
-
Shrunk in the limit.
-
It's an infinitesimal element.
-
This shadow is going
to be a rectangle.
-
Say that again, Magdalena.
-
This is not just
delta X and delta Y.
-
This is DX and DY
because I shrink them
-
until it become
infinitesimally small.
-
So you can imagine,
which one is bigger?
-
DS is bigger, or DA is bigger?
-
STUDENT: DS is bigger.
-
PROFESSOR: DS is bigger.
-
DS is bigger.
-
And can I see it's true?
-
Yes.
-
Because for God's sake, this
is greater than 1, right?
-
And if I multiply the
little orange area, by that,
-
I'm going to get this,
which is greater than 1.
-
They could be equal when
both would be plainer, right?
-
If you have a plane or surface
on top of a plane or surface,
-
then you have two
tiny rectangles
-
and you have like a prism
between them, goes down.
-
But in general, the
curve in your [INAUDIBLE]
-
here-- let me make
him more curvilinear.
-
He looks so-- so square.
-
But he's between two lines,
but he's a curvilinear.
-
Dinah says that he belongs to a
curved surface, not a flat one.
-
All right.
-
When he could be flat,
these guys go away.
-
Zero and zero.
-
And that would be it.
-
If somebody else, they--
well, this is hard to imagine,
-
but what if it could
be a tiny-- this
-
would not be curvilinear, right?
-
But it would be something like
a rectangular patch of a plane.
-
You have a grid in that plane.
-
And then it's just-- DS
would be itself a rectangle.
-
When you project
that rectangle here,
-
it will still be a rectangle.
-
When we were little-- I mean,
little, we were in K-12,
-
we're smart in math better
than other people in class--
-
did you ever have to do
anything with the two areas?
-
I did.
-
This was the shadow.
-
The projection in this
was that [INAUDIBLE].
-
And do you know what
the relationship
-
would be if I have a plane.
-
I'm doing that for-- actually,
I'm doing that for Casey
-
because she has something
similar to that.
-
So imagine that
you have to project
-
a rectangle that's in plane to
a rectangle that is the shadow.
-
The rectangle is on the ground.
-
The flat ground.
-
What's the relationship
between the two ends?
-
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: No matter
what it is, but assume
-
it's like a rectangle up
here and the shadow is also
-
a rectangle down here.
-
Obviously, the rectangle
down here, the shadow
-
will be much smaller than
this because this is oblique.
-
It's an oblique.
-
And assume that I
have this plane making
-
an angle, a fixed angle with
this laying on the table.
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Excellent.
-
STUDENT: --cosine--
-
PROFESSOR: Which one
is cosine of what?
-
So the S would be the
the equal sign of theta,
-
or the A will be the
S cosine of theta?
-
STUDENT: [INAUDIBLE] DA.
-
PROFESSOR: DA is the S
cosine of theta, a very smart
-
[INAUDIBLE].
-
How does she know [INAUDIBLE]?
-
STUDENT: Because it's
got to be less than one.
-
PROFESSOR: It's less
than one, right?
-
Cosine theta is
between zero and one,
-
so you think which one is less.
-
All right, very good.
-
So when you have a
simple example like that,
-
you were back to
your K-12, and you
-
were happy-- I
just meant we were
-
avoiding three years of exams.
-
We only have [INAUDIBLE].
-
But now exams became
serious, and look.
-
This is curvilinear
elemental variant.
-
So let me write it how
people call the S's then.
-
Some people call it
curvilinear elemental variant.
-
Yeah?
-
Many engineers I
talk to do that.
-
-
Now, I think we should just
call it surface area element.
-
-
[? I'm ?] a physicist, so you
also say surface area element.
-
So I think we should just
learn each other's language.
-
We are doing the same things.
-
We just-- we have a language
barrier between-- it's
-
not writing interdisciplinary,
so if we could establish
-
a little bit more work in
common, because there are so
-
many applications to
engineering of this thing,
-
you have no idea yet.
-
OK, let's pick a problem like
the ones we wrote in the book,
-
and see how hard it gets.
-
It shouldn't get very hard.
-
I'll start with one, the
only one, that is naturally
-
coming to your mind
right now, which would
-
be the one where G would be 1.
-
Somebody has to tell
me what that would be.
-
-
So guys, what if G would be 1?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Very good.
-
It would be the
A of the surface.
-
I'm going to look for
some simple application.
-
Nothing is simple.
-
Why did we make this problem,
this book, so complicated?
-
OK, it' s good.
-
We can pick-- I can make
up a problem like this one.
-
-
But I can do a better job.
-
I can give you an
better example.
-
I'm looking at the
example 1 in section 13.5.
-
I'll give you
something like that
-
if I were to write an exam 1.
-
I put on it something
like Z equals
-
X squared plus 1 squared.
-
You know is my favorite
eggshell which is a [INAUDIBLE].
-
-
And somebody says,
I'm not interested
-
in the whole surface,
which is infinitely large.
-
I'm only interested in a
piece of a surface that
-
is above the disk D of
center O and radius 1.
-
So say, what, Magdalena?
-
Say that I want just
that part of the surface
-
that he's sitting above the
disk of center O and radius 1.
-
And I want to know how to
set up the surface integral.
-
Set up main surface
area integral.
-
-
And of course, when you
first see that you freak out
-
for a second, and then you say,
no, no, that's not a problem.
-
I know how to do that.
-
So example 1 out of
this section would
-
be a double integral over your
S. You have to call it names.
-
S. 1 instead of G and DS.
-
-
But then you say wait a minute.
-
I know that is true, but I
have to change it accordingly.
-
The same thing is here.
-
So I'm going to have it
over D. And D is the shadow,
-
DS is the plane of what?
-
Of 1 times.
-
I know I'm silly saying 1
times, but that's what it is.
-
Square root of-- S of X squared
plus S of Y squared plus 1.
-
DS, DY or DA as Rachel
said, somebody said.
-
Aaron said.
-
I don't know, you just
whispered, I should say.
-
-
All right.
-
So first of all, this
looks a little bit bad.
-
It makes me a
little bit nervous.
-
But in the end, with your
help, I'm going to do it.
-
And I'm going to do it by
using what kind of coordinates?
-
I'm--
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Former
coordinates of the Y and Z.
-
It would be a killer.
-
Double, double, square root
of 1 plus-- who's telling me
-
what's coming next?
-
STUDENT: 4X squared.
-
4X squared, excellent.
-
4R squared, you say.
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: OK.
-
Let me write it with
X and Y, and then
-
realize that this is our square.
-
How about that?
-
And then I have DX,
DY over the domain D,
-
and now I finally become
smart and say I just
-
fooled around here.
-
I want to do it in four
coordinates finally.
-
And that means I'll say
zero to 2 pi for theta.
-
So that theta will be the
last of the [INAUDIBLE].
-
R will be from zero to 1.
-
-
And So what?
-
This is an ugly, fairly
ugly, I just [INAUDIBLE].
-
I don't know what
I'm going to do yet.
-
I reduced our confusion, right?
-
But I'm not done.
-
STUDENT: R.
-
PROFESSOR: R. Never forget it.
-
So if I didn't have this
R, I would be horrible.
-
Why would it be horrible?
-
Imagine you couldn't have the R.
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: We have to look that
this thing in integral table
-
or some-- use the calculator,
which we are not allowed
-
to do in this kind of course.
-
So what do we do?
-
We say it's a new substitution.
-
I have an R. That's a blessing.
-
So U equals 4 squared plus 1.
-
DU equals 8R, DR. I
think R, DR is a block.
-
And I know that's what I'm
going to do is a U substitution.
-
And I'm almost there.
-
-
It's a pretty good
example, but the one
-
you have as a first example
in this section, 13.5,
-
it's a little bit
too computational.
-
It's not smart at all.
-
It has a similar function over a
rectangle, something like that.
-
But it's a little bit
too confrontational.
-
We are looking
for something that
-
is not going-- examples that are
going to be easy to do and not
-
involve too much heavy
competition by him, because you
-
do everything by him.
-
Not-- like you don't have
a calculator, et cetera.
-
And the exam is very
limited in time, DU over 8.
-
So you say OK, I'm
know what that is.
-
That's going to be the A of S.
And that is going to be 2 pi.
-
Why can't I be so confident
and pull 2 pi out?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Because there
is no dependence on theta.
-
All right?
-
So I have that one.
-
And then you go all right,
integral, square of you
-
times the U over 8-- 1 over 8DU.
-
And I have to be careful
because when R is zero--
-
if I put zero and 1 here
like some of my students,
-
I'm dead meat, because I'm going
to lose a lot of credit, right?
-
So I have to pay attention.
-
R is 0, and U equals?
-
STUDENT: 1.
-
PROFESSOR: 1.
-
R equals 1.
-
U equals 5.
-
And I worked this out
and I should be done.
-
And that's-- you should
expect something like that.
-
Nice, not computational,
you kind of looking.
-
What is integral of square of U?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: So you have--
you do the three halves,
-
and you pull out the 2/3, right?
-
That's what you do.
-
And then you go between U equals
1 down, and U equals 5 up.
-
And it's like one of those
examples we worked before.
-
Remember, and more
important, you
-
had something like
that for surface area?
-
Oh, my god.
-
4 over 8.
-
How much is 4 over 8?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: One half.
-
-
Right?
-
So we will have 1 over
6, and write pi times 5
-
to the three halves minus 1.
-
So do I like it?
-
I would leave it like that.
-
I'm fine.
-
I'll forget about it.
-
I have people who care.
-
I don't care how some people
write it-- 5 with 5 minus 1
-
because they think
it looks better.
-
It doesn't.
-
That's the scientific
equation, and I'm fine with it.
-
Right?
-
OK.
-
So expect something like--
maybe I'm talking too much,
-
but maybe it's a good thing
to tell you what to expect
-
because we have to [INAUDIBLE].
-
At the same time, we're
teaching new things
-
as staff instructors doing
review of what's important.
-
I'm thinking if I'm doing things
right and at the same pace,
-
I should be finished
with chapter 13
-
at the end of next week.
-
Because after 13.5,
we have 13.6 which
-
is a generalization
of Green's Theorem.
-
13.6 as you recall is
called Stokes' Theorem.
-
13.7 is also a generalization
of Green's Theorem.
-
And they are all related.
-
It's like the trinity
on [INAUDIBLE].
-
That's the Divergence Theorem.
-
That is the last section,
13.7, Divergence Theorem.
-
So if I am going
at the right pace,
-
by-- what is next
wee on Thursday?
-
The-- 23rd?
-
I should be more or less
done with the chapter.
-
And I'm thinking I have
all the time in the world
-
to review with you
from that moment on.
-
In which sense are
we going to review?
-
We are going to review
by solving past finals.
-
Right?
-
That's what we are-- that's
what I'm planning to do.
-
I'm going to erase this
and move on to something
-
more spectacular.
-
Many-- OK.
-
This second part that
I want to teach you
-
now about, many instructors
in regular courses
-
just skip it because they do
not want to teach you-- not you,
-
you are honor students.
-
But they don't want
to teach the students
-
about some more general
ways to look at a surface.
-
Remember, guys, a surface
that is written like that
-
is called a graph.
-
But not all the
surfaces were graphs.
-
And actually for a surface
S, what the most general way
-
to represent the presentation
would be a parameterization.
-
-
And I'll do a little bit
of a review for those.
-
R-- little R or big R--
big R, because that's
-
the position vector the
way I serve it to you
-
on a plate, whether,
for curves in space.
-
I say that's R of P. And when
we moved on curves to surfaces,
-
I said you move your path
two directions of motion.
-
You have two-- what are
those called in mechanics?
-
Degrees of freedom.
-
So you have two
degrees of freedom
-
like latitude and longitude.
-
Then R belongs--
the position vector
-
is a function of two variables,
and it belongs to R3,
-
because it's a vector in R3.
-
And want to have-- imagine
that my hand is a surface.
-
Well, OK.
-
This is the position vector, I'm
just kind of sweeping my hand,
-
going this way, one
degree of freedom.
-
Or going that way, the
other degree of freedom.
-
This is what
parameterization is.
-
So for a sphere, if you want to
parameterize the whole sphere--
-
I'll be done in a second.
-
I need you to see
if you remember
-
how to parameterize a sphere.
-
I'm testing you.
-
I'm mean today.
-
So examples.
-
Example 1 is
parameterize a sphere.
-
-
Was it hard?
-
That was a long
time ago, my god.
-
X, Y, and Z are what?
-
Latitude from Santa Clause.
-
Always latitude from
the North Pole is 5.
-
Longitude is from zero to 5.
-
The meridian is zero to 5.
-
That was theta, the
parameter of theta.
-
R was the distance
from this to a point.
-
But R was allowed to be
from-- take many values.
-
Now if I'm moving on
a sphere of radius
-
A-- let me make
that radius a just
-
to make your life miserable.
-
Assume that A would
be a sample, A.
-
How am I going to write
that parameterization?
-
STUDENT: X equals
A plus [INAUDIBLE]?
-
PROFESSOR: A something,
A something, A something.
-
STUDENT: A [INAUDIBLE]
-
PROFESSOR: He is right.
-
I have to move on.
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Go slow.
-
So I have-- the last
one-- you were right,
-
Buddy, you have the
memory of a medical doctor
-
and some day you will
be a medical doctor.
-
Not everybody has a good memory.
-
So the way you can do that
is, wait a minute, this is pi,
-
right?
-
This [INAUDIBLE].
-
If you want the Z, you
start with that first.
-
And since Z is adjacent, you
go R, cosine, sine, phi equals
-
sine phi.
-
Now we started with X
because he's worked on this
-
and remembers everything.
-
He has it memorized.
-
Sine phi for both.
-
And times what in both cases?
-
He's just the guy
who's not here.
-
So sine phi.
-
It helps to memorize N
cosine theta, and sine theta.
-
Is that really easy to memorize?
-
So where phi was the
latitude from the North
-
Pole between zero
and phi, it theta
-
was the longitude-- excuse
me, guys-- longitude from zero
-
to 2 pi, all around one more.
-
So you say wait a
minute, Magdalena,
-
these are Euler's angle.
-
What do they call in mechanics?
-
I think they call
them Euler angles.
-
But anyway, for
phi theta, we call
-
them latitude and longitude.
-
I'll replace them, because look,
I want R to be in terms of U,V.
-
So in mathematics, it's
not about location.
-
We can call them
whatever we want.
-
Mathematics is about the freedom
to call people names-- no--
-
to call things names
and people names--
-
STUDENT: Could U not equal zero?
-
PROFESSOR: Who?
-
STUDENT: U.
-
PROFESSOR: Yes.
-
So U can--
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: --yeah, but
why didn't I write zero?
-
Well--
-
STUDENT: [INAUDIBLE]
makes sense.
-
PROFESSOR: --because,
yeah, you can take both.
-
If I want to study
differentiability,
-
I usually have to take it less
than and less than and less
-
than and less than because we
studied differentiability on
-
[INAUDIBLE].
-
But right now, I can take them
from the North Pole itself
-
to the South Pole itself-- so.
-
I'm not deleting any meridian.
-
If I were-- suppose
I were to delete it.
-
By the way, what does this mean?
-
I'm just kidding.
-
I'll put it back.
-
But Alex had a smart
question over there,
-
and he made me thinking.
-
It's a dangerous thing
when people make you think.
-
So it goes from zero to 2 pi.
-
Why would that be?
-
Imagine you have all the
meridians in the world
-
except for one.
-
From the sphere, you cut it and
remove the Greenwich meridian,
-
the one that passes
through Greenwich Village.
-
The one-- not the one in New
York, the one next to London,
-
right?
-
So put it back.
-
Put that meridian back.
-
It's like you take an
orange, and you make a slice.
-
I am-- OK.
-
Stop with the fruit
because I'm hungry.
-
Now, example two.
-
Now, imagine another surface
area you're used to, the what?
-
The paraboloid is one of our
favorite guys this semester.
-
X squared plus Y squared.
-
What is the
parameterization of that?
-
-
Well, if I write it
like that, it's a graph.
-
But if I don't want to
write it as a graph,
-
I have to write
it as a parameter.
-
What am I going to do?
-
I really know X to be U, right?
-
That's the simplest
choice possible.
-
Y could be V. And then Z will
be U squared plus V squared.
-
And there I am.
-
[SNEEZE]
-
So I'm going to write-- bless
your heart, [INAUDIBLE].
-
-
V plus J plus U squared
plus V squared, K. So this
-
is the parameterization
of a paraboloid.
-
That one of them--
there are infinitely
-
many-- the one that comes
to mind because it's
-
the easiest one to think about.
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Good.
-
For a minute, guys,
you didn't need me.
-
You didn't need me at all
to come up with those.
-
But maybe you would need me
to remember, or maybe not--
-
to remind you of the helicoid.
-
Helicoid.
-
Did you go to the,
as I told you to go
-
to the [INAUDIBLE] spectrum--
what was that called?
-
The--
-
STUDENT: Science spectrum.
-
PROFESSOR: Science spectrum.
-
And dip into soap solution
the thingy was-- a metal
-
rod with a-- with a what?
-
With; a helix made
of metal so the soap
-
film would take which shape?
-
The shape of this spiral that's
going to go inside here, right?
-
That's called a helicoid.
-
OK.
-
All right.
-
You're not mad at me.
-
STUDENT: No.
-
PROFESSOR: OK, good.
-
So in this case, R
of UV will be what?
-
It was a long time ago, once
upon a time I gave it to you.
-
It's extremely hard to
memorize if you don't work
-
with it on a regular basis.
-
If it were a helix,
what would it be?
-
If it were a helix, it
would be R of T right?
-
It would be like equal
sign T, A sine T, BT.
-
Say it again, Magdalena,
that was a long time ago,
-
chapter 10.
-
Chapter 10.
-
Equal sign, T, A sine
T, MBT, standard helix.
-
This is not going to be that.
-
It's going to be-- U cosine B.
U sine B. Look at the picture.
-
And imagine that these guys
are extended to infinity.
-
It's not just the
stairs themselves,
-
or whatever they are.
-
There are infinite lines,
straight lines, and busy.
-
This is done.
-
NB is a positive constant.
-
-
But your parameters are
U and V. Any other guy
-
that comes to mind, I'm out
of imagination right now.
-
You can do a torus on the
fold that looks like a donut.
-
You will have two parameters.
-
Imagine a donut.
-
How do you-- I'm not
going to write that.
-
Eventually I could give you
that as an extra credit thing.
-
What are the two degrees of
freedom of moving on the donut,
-
assuming that you would
like to move in circles?
-
-
STUDENT: [INAUDIBLE]
-
-
PROFESSOR: Let me draw a
donut, because I'm hungry,
-
and I really-- I cannot help it.
-
I just have to-- this is
called a torus in mathematics.
-
And you'll have-- one degree
of freedom will be like this,
-
the other degree of
freedom will be like that.
-
This is U and B.
Instead of U and B,
-
mathematicians,
apologists, geometers,
-
they call those angles phi
and theta because they really
-
are between zero and 2 pi.
-
It has a rotation like
that along the donut.
-
You can cut, slice
the donut, or if they
-
don't put cheese filling in it.
-
That was a bad idea not
having anything to eat.
-
And the other angle will be your
2 pi along this little circle.
-
So you still have two degrees
of freedom on a donut.
-
It's a surface.
-
You can write the
parameterization.
-
Yes?
-
STUDENT: Why is a
pie this way around.
-
Why is it like [INAUDIBLE].
-
PROFESSOR: It
doesn't have to be.
-
STUDENT: Or is it just
kind of like [INAUDIBLE]?
-
PROFESSOR: That's
what they call it.
-
Yeah.
-
So they are between 2 and 2 pi.
-
While I erase-- or should
I-- enough expectation
-
in terms of parameterization,
I have to night
-
teach you something about that.
-
If somebody would say I'm
giving you a patch of a surface,
-
but that patch of a
surface is in a frame--
-
it's a nice parameterization.
-
This is the P on the surface.
-
-
And you say, well,
the parameterization
-
is going to be R
of U and V equals
-
X of UVI plus Y of
UVJ plus Z of UVK.
-
-
And suppose that somebody says
this is you favorite test.
-
-
Find V. Well, that
would be absurd.
-
My god, how do we do that?
-
Find the flux
corresponding to-- do
-
we say restart--
just a second-- just
-
to restart with applications.
-
[INAUDIBLE]
-
-
We don't say what kind of
vector field that it is,
-
but we will say plus
corresponding to the vector
-
field.
-
F [INAUDIBLE].
-
-
And this vector
field, I'll tell you
-
in a second what's expected
from this to be a vector field.
-
Through, on the surface, we
find on the surface-- yes.
-
Mathematicians say
define normal surface S.
-
But a physicist will
say flux through,
-
the flux corresponding
to F through the surface.
-
-
Yes.
-
So you'll say why would that
be, and what is the flux?
-
By definition, how
should we denote it?
-
Let's make a beautiful script
F. That's crazy, right?
-
And then it goes doubling
over the surface F test.
-
Is anybody mechanical
engineering here?
-
Do you know the flux formula?
-
It's going to be [INAUDIBLE]
over S of F, this magic thing.
-
Not DN, DS.
-
Do you know what N means?
-
What it is N for
mechanical engineering,
-
[INAUDIBLE] engineers?
-
N to would be the unit normal
vector field to the surface S.
-
How would you want
to imagine that?
-
You would have a surface, and
you have this velocity vectors
-
here at the bottom that goes
to S. And this field goes up.
-
You'll have a force and
acceleration, velocity,
-
you have everything
going this way.
-
And you want to find
out what happens.
-
You introduce this notion
of flux through the surface.
-
Another way to have a
flux through the surface
-
maybe through the same
surface but associated
-
through another
kind of concept--
-
if there could be
something else.
-
In electromagnetism, F would be
something else, some other type
-
of vector field.
-
Yes, sir.
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: So find out, by
the way until next time,
-
if you were an electrical
engineering major, what
-
would flux be for you guys?
-
Two surfaces, one would be the
meaning of the vector field
-
F for you, and
why would you care
-
about the electromagnetic
flux or something like that.
-
I don't want to talk
too much about it.
-
It's for you to do the
search and find out.
-
So suppose that
somebody gives you
-
this notion that says you
have a parameteric surface.
-
Give an application
of that and find out
-
how you're going
be deal with it.
-
I'll give you an example
that shouldn't be too hard.
-
-
I'll make up my own example.
-
And looks like example 6, but
it's going to be different.
-
-
Example.
-
-
Find the flux F if F will
be a simple function.
-
Let's say something equals X, I
plus Y,J Z, K at every point X,
-
Y-- at every point
of the space XYZ.
-
That means you could have this
vector field defined everywhere
-
in space in [INAUDIBLE].
-
But you only care about
this acting on the surface.
-
So it's acting on the surface.
-
-
And then what will the flux be?
-
On the surface, which surface?
-
My favorite one, Z equals
X squared plus Y squared.
-
-
First of all, you say
wait, wait, Magdalena,
-
do you want to do it like that?
-
Do you want to say F
over XY to be a graph?
-
Or do you want to consider it
as a parameterized surface?
-
And that means it's the same
thing, equivalent to or if
-
and only if, who tells me again
what R was for such a surface?
-
STUDENT: XI.
-
PROFESSOR: X is
U. Y is V, so U--
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: --I, that
would be J, then good.
-
U squared plus U squared UK.
-
Well, when you say that,
we have-- first of all,
-
we have no idea what
the heck we need to do,
-
because do we want to do it
in this form like a graph?
-
Or do we want to do
it parameterized?
-
We have to set up
formulas for the flats.
-
It's not so easy.
-
So assume that we are brave
enough and we start everything.
-
I want to understand what
flux really is as an integral.
-
And let me set it up for the
first case, the case of Z
-
equals F of X and Y.
And I'm happy with it
-
because that's
the simplest case.
-
Who's going to teach
me what I have to do?
-
You are confusing.
-
I have double integral over S
minus theory of F in general.
-
This is a general
vector value field.
-
-
It could be anything.
-
Could be anything.
-
But then I have to [INAUDIBLE],
because N corresponds
-
to the normal to the surface.
-
So I-- it's not so easy, right?
-
I have to be a little bit smart.
-
If I'm not smart--
-
STUDENT: [INAUDIBLE]
-
-
PROFESSOR: That-- you
are getting close.
-
So guys, the normal
two-way surface-- somebody
-
gave you a surface, OK?
-
And normal to a surface
is normal to the plane--
-
the tangent plane
of the surface.
-
So how did we get that?
-
There were many ways to do it.
-
Either you write
the tangent plane
-
and you know it by heart--
that was Z minus Z zero
-
equals-- what the heck was
that-- S of X times X minus X
-
equals-- plus X of Y
times Y minus Y zero.
-
And from here you collect--
what do you collect?
-
You move everybody--
it's a moving sale.
-
You move everybody to the
left hand side and that's it.
-
[INAUDIBLE] moving sale.
-
OK?
-
And everybody will be
giving you some components.
-
You're going to have minus S
of X-- S minus X zero-- minus S
-
of Y, Y minus Y zero, plus
1-- this is really funny.
-
1 times Z minus Z, Z.
-
Your normal will
be given by what?
-
The normal-- how do
you collect the normal?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Pi is A, B, C. A,
B, and C will be the normal.
-
Except it's not unitary.
-
And the mechanical engineer
tells you, yeah, you're
-
stupid-- well, they
never say that.
-
They will stay look, you have
to be a little more careful.
-
Not say they are equal.
-
What do they mean?
-
They say for us,
in fluid mechanics,
-
solid mechanics, when we write
N, we mean you mean vector.
-
You are almost there.
-
What's missing?
-
STUDENT: Magnitude. [INAUDIBLE].
-
PROFESSOR: Very
good, the magnitude.
-
So they will say, go ahead and
you [INAUDIBLE] the magnitude.
-
And you are lucky now that
you know what N will be.
-
On the other hand--
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: This is excellent.
-
The one on the bottom-- Alex
is thinking like in chess, two
-
or three moves ahead.
-
You should get two extra
credit points with that.
-
STUDENT: All right.
-
PROFESSOR: You already got it.
-
DS is 1 plus S of X squared
plus F of X squared.
-
The 1 on the bottom and the
1 on the top will simplify.
-
So say it again, Magdalena.
-
Let me write it down here.
-
1 S of X, minus S of Y
1 over all this animal,
-
S of X squared plus S
of Y squared plus 1.
-
This is the thinking
like the early element
-
times the early element
will be the same thing.
-
I'll write it twice even if you
laugh at me because we are just
-
learning together,
and now you finally
-
see-- everybody can
see that desimplifies.
-
-
So it's going to be easy to
solve this integral in the end,
-
right?
-
So let's do the
problem, finally.
-
I'm going to erase it.
-
Let's do this problem
just for us, at any point.
-
I didn't say where.
-
Over the same thing.
-
The DS was over V01.
-
So the picture is
the same as before.
-
The S will be the
nutshell, the eggshell--
-
I don't know what it was--
over the domain D plane.
-
The domain D plane
was D of zero 1.
-
And I say that I need
to use another color.
-
This it's going to be my
shell, my surface S. Z
-
equals X squared
plus [INAUDIBLE].
-
How do you compute the flux?
-
Well, this is that.
-
So if we have to be a
little bit careful and smart
-
and say double integral over
S, and now without rushing,
-
we have to do a good job.
-
First of all, how do
you do the dot product?
-
The dot product--
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Right.
-
So first component
times first component,
-
a second component,
second component times
-
second component plus
that component times
-
third component.
-
So if 1 is X, F2 is 1.
-
Good.
-
Z, though, he's not free.
-
He's married.
-
Why is he married?
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Because he
depends on X and Y.
-
So Z was even here,
because I'm on the surface.
-
I don't care what F does
away from the surface,
-
but when he sticks
to the surface, when
-
he's origin is on
the surface, then he
-
has to listen to the surface.
-
And that Z is not independent.
-
The Z is X squared
by Y squared here.
-
In a bracket, we are
over the surface.
-
That product minus S
of X, minus S of Y. I
-
know you're going to
laugh at me because I
-
haven't written where they are.
-
But that's what I
need your help for.
-
DA.
-
-
Who are they?
-
Who is this guy?
-
STUDENT: The [INAUDIBLE].
-
PROFESSOR: What?
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Negative 2X.
-
Is it?
-
STUDENT: No.
-
PROFESSOR: How about this guy?
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Negative 2Y.
-
How about this guy?
-
I'm just kidding.
-
OK.
-
So finally we should be able
to compute this integral.
-
That looks awful.
-
Over D.
-
So instead of S, we
have the D, which
-
is the disk of
radius one in plane.
-
And we say, OK, I have,
oh my god, it's OK.
-
This times that is how much?
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Minus 2X squared.
-
Right?
-
There.
-
Take the green.
-
This times that is how much?
-
Minus the Y squared.
-
And this times that is finally
just X squared plus Y squared.
-
Very nice think.
-
I think that at
first, but now I see
-
that life is beautiful
again-- DX, DY--
-
that I can go ahead and do it.
-
I can get a hold of this.
-
And inside that, what do I--
what am I left with in the end?
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Minus 2
times this animal,
-
called X squared plus Y squared,
which is going to be R squared.
-
So the flux-- the flux for
this problem in the end
-
is going to be very
nice and sassy.
-
Look at that.
-
F would be--
-
STUDENT: There
would not be any--
-
STUDENT: [INAUDIBLE]
-
-
PROFESSOR: What?
-
STUDENT: You've got
minus 2 and the plus 1.
-
PROFESSOR: Oh, thank God.
-
Thank God you exist.
-
So I thought about
it before, but then I
-
said-- I don't know why.
-
I messed up.
-
So we have minus R squared.
-
Very good.
-
It's easy.
-
Times an R from the
Jacobian, DR is theta.
-
And theta is between 0 and 2 pi.
-
And R between 0 and 1.
-
And now I will need
a plumber to tell me
-
what I do the limits
of the integrals,
-
because I think I'm getting
a negative answer, so.
-
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: I'll do
it, and then you
-
tell me why I got what I got.
-
I have a minus
pulled out by nature.
-
And then I have integral--
-
STUDENT: R [INAUDIBLE].
-
PROFESSOR: R to the
fourth of a fourth.
-
Very good.
-
But you have your [INAUDIBLE] so
when I do between zero and 1--
-
STUDENT: It's [INAUDIBLE].
-
PROFESSOR: 1 over
4-- you are too
-
fast-- as 2 pi-- that's a
good thing-- minus pi over 2,
-
you said, Gus.
-
And I could see it
coming straight at me
-
and hit me between the eyes.
-
What is the problem.
-
Is there a problem?
-
Without an area as a flux, would
that say, what is the negative?
-
Yes.
-
How can I make it positive?
-
This is my question.
-
STUDENT: Change the direction.
-
PROFESSOR: Change
the direction of who?
-
STUDENT: The flux.
-
PROFESSOR: The flux.
-
I could change the direction.
-
So what is it that
doesn't match?
-
[INAUDIBLE]
-
If I want to keep-- the
flux will be the same.
-
When I can change the
orientation of the service.
-
And instead I get a minus then.
-
My N was it sticking
in-- oh, my god.
-
So is it sticking
in or sticking out?
-
Look at it.
-
Think about it.
-
I have minus the positive guy
minus another positive guy,
-
and 1 sticking out.
-
But it goes with
the holes inside.
-
This is the paraboloid
[INAUDIBLE].
-
If I have something I minus I
minus J, does it go out or in?
-
STUDENT: In.
-
PROFESSOR: It goes in.
-
It goes in, and it'll be up.
-
So it's going to be like
all these normals are
-
going to be like a vector
field like that, like amoebas.
-
But they are pointing
towards inside.
-
Do I like that?
-
Yes, because I'm a
crazy mathematician.
-
Does the engineer like that?
-
No.
-
Why?
-
The flux is pointing in or out?
-
The flux.
-
The flux.
-
The flux, the flux
is pointing out.
-
Are you guys with me?
-
X plus Y-- X plus I plus J.
It's like this pointing out.
-
So the flux get
out of the surface.
-
It's like to pour water
inside, and the water's
-
just a net-- not a net, but
like something that holds it in.
-
And like a--
-
STUDENT: Like a [INAUDIBLE]?
-
PROFESSOR: --pasta strainer.
-
And the water goes up
[SPRAYING NOISE], well,
-
like a jet.
-
Like that.
-
So that is your flux
going through the surface.
-
Are you happy that I took
the normal pointing inside?
-
No.
-
That was crazy.
-
So here comes you, the
mechanical engineer majoring
-
in solid or [INAUDIBLE]
and say Magdalena,
-
you should have taken
the outer normal,
-
because look at the
flux pointing out.
-
Take the outer of normal,
and things are going
-
to looks right and nice again.
-
So if I were to
change the normal,
-
I would put the
plus, plus, minus.
-
I'll take the outer normal.
-
And in the end I
get plus 5 over 2.
-
So no remark.
-
If I change N to minus N, this
would become the outer normal.
-
Then the flux would
become pi over 2. solar
-
flux depends on the what?
-
The match between
the flux, the angles,
-
sort of between the flux if
function, vector [INAUDIBLE]
-
function, and the normal
that I take to the surface.
-
Right?
-
I can change the normal and
I get the opposite answer.
-
In absolute values,
the same flux.
-
So flux should be equal
[INAUDIBLE] the absolute value.
-
Unlike the area that should
be always a positive number.
-
Volume, that should always
be a positive number.
-
So if I get a limited area,
that means I messed up.
-
If I get a negative
on all of them,
-
it means messed up in my
computation somewhere.
-
But that doesn't mean
I messed up here.
-
I just chose the other normal.
-
It's possible.
-
So the flux can be taken as
is and put in absolute value.
-
All right.
-
OK.
-
We have to think of it like
the surface, and stuff that
-
goes through surface
in electric circuits.
-
Can you do some research
for you about flux
-
and electrical engineering?
-
And next time somebody
tells me a story about it.
-
Who is-- again-- who is
electrical engineering major
-
here?
-
Oh, so five people.
-
You're going to get four
extra credit points.
-
You guys are jealous.
-
I'm going to give you four extra
credit points if in 10 minutes
-
you can tell us a little bit
about where flux can be seen.
-
Well, you don't have
to come to the board.
-
You can just talk to us
from outside if you want,
-
or down inside the classroom.
-
Tell us where the
notion of flux appears
-
in the electric
circuits and why it
-
would be important for
Calculus 3 as well.
-
OK.
-
Now a big question
before I let you go.
-
Can I have a flux
that corresponds
-
to a parameterization?
-
That is my big worry, that
I have to do that as well.
-
Eventually, could I
have solved this problem
-
if the surface that
is parameterized
-
was my friend--
who was my friend?
-
I don't remember.
-
UI plus VJ plus U
squared plus-- you
-
gave it to me-- OK, that
was the previous example,
-
and that's the last
example on the board.
-
So you have double integral
of force field times NDS.
-
Now, what if I say I don't
want to do it like this-- Z
-
equals F of XY.
-
So I don't want to
do it like that.
-
I want to do it in
a different way.
-
That means you pulling out of
your brain some old memories.
-
F was F, right?
-
You need to leave F alone,
poor fellow, because he
-
has no better way to do it.
-
This is becoming
complicated, the [INAUDIBLE]
-
mechanical engineering.
-
-
And what's given to you
before, but you don't remember?
-
R was given to you
as position vector.
-
R sub U and R sub V,
you may not remember--
-
that was a long time ago-- we
proved that R sub U and R sub
-
V were on the surface.
-
They are both tensions
of the surface.
-
It was a long time ago.
-
So the normal is
[INAUDIBLE], and that's
-
exactly what I wanted to
say the normal will be.
-
Not quite pressed product,
but just like before,
-
pressed product
divided by the norm,
-
because then the unit normal
vector has to be length 1.
-
So I have to divide
by the number.
-
[SNEEZE]
-
The DS--
-
STUDENT: Thank you.
-
PROFESSOR: --is going to--
OK, now it's up to you guys.
-
You're smart.
-
You know what I want to say.
-
So I'll pretend that you
know what DS is in terms
-
of the parameterization.
-
What's coming?
-
We said that.
-
It was a long time ago.
-
You can guess it by
just being smart--
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: --or you can--
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Yes, exactly.
-
And you got another
one extra credit point.
-
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: So since before,
they were simplified,
-
for god's sake.
-
Now we have the new kind
of writing area element DS.
-
They also have to simplify.
-
It wasn't hard to see.
-
So you could have
done it like that.
-
You could have done
it like that, how?
-
Somebody need to help me,
because I have no idea what
-
I'm going to do here.
-
Do we get the same thing or not?
-
This is the question.
-
And I'm going to
finish with that,
-
but I don't want to
go home-- I'm not
-
going to let you go home
until you finish this.
-
F was a simple,
beautiful vector field.
-
Given-- like that.
-
This is a force.
-
May the force be
with you like that.
-
But we changed it in U,V because
we are acting on the surface S,
-
what is the pressure
in V, right?
-
So you have UI plus VJ
plus-- you gave it to me--
-
U squared plus V squared.
-
Am I right, or am
I talking nonsense?
-
-
All right.
-
So now again I have
to be seeing them.
-
Am I getting the same thing?
-
If I'm not getting
the same thing,
-
I can just go home and
get drunk and be sad.
-
But I have to get
the same thing.
-
Otherwise, there is something
wrong with my setup.
-
So I have to have U, V.
U squared plus V squared.
-
Close.
-
Dot product.
-
This guy over on top-- say what?
-
Magdalena, this guy over on top
has to be-- has to be a what?
-
Well, I didn't say what it was.
-
I should do it now.
-
Right?
-
So how will we do that?
-
We were saying R of
UV will be UI plus VJ
-
plus U squared plus V squared.
-
OK.
-
So R sub U will be--
you teach me quickly,
-
and R sub [INAUDIBLE]
is-- voila.
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: 1--
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Plus zero--
thank you-- plus 2U, OK.
-
0 plus 1J plus 2VK Am I done?
-
I'm done.
-
No, I'm not done.
-
What do I have to do?
-
Cross them.
-
-
Cross multiply IJK.
-
This looks nice.
-
Look, it's not so ugly.
-
I thought it would
be uglier, right?
-
OK.
-
What it is?
-
What it this thing?
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Minus the
U, I. Minus-- plus.
-
Minus, plus 1.
-
2V minus because it's--
-
STUDENT: Minus.
-
PROFESSOR: --minus in front.
-
Right.
-
So I'm alternating.
-
And 1K.
-
So again, I get minus
X of S minus XY and 1,
-
and again, I'm pointing
in, and that's bad.
-
So my normal will point
inside the surface
-
like needles that are
perpendicular to the surface
-
pointing inside.
-
But that's OK.
-
In the end, I take
everything in absolute value.
-
Right?
-
-
So again, I do the same math.
-
So I get minus-- I don't
want to do it anymore.
-
Minus 2A squared, minus 2B
squared, plus your squared,
-
plus this squared,
then you save me
-
and you said minus 2
squared [INAUDIBLE] squared.
-
DUDV.
-
But DUDV means that UV is a
pair, a point in this, guys.
-
UV.
-
It's a pair in the
disk of radius one.
-
So I'm getting exactly,
what exactly the same thing
-
as before.
-
Because this is
minus R squared, so I
-
get integral, integral, minus
R squared times R. DR, D theta.
-
From zero to 1,
from zero to 2 pi,
-
and I get the same
answer, which was?
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: Minus what?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: Pi over--
-
STUDENT: [INAUDIBLE].
-
PROFESSOR: You see?
-
I already forgot.
-
STUDENT: 2.
-
PROFESSOR: So what
matters is that we
-
take the flux in
absolute value because it
-
depends on the
orientation of the normal.
-
If we take the
normal [INAUDIBLE].
-
Please, one thing I want you
to do when you go home now,
-
open the book which
maybe you rarely do,
-
but now it's
really-- the material
-
became complicated enough.
-
We are not just doing math,
calculus, we are doing physics,
-
we are doing mechanics, we are
dealing with surface integrals
-
and flux.
-
I want you to open the book
at page-- I don't know.
-
At surface integrals
starts at page 1,063.
-
Section 13.5.
-
And it keeps going like that,
pretty pictures of surfaces
-
and fluxes and so on.
-
Vector fields.
-
And it keeps going like that.
-
But it doesn't cover anything
new except what I said today.
-
It's just that it shows
you examples that are not
-
as beautiful as the ones I
gave, but they are essentially
-
the same, only a little
bit nastier to complete.
-
So up to 1,072.
-
So that is what you're
going to do this weekend,
-
plus the homework.
-
Keep on the homework.
-
Now, if you get stuck
Saturday, Sunday,
-
whenever you try your homework
you get stuck, what do you do?
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: You email me.
-
So you say what in the world
is going on with this problem
-
because I tried it seven
times and-- 88 times.
-
And then you got
the brownie points.
-
STUDENT: [INAUDIBLE]
-
PROFESSOR: [INAUDIBLE] problem.
-
STUDENT: [INAUDIBLE] by 32.
-
PROFESSOR: There
was a problem, guys.
-
There are not so many problems.
-
But the only part, serious
part that we would catch,
-
he found it first, and
he tried it 88 times.
-
-
I'll never forget you, though,
because you are unique,
-
and that-- I appreciated
that very much.
-
So doing this weekend, do
not hesitate to pester.
-
I will answer all the web
work problems you have.
-
I want you to do well.
-
Next week is the last
week on new theory,
-
and then we start
working for the final,
-
so by the time of the final,
you'll be [INAUDIBLE].
-
STUDENT: [INAUDIBLE]?
-
PROFESSOR: Yes, sir.
-
Oh, I appreciated
that you did that.
-
STUDENT: [INAUDIBLE]
-
-
PROFESSOR: Again,
I forgot these.
-
-
With the extra points you
got, you shouldn't care.
-