## TTU Math2450 Calculus3 Secs 13.4 -13.5

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PROFESSOR: I would like to
review just briefly what
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we discussed last time.
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We gave very important results,
and that was Green's Theorem.
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And I would like to
know if you remember
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settling for this problem.
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So we'll assume we have
a domain without a hole,
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D. D is a domain
without a hole inside,
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without punctures or holes.
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There is a scientific name in
mathematics for such a domain.
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This is going to be
simply connected.
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And this is a difficult
topological theorem,
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but this is what we expect, OK?
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And what does it mean?
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What does it mean?
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It means that in the C being
a Jordan curve was what?
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How?
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This was continuous,
no self intersections.
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In such a case, we set up
M and N to be C1 functions.
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And then we proceed through
the path integral of C.
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Do you like this kind of C,
or you prefer a straight C?
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The path integral of C of M
of xy dx class, N of xy, dy,
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everything is in plane.
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I'm sorry that I
cannot repeat that,
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but we discussed that
time, is in the plane of 2.
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And then what-- do
you remember in terms
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of how this path integral,
[INAUDIBLE] inside,
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is connected to a double
integral over the whole domain.
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In particular, do you remember--
this is easy to memorize--
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but do you remember
what's inside?
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Because for the final, you are
expected to know his result.
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STUDENT: [INAUDIBLE]
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PROFESSOR: N sub X.
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STUDENT: Minus M sub Y.
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PROFESSOR: Minus M
sub Y. [INAUDIBLE]
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must M-- M and N-- M sub Y.
Here is the Y. Of course this
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would be dA in plane,
and in the-- if you
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want to represent this
in the general format,
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the MdX minus the MdY.
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Feel free to do that.
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One was a correlary
or a consequence.
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This theorem was that if I
were to take this big M to be
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the minus Y as a function,
then this function N will
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be plus X, what will I get?
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I would get that minus
YdX plus NdY will be what?
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STUDENT: [INAUDIBLE]
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PROFESSOR: Two times, excellent.
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You are very awake.
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So I wanted to catch you.
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I couldn't catch you.
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I thought you would say
the A of the domain,
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but you said it right.
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You said Y is the
area of the domain.
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You probably
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did the math saying X sub X
is one, minus Y sub 1 is 1.
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1 plus 1 is two, so the
two part [INAUDIBLE].
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OK, so what did we do with it?
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We just stared at it?
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No.
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We didn't just stare at it.
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We did something nice
with it last time.
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We proved that, finally,
that the area, this radius R
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will be pi R
squared, and we also
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proved that the area
[INAUDIBLE] is what?
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I'm testing you to
see if you remember.
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STUDENT: AB pi.
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PROFESSOR: AB pi.
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Very good.
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Or pi AB.
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It's more, I like it the
way you said it, AB pi,
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because pi is a transcendental
number, and you go around
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and it's like partly
variable to put at the end.
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And the real numbers
that could be anything,
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so [INAUDIBLE] they are the
semi axes of the ellipse.
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So we gain new knowledge and
we are ready to move forward.
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And we're going to move
forward to something
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called section 13.5, which
is the surface integral.
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We will come back
to Green's Theorem
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because there are
generalizations
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of the Green's Theorem to
more complicate the case.
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But in order to
see those, we have
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to learn a little bit more.
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In mathematics, you need to
know many things, many pieces
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of the puzzle, and then
you put them together
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to get the whole picture.
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All right, so what
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This is just review.
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13.5, if should be looking like
a friend, old friend, to you.
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And I'll show you
in a minute why this
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is called the surface integral.
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I saw that US natives
don't pronounce integral,
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they pronounce in-negral.
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And everybody that I heard
in romance language-speaking
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countries like Spanish,
Italian, Portuguese,
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they put the T there
out, very visibly.
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So it doesn't matter.
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Even some accent difference
in different parts
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of the United States
pronounce it differently.
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So what is the surface
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function, not a vector value,
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but a real value function.
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Let's say you have G or XY being
a nice interglobal function
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over some surfaces.
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And you say, I'm
going to take it,
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double integral, over S of GDS,
where DS will be area level.
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I had a student one time who
looked at two different books
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and said, I have a problem
with this, [INAUDIBLE].
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In one book it shows a
big, fat snake over S.
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And in another book, a
double integral over it,
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and I don't know which one it
is because I don't understand.
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No matter how you denote it,
it's still a double integral.
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You know why?
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Because it's an
integral over a surface.
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The same thing, integral over
a surface or a domain plane,
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or anything two-dimensional
will be a double integral.
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So it doesn't matter
how you denote it.
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In the end, it's going
to be a double integral.
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Now, what in the world
do we mean by that?
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DS is an old friend of
yours, and I don't know
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if you remember him at all.
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He was infinitesimal element
on some curved or linear patch.
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Let's assume it's a graph.
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It doesn't have to be a graph,
but let's assume it's a graph.
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And that's your
favorite surface S.
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And then you draw
coordinate lines,
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and you are looking at a patch.
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And this patch looks small,
but it's not small enough.
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I want this to be
infinitesimally small.
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Imagine that these curvature
lines become closer and closer
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to one another.
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And then we look in the
directions of DX and DY,
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and then you say, wait a
minute, I'm not in plane.
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If I were in plane,
DA will be DX, DY.
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If you work with [INAUDIBLE],
I will be DX with DY.
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So we've matched
the orientation.
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If you would change
DY, [INAUDIBLE]
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put the minus in front.
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But this happens
because-- thank God this
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will be a rectangular 1 patch
in plane, in the plane of 2.
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But what if you
were on the surface?
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On the surface, you
don't have this animal.
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You will have-- which animal--
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I'm doing review with you.
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For sure, you will
see something that
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involves the S in the final.
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Have you started browsing
through those finals
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I sent you?
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Just out of curiosity.
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And do they look awful to you?
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They look awful to you.
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Come on.
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I'm going to work with
you on some of those.
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I don't want you to
have-- I don't want
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you to be afraid of this final.
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Because compared to
other exams that you'll
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have in other courses,
where a lot of memorization
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is emphasized, this
should not be a problem.
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So you could go over
the types of problems
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that are significant
in this course,
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you will not have any-- you
shouldn't have any problem.
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And I sent you three samples.
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Didn't I send you three
samples with solutions?
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Those are going to help
you once you read the exam
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and try the exam
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If I give you more of that, then
you should be doctors in those,
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and you would be able to
solve them yourselves.
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This is not DA.
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It's a DA times something.
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There is some factor in front
of that, and why is that?
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In case of Z equals
F of X and Y,
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you should know that by heart,
and I know that some of you
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know it.
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You just have to ring
the bell, and I'll
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start ringing the bell.
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Look at my first step.
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And now you know, right?
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STUDENT: [INAUDIBLE] 1--
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You said it right.
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1 plus--
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STUDENT: F of X.
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PROFESSOR: F of X--
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STUDENT: F squared.
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PROFESSOR: Squared plus--
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STUDENT: [INAUDIBLE]
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PROFESSOR: --SY squared.
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So this what you're doing.
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What are you going to do?
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You're going to
do wait a minute.
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This animal of mine,
that looks so scary,
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this is nothing but what?
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It's the same thing as, not
the picture, my picture.
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It's going to be double integral
over a plane or domain D.
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Well, I just said
goodbye to the picture,
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but I find you are really smart.
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I would have drawn the
[INAUDIBLE] of a picture here.
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This is S and this
is D. What is D?
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It's the projection,
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The projection of S
on the plane XY when
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I have to deal with a graph.
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So when I have to deal with a
graph, my life is really easy.
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And I said I'd get double
integral over D of G of God
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knows what in the end will
be a function of X and Y. OK?
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And here I'm going to have
square root of this animal.
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Let me change it,
F sub X squared
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like-- because in this
one it is like that.
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Plus 1.
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It doesn't matter
where I put the 1.
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DXDY.
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DXDY will be like the area of an
infinitesimally small rectangle
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based on displacement
DX and displacement DY
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and disintegration.
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So this is DA.
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Make the distinction
between the DA and the DX.
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Can I draw the two animals?
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Let me try again.
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So you have the
direction of X and Y.
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You have to be imaginative and
see that some coordinate lines
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are [INAUDIBLE] for fixing Y.
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When I fix Y, I sliced a
lot like that very nicely.
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That's the same piece of
cheese that I've been dreaming
• 13:12 - 13:13
because I didn't have lunch.
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I was too busy not
to have any lunch.
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So you slice it
up like that where
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Y equals constant to slice
it up like that for X
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equals constant.
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What you get are so-called
coordinate lines.
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So the coordinate
lines are [INAUDIBLE].
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Y equals my zeros, and
X equals the zeros.
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And when they get to be
many dense and refined,
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this-- between two curves
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like this two curves like that.
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Shrunk in the limit.
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It's an infinitesimal element.
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to be a rectangle.
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Say that again, Magdalena.
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This is not just
delta X and delta Y.
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This is DX and DY
because I shrink them
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until it become
infinitesimally small.
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So you can imagine,
which one is bigger?
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DS is bigger, or DA is bigger?
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STUDENT: DS is bigger.
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PROFESSOR: DS is bigger.
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DS is bigger.
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And can I see it's true?
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Yes.
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Because for God's sake, this
is greater than 1, right?
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And if I multiply the
little orange area, by that,
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I'm going to get this,
which is greater than 1.
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They could be equal when
both would be plainer, right?
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If you have a plane or surface
on top of a plane or surface,
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then you have two
tiny rectangles
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and you have like a prism
between them, goes down.
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But in general, the
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here-- let me make
him more curvilinear.
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He looks so-- so square.
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But he's between two lines,
but he's a curvilinear.
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Dinah says that he belongs to a
curved surface, not a flat one.
• 15:07 - 15:08
All right.
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When he could be flat,
these guys go away.
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Zero and zero.
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And that would be it.
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If somebody else, they--
well, this is hard to imagine,
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but what if it could
be a tiny-- this
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would not be curvilinear, right?
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But it would be something like
a rectangular patch of a plane.
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You have a grid in that plane.
• 15:40 - 15:46
And then it's just-- DS
would be itself a rectangle.
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When you project
that rectangle here,
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it will still be a rectangle.
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When we were little-- I mean,
little, we were in K-12,
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we're smart in math better
than other people in class--
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did you ever have to do
anything with the two areas?
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I did.
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The projection in this
was that [INAUDIBLE].
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And do you know what
the relationship
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would be if I have a plane.
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I'm doing that for-- actually,
I'm doing that for Casey
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because she has something
similar to that.
• 16:20 - 16:23
So imagine that
you have to project
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a rectangle that's in plane to
a rectangle that is the shadow.
• 16:28 - 16:30
The rectangle is on the ground.
• 16:30 - 16:34
The flat ground.
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What's the relationship
between the two ends?
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STUDENT: [INAUDIBLE]
• 16:43 - 16:45
PROFESSOR: No matter
what it is, but assume
• 16:45 - 16:48
it's like a rectangle up
here and the shadow is also
• 16:48 - 16:50
a rectangle down here.
• 16:50 - 16:52
Obviously, the rectangle
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will be much smaller than
this because this is oblique.
• 16:56 - 16:57
It's an oblique.
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And assume that I
have this plane making
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an angle, a fixed angle with
this laying on the table.
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STUDENT: [INAUDIBLE]
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PROFESSOR: Excellent.
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STUDENT: --cosine--
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PROFESSOR: Which one
is cosine of what?
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So the S would be the
the equal sign of theta,
• 17:12 - 17:19
or the A will be the
S cosine of theta?
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STUDENT: [INAUDIBLE] DA.
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PROFESSOR: DA is the S
cosine of theta, a very smart
• 17:23 - 17:23
[INAUDIBLE].
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How does she know [INAUDIBLE]?
• 17:25 - 17:26
STUDENT: Because it's
got to be less than one.
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PROFESSOR: It's less
than one, right?
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Cosine theta is
between zero and one,
• 17:29 - 17:32
so you think which one is less.
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All right, very good.
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So when you have a
simple example like that,
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you were back to
• 17:40 - 17:42
were happy-- I
just meant we were
• 17:42 - 17:45
avoiding three years of exams.
• 17:45 - 17:48
We only have [INAUDIBLE].
• 17:48 - 17:50
But now exams became
serious, and look.
• 17:50 - 17:53
This is curvilinear
elemental variant.
• 17:53 - 17:59
So let me write it how
people call the S's then.
• 17:59 - 18:01
Some people call it
curvilinear elemental variant.
• 18:01 - 18:02
Yeah?
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Many engineers I
talk to do that.
• 18:06 - 18:09
• 18:09 - 18:12
Now, I think we should just
call it surface area element.
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• 18:15 - 18:18
[? I'm ?] a physicist, so you
also say surface area element.
• 18:18 - 18:23
So I think we should just
learn each other's language.
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We are doing the same things.
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We just-- we have a language
barrier between-- it's
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not writing interdisciplinary,
so if we could establish
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a little bit more work in
common, because there are so
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many applications to
engineering of this thing,
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you have no idea yet.
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OK, let's pick a problem like
the ones we wrote in the book,
• 18:47 - 18:51
and see how hard it gets.
• 18:51 - 18:54
It shouldn't get very hard.
• 18:54 - 18:57
only one, that is naturally
• 18:57 - 19:00
right now, which would
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be the one where G would be 1.
• 19:04 - 19:06
Somebody has to tell
me what that would be.
• 19:06 - 19:10
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So guys, what if G would be 1?
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STUDENT: [INAUDIBLE]
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PROFESSOR: Very good.
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It would be the
A of the surface.
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I'm going to look for
some simple application.
• 19:22 - 19:23
Nothing is simple.
• 19:23 - 19:28
Why did we make this problem,
this book, so complicated?
• 19:28 - 19:28
OK, it' s good.
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We can pick-- I can make
up a problem like this one.
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• 19:39 - 19:41
But I can do a better job.
• 19:41 - 19:43
I can give you an
better example.
• 19:43 - 19:47
I'm looking at the
example 1 in section 13.5.
• 19:47 - 19:49
I'll give you
something like that
• 19:49 - 19:51
if I were to write an exam 1.
• 19:51 - 19:56
I put on it something
like Z equals
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X squared plus 1 squared.
• 19:58 - 20:02
You know is my favorite
eggshell which is a [INAUDIBLE].
• 20:02 - 20:16
• 20:16 - 20:20
And somebody says,
I'm not interested
• 20:20 - 20:25
in the whole surface,
which is infinitely large.
• 20:25 - 20:30
I'm only interested in a
piece of a surface that
• 20:30 - 20:40
is above the disk D of
• 20:40 - 20:41
So say, what, Magdalena?
• 20:41 - 20:48
Say that I want just
that part of the surface
• 20:48 - 20:54
that he's sitting above the
disk of center O and radius 1.
• 20:54 - 20:58
And I want to know how to
set up the surface integral.
• 20:58 - 21:02
Set up main surface
area integral.
• 21:02 - 21:07
• 21:07 - 21:10
And of course, when you
first see that you freak out
• 21:10 - 21:13
for a second, and then you say,
no, no, that's not a problem.
• 21:13 - 21:16
I know how to do that.
• 21:16 - 21:19
So example 1 out of
this section would
• 21:19 - 21:27
be a double integral over your
S. You have to call it names.
• 21:27 - 21:30
S. 1 instead of G and DS.
• 21:30 - 21:34
• 21:34 - 21:36
But then you say wait a minute.
• 21:36 - 21:40
I know that is true, but I
have to change it accordingly.
• 21:40 - 21:42
The same thing is here.
• 21:42 - 21:48
So I'm going to have it
over D. And D is the shadow,
• 21:48 - 21:53
DS is the plane of what?
• 21:53 - 21:55
Of 1 times.
• 21:55 - 22:00
I know I'm silly saying 1
times, but that's what it is.
• 22:00 - 22:07
Square root of-- S of X squared
plus S of Y squared plus 1.
• 22:07 - 22:14
DS, DY or DA as Rachel
said, somebody said.
• 22:14 - 22:15
Aaron said.
• 22:15 - 22:17
I don't know, you just
whispered, I should say.
• 22:17 - 22:20
• 22:20 - 22:22
All right.
• 22:22 - 22:28
So first of all, this
• 22:28 - 22:31
It makes me a
little bit nervous.
• 22:31 - 22:35
But in the end, with your
help, I'm going to do it.
• 22:35 - 22:38
And I'm going to do it by
using what kind of coordinates?
• 22:38 - 22:38
I'm--
• 22:38 - 22:40
STUDENT: [INAUDIBLE]
• 22:40 - 22:42
PROFESSOR: Former
coordinates of the Y and Z.
• 22:42 - 22:43
It would be a killer.
• 22:43 - 22:49
Double, double, square root
of 1 plus-- who's telling me
• 22:49 - 22:50
what's coming next?
• 22:50 - 22:51
STUDENT: 4X squared.
• 22:51 - 22:53
4X squared, excellent.
• 22:53 - 22:56
4R squared, you say.
• 22:56 - 22:57
STUDENT: [INAUDIBLE]
• 22:57 - 22:58
PROFESSOR: OK.
• 22:58 - 23:01
Let me write it with
X and Y, and then
• 23:01 - 23:03
realize that this is our square.
• 23:03 - 23:05
• 23:05 - 23:08
And then I have DX,
DY over the domain D,
• 23:08 - 23:12
and now I finally become
smart and say I just
• 23:12 - 23:14
fooled around here.
• 23:14 - 23:17
I want to do it in four
coordinates finally.
• 23:17 - 23:22
And that means I'll say
zero to 2 pi for theta.
• 23:22 - 23:26
So that theta will be the
last of the [INAUDIBLE].
• 23:26 - 23:29
R will be from zero to 1.
• 23:29 - 23:33
• 23:33 - 23:35
And So what?
• 23:35 - 23:41
This is an ugly, fairly
ugly, I just [INAUDIBLE].
• 23:41 - 23:43
I don't know what
I'm going to do yet.
• 23:43 - 23:45
I reduced our confusion, right?
• 23:45 - 23:46
But I'm not done.
• 23:46 - 23:47
STUDENT: R.
• 23:47 - 23:52
PROFESSOR: R. Never forget it.
• 23:52 - 23:57
So if I didn't have this
R, I would be horrible.
• 23:57 - 23:59
Why would it be horrible?
• 23:59 - 24:01
Imagine you couldn't have the R.
• 24:01 - 24:03
STUDENT: [INAUDIBLE]
• 24:03 - 24:06
PROFESSOR: We have to look that
this thing in integral table
• 24:06 - 24:10
or some-- use the calculator,
which we are not allowed
• 24:10 - 24:12
to do in this kind of course.
• 24:12 - 24:14
So what do we do?
• 24:14 - 24:17
We say it's a new substitution.
• 24:17 - 24:19
I have an R. That's a blessing.
• 24:19 - 24:23
So U equals 4 squared plus 1.
• 24:23 - 24:30
DU equals 8R, DR. I
think R, DR is a block.
• 24:30 - 24:35
And I know that's what I'm
going to do is a U substitution.
• 24:35 - 24:36
And I'm almost there.
• 24:36 - 24:39
• 24:39 - 24:41
It's a pretty good
example, but the one
• 24:41 - 24:48
you have as a first example
in this section, 13.5,
• 24:48 - 24:50
it's a little bit
too computational.
• 24:50 - 24:53
It's not smart at all.
• 24:53 - 24:58
It has a similar function over a
rectangle, something like that.
• 24:58 - 25:00
But it's a little bit
too confrontational.
• 25:00 - 25:01
We are looking
for something that
• 25:01 - 25:05
is not going-- examples that are
going to be easy to do and not
• 25:05 - 25:09
involve too much heavy
competition by him, because you
• 25:09 - 25:11
do everything by him.
• 25:11 - 25:15
Not-- like you don't have
a calculator, et cetera.
• 25:15 - 25:20
And the exam is very
limited in time, DU over 8.
• 25:20 - 25:23
So you say OK, I'm
know what that is.
• 25:23 - 25:28
That's going to be the A of S.
And that is going to be 2 pi.
• 25:28 - 25:32
Why can't I be so confident
and pull 2 pi out?
• 25:32 - 25:33
STUDENT: [INAUDIBLE]
• 25:33 - 25:36
PROFESSOR: Because there
is no dependence on theta.
• 25:36 - 25:38
All right?
• 25:38 - 25:42
So I have that one.
• 25:42 - 25:46
And then you go all right,
integral, square of you
• 25:46 - 25:52
times the U over 8-- 1 over 8DU.
• 25:52 - 25:56
And I have to be careful
because when R is zero--
• 25:56 - 26:00
if I put zero and 1 here
like some of my students,
• 26:00 - 26:05
I'm dead meat, because I'm going
to lose a lot of credit, right?
• 26:05 - 26:07
So I have to pay attention.
• 26:07 - 26:09
R is 0, and U equals?
• 26:09 - 26:10
STUDENT: 1.
• 26:10 - 26:11
PROFESSOR: 1.
• 26:11 - 26:13
R equals 1.
• 26:13 - 26:17
U equals 5.
• 26:17 - 26:23
And I worked this out
and I should be done.
• 26:23 - 26:28
And that's-- you should
expect something like that.
• 26:28 - 26:34
Nice, not computational,
you kind of looking.
• 26:34 - 26:37
What is integral of square of U?
• 26:37 - 26:38
STUDENT: [INAUDIBLE]
• 26:38 - 26:42
PROFESSOR: So you have--
you do the three halves,
• 26:42 - 26:44
and you pull out the 2/3, right?
• 26:44 - 26:45
That's what you do.
• 26:45 - 26:49
And then you go between U equals
1 down, and U equals 5 up.
• 26:49 - 26:52
And it's like one of those
examples we worked before.
• 26:52 - 26:53
Remember, and more
important, you
• 26:53 - 26:58
that for surface area?
• 26:58 - 27:00
Oh, my god.
• 27:00 - 27:01
4 over 8.
• 27:01 - 27:03
How much is 4 over 8?
• 27:03 - 27:04
STUDENT: [INAUDIBLE]
• 27:04 - 27:05
PROFESSOR: One half.
• 27:05 - 27:09
• 27:09 - 27:09
Right?
• 27:09 - 27:15
So we will have 1 over
6, and write pi times 5
• 27:15 - 27:20
to the three halves minus 1.
• 27:20 - 27:22
So do I like it?
• 27:22 - 27:23
I would leave it like that.
• 27:23 - 27:23
I'm fine.
• 27:23 - 27:25
• 27:25 - 27:27
I have people who care.
• 27:27 - 27:32
I don't care how some people
write it-- 5 with 5 minus 1
• 27:32 - 27:34
because they think
it looks better.
• 27:34 - 27:34
It doesn't.
• 27:34 - 27:37
That's the scientific
equation, and I'm fine with it.
• 27:37 - 27:38
Right?
• 27:38 - 27:39
OK.
• 27:39 - 27:43
So expect something like--
maybe I'm talking too much,
• 27:43 - 27:47
but maybe it's a good thing
to tell you what to expect
• 27:47 - 27:48
because we have to [INAUDIBLE].
• 27:48 - 27:50
At the same time, we're
teaching new things
• 27:50 - 27:55
as staff instructors doing
review of what's important.
• 27:55 - 28:03
I'm thinking if I'm doing things
right and at the same pace,
• 28:03 - 28:10
I should be finished
with chapter 13
• 28:10 - 28:12
at the end of next week.
• 28:12 - 28:15
Because after 13.5,
we have 13.6 which
• 28:15 - 28:18
is a generalization
of Green's Theorem.
• 28:18 - 28:21
13.6 as you recall is
called Stokes' Theorem.
• 28:21 - 28:25
13.7 is also a generalization
of Green's Theorem.
• 28:25 - 28:27
And they are all related.
• 28:27 - 28:32
It's like the trinity
on [INAUDIBLE].
• 28:32 - 28:33
That's the Divergence Theorem.
• 28:33 - 28:38
That is the last section,
13.7, Divergence Theorem.
• 28:38 - 28:42
So if I am going
at the right pace,
• 28:42 - 28:44
by-- what is next
wee on Thursday?
• 28:44 - 28:48
The-- 23rd?
• 28:48 - 28:51
I should be more or less
done with the chapter.
• 28:51 - 28:54
And I'm thinking I have
all the time in the world
• 28:54 - 28:57
to review with you
from that moment on.
• 28:57 - 28:59
In which sense are
we going to review?
• 28:59 - 29:05
We are going to review
by solving past finals.
• 29:05 - 29:06
Right?
• 29:06 - 29:09
That's what we are-- that's
what I'm planning to do.
• 29:09 - 29:12
I'm going to erase this
and move on to something
• 29:12 - 29:15
more spectacular.
• 29:15 - 29:16
Many-- OK.
• 29:16 - 29:18
This second part that
I want to teach you
• 29:18 - 29:23
in regular courses
• 29:23 - 29:29
just skip it because they do
not want to teach you-- not you,
• 29:29 - 29:30
you are honor students.
• 29:30 - 29:31
But they don't want
to teach the students
• 29:31 - 29:36
ways to look at a surface.
• 29:36 - 29:40
Remember, guys, a surface
that is written like that
• 29:40 - 29:43
is called a graph.
• 29:43 - 29:48
But not all the
surfaces were graphs.
• 29:48 - 29:57
And actually for a surface
S, what the most general way
• 29:57 - 30:00
to represent the presentation
would be a parameterization.
• 30:00 - 30:06
• 30:06 - 30:10
And I'll do a little bit
of a review for those.
• 30:10 - 30:18
R-- little R or big R--
big R, because that's
• 30:18 - 30:21
the position vector the
way I serve it to you
• 30:21 - 30:24
on a plate, whether,
for curves in space.
• 30:24 - 30:29
I say that's R of P. And when
we moved on curves to surfaces,
• 30:29 - 30:34
I said you move your path
two directions of motion.
• 30:34 - 30:37
You have two-- what are
those called in mechanics?
• 30:37 - 30:38
Degrees of freedom.
• 30:38 - 30:41
So you have two
degrees of freedom
• 30:41 - 30:42
like latitude and longitude.
• 30:42 - 30:47
Then R belongs--
the position vector
• 30:47 - 30:53
is a function of two variables,
and it belongs to R3,
• 30:53 - 30:54
because it's a vector in R3.
• 30:54 - 30:58
And want to have-- imagine
that my hand is a surface.
• 30:58 - 31:00
Well, OK.
• 31:00 - 31:02
This is the position vector, I'm
just kind of sweeping my hand,
• 31:02 - 31:05
going this way, one
degree of freedom.
• 31:05 - 31:07
Or going that way, the
other degree of freedom.
• 31:07 - 31:11
This is what
parameterization is.
• 31:11 - 31:17
So for a sphere, if you want to
parameterize the whole sphere--
• 31:17 - 31:20
I'll be done in a second.
• 31:20 - 31:23
I need you to see
if you remember
• 31:23 - 31:25
how to parameterize a sphere.
• 31:25 - 31:26
I'm testing you.
• 31:26 - 31:28
I'm mean today.
• 31:28 - 31:30
So examples.
• 31:30 - 31:32
Example 1 is
parameterize a sphere.
• 31:32 - 31:37
• 31:37 - 31:38
Was it hard?
• 31:38 - 31:41
That was a long
time ago, my god.
• 31:41 - 31:46
X, Y, and Z are what?
• 31:46 - 31:48
Latitude from Santa Clause.
• 31:48 - 31:52
Always latitude from
the North Pole is 5.
• 31:52 - 31:54
Longitude is from zero to 5.
• 31:54 - 31:57
The meridian is zero to 5.
• 31:57 - 32:00
That was theta, the
parameter of theta.
• 32:00 - 32:05
R was the distance
from this to a point.
• 32:05 - 32:08
But R was allowed to be
from-- take many values.
• 32:08 - 32:12
Now if I'm moving on
• 32:12 - 32:16
A-- let me make
• 32:16 - 32:17
• 32:17 - 32:20
Assume that A would
be a sample, A.
• 32:20 - 32:22
How am I going to write
that parameterization?
• 32:22 - 32:25
STUDENT: X equals
A plus [INAUDIBLE]?
• 32:25 - 32:29
PROFESSOR: A something,
A something, A something.
• 32:29 - 32:30
STUDENT: A [INAUDIBLE]
• 32:30 - 32:31
PROFESSOR: He is right.
• 32:31 - 32:32
I have to move on.
• 32:32 - 32:33
STUDENT: [INAUDIBLE]
• 32:33 - 32:35
PROFESSOR: Go slow.
• 32:35 - 32:37
So I have-- the last
one-- you were right,
• 32:37 - 32:40
Buddy, you have the
memory of a medical doctor
• 32:40 - 32:42
and some day you will
be a medical doctor.
• 32:42 - 32:45
Not everybody has a good memory.
• 32:45 - 32:50
So the way you can do that
is, wait a minute, this is pi,
• 32:50 - 32:51
right?
• 32:51 - 32:52
This [INAUDIBLE].
• 32:52 - 32:55
If you want the Z, you
• 32:55 - 33:01
And since Z is adjacent, you
go R, cosine, sine, phi equals
• 33:01 - 33:02
sine phi.
• 33:02 - 33:05
Now we started with X
because he's worked on this
• 33:05 - 33:06
and remembers everything.
• 33:06 - 33:08
He has it memorized.
• 33:08 - 33:10
Sine phi for both.
• 33:10 - 33:14
And times what in both cases?
• 33:14 - 33:15
He's just the guy
who's not here.
• 33:15 - 33:17
So sine phi.
• 33:17 - 33:21
It helps to memorize N
cosine theta, and sine theta.
• 33:21 - 33:23
Is that really easy to memorize?
• 33:23 - 33:26
So where phi was the
latitude from the North
• 33:26 - 33:30
Pole between zero
and phi, it theta
• 33:30 - 33:36
was the longitude-- excuse
me, guys-- longitude from zero
• 33:36 - 33:41
to 2 pi, all around one more.
• 33:41 - 33:44
So you say wait a
minute, Magdalena,
• 33:44 - 33:45
these are Euler's angle.
• 33:45 - 33:47
What do they call in mechanics?
• 33:47 - 33:50
I think they call
them Euler angles.
• 33:50 - 33:53
But anyway, for
phi theta, we call
• 33:53 - 33:56
them latitude and longitude.
• 33:56 - 34:00
I'll replace them, because look,
I want R to be in terms of U,V.
• 34:00 - 34:02
So in mathematics, it's
• 34:02 - 34:05
We can call them
whatever we want.
• 34:05 - 34:10
to call people names-- no--
• 34:10 - 34:14
to call things names
and people names--
• 34:14 - 34:16
STUDENT: Could U not equal zero?
• 34:16 - 34:17
PROFESSOR: Who?
• 34:17 - 34:18
STUDENT: U.
• 34:18 - 34:19
PROFESSOR: Yes.
• 34:19 - 34:19
So U can--
• 34:19 - 34:20
STUDENT: [INAUDIBLE]
• 34:20 - 34:25
PROFESSOR: --yeah, but
why didn't I write zero?
• 34:25 - 34:25
Well--
• 34:25 - 34:27
STUDENT: [INAUDIBLE]
makes sense.
• 34:27 - 34:30
PROFESSOR: --because,
yeah, you can take both.
• 34:30 - 34:33
If I want to study
differentiability,
• 34:33 - 34:36
I usually have to take it less
than and less than and less
• 34:36 - 34:39
than and less than because we
studied differentiability on
• 34:39 - 34:40
[INAUDIBLE].
• 34:40 - 34:43
But right now, I can take them
from the North Pole itself
• 34:43 - 34:47
to the South Pole itself-- so.
• 34:47 - 34:51
I'm not deleting any meridian.
• 34:51 - 34:54
If I were-- suppose
I were to delete it.
• 34:54 - 34:56
By the way, what does this mean?
• 34:56 - 34:57
I'm just kidding.
• 34:57 - 34:58
I'll put it back.
• 34:58 - 35:01
question over there,
• 35:01 - 35:04
• 35:04 - 35:07
It's a dangerous thing
when people make you think.
• 35:07 - 35:10
So it goes from zero to 2 pi.
• 35:10 - 35:12
Why would that be?
• 35:12 - 35:15
Imagine you have all the
meridians in the world
• 35:15 - 35:18
except for one.
• 35:18 - 35:22
From the sphere, you cut it and
remove the Greenwich meridian,
• 35:22 - 35:26
the one that passes
through Greenwich Village.
• 35:26 - 35:30
The one-- not the one in New
York, the one next to London,
• 35:30 - 35:31
right?
• 35:31 - 35:32
So put it back.
• 35:32 - 35:35
Put that meridian back.
• 35:35 - 35:39
It's like you take an
orange, and you make a slice.
• 35:39 - 35:40
I am-- OK.
• 35:40 - 35:43
Stop with the fruit
because I'm hungry.
• 35:43 - 35:47
Now, example two.
• 35:47 - 35:53
Now, imagine another surface
area you're used to, the what?
• 35:53 - 35:58
The paraboloid is one of our
favorite guys this semester.
• 35:58 - 36:00
X squared plus Y squared.
• 36:00 - 36:02
What is the
parameterization of that?
• 36:02 - 36:07
• 36:07 - 36:10
Well, if I write it
like that, it's a graph.
• 36:10 - 36:12
But if I don't want to
write it as a graph,
• 36:12 - 36:14
I have to write
it as a parameter.
• 36:14 - 36:16
What am I going to do?
• 36:16 - 36:19
I really know X to be U, right?
• 36:19 - 36:21
That's the simplest
choice possible.
• 36:21 - 36:25
Y could be V. And then Z will
be U squared plus V squared.
• 36:25 - 36:26
And there I am.
• 36:26 - 36:27
[SNEEZE]
• 36:27 - 36:29
So I'm going to write-- bless
• 36:29 - 36:32
• 36:32 - 36:38
V plus J plus U squared
plus V squared, K. So this
• 36:38 - 36:44
is the parameterization
of a paraboloid.
• 36:44 - 36:47
That one of them--
there are infinitely
• 36:47 - 36:49
many-- the one that comes
to mind because it's
• 36:49 - 36:53
the easiest one to think about.
• 36:53 - 36:54
STUDENT: [INAUDIBLE].
• 36:54 - 36:55
PROFESSOR: Good.
• 36:55 - 36:59
For a minute, guys,
you didn't need me.
• 36:59 - 37:02
You didn't need me at all
to come up with those.
• 37:02 - 37:06
But maybe you would need me
to remember, or maybe not--
• 37:06 - 37:08
to remind you of the helicoid.
• 37:08 - 37:10
Helicoid.
• 37:10 - 37:13
Did you go to the,
as I told you to go
• 37:13 - 37:16
to the [INAUDIBLE] spectrum--
what was that called?
• 37:16 - 37:16
The--
• 37:16 - 37:17
STUDENT: Science spectrum.
• 37:17 - 37:18
PROFESSOR: Science spectrum.
• 37:18 - 37:25
And dip into soap solution
the thingy was-- a metal
• 37:25 - 37:28
rod with a-- with a what?
• 37:28 - 37:34
of metal so the soap
• 37:34 - 37:37
film would take which shape?
• 37:37 - 37:42
The shape of this spiral that's
going to go inside here, right?
• 37:42 - 37:46
That's called a helicoid.
• 37:46 - 37:46
OK.
• 37:46 - 37:47
All right.
• 37:47 - 37:48
• 37:48 - 37:48
STUDENT: No.
• 37:48 - 37:49
PROFESSOR: OK, good.
• 37:49 - 37:54
So in this case, R
of UV will be what?
• 37:54 - 37:58
It was a long time ago, once
upon a time I gave it to you.
• 37:58 - 38:00
It's extremely hard to
memorize if you don't work
• 38:00 - 38:04
with it on a regular basis.
• 38:04 - 38:07
If it were a helix,
what would it be?
• 38:07 - 38:10
If it were a helix, it
would be R of T right?
• 38:10 - 38:14
It would be like equal
sign T, A sine T, BT.
• 38:14 - 38:17
Say it again, Magdalena,
that was a long time ago,
• 38:17 - 38:18
chapter 10.
• 38:18 - 38:20
Chapter 10.
• 38:20 - 38:24
Equal sign, T, A sine
T, MBT, standard helix.
• 38:24 - 38:25
This is not going to be that.
• 38:25 - 38:33
It's going to be-- U cosine B.
U sine B. Look at the picture.
• 38:33 - 38:38
And imagine that these guys
are extended to infinity.
• 38:38 - 38:39
It's not just the
stairs themselves,
• 38:39 - 38:41
or whatever they are.
• 38:41 - 38:46
There are infinite lines,
straight lines, and busy.
• 38:46 - 38:48
This is done.
• 38:48 - 38:49
NB is a positive constant.
• 38:49 - 38:52
• 38:52 - 38:57
U and V. Any other guy
• 38:57 - 39:00
that comes to mind, I'm out
of imagination right now.
• 39:00 - 39:05
You can do a torus on the
fold that looks like a donut.
• 39:05 - 39:06
You will have two parameters.
• 39:06 - 39:08
Imagine a donut.
• 39:08 - 39:11
How do you-- I'm not
going to write that.
• 39:11 - 39:13
Eventually I could give you
that as an extra credit thing.
• 39:13 - 39:19
What are the two degrees of
freedom of moving on the donut,
• 39:19 - 39:21
assuming that you would
like to move in circles?
• 39:21 - 39:25
• 39:25 - 39:26
STUDENT: [INAUDIBLE]
• 39:26 - 39:30
• 39:30 - 39:32
PROFESSOR: Let me draw a
donut, because I'm hungry,
• 39:32 - 39:35
and I really-- I cannot help it.
• 39:35 - 39:38
I just have to-- this is
called a torus in mathematics.
• 39:38 - 39:43
And you'll have-- one degree
of freedom will be like this,
• 39:43 - 39:45
the other degree of
freedom will be like that.
• 39:45 - 39:47
This is U and B.
• 39:47 - 39:50
mathematicians,
apologists, geometers,
• 39:50 - 39:53
they call those angles phi
and theta because they really
• 39:53 - 39:56
are between zero and 2 pi.
• 39:56 - 40:00
It has a rotation like
that along the donut.
• 40:00 - 40:02
You can cut, slice
the donut, or if they
• 40:02 - 40:04
don't put cheese filling in it.
• 40:04 - 40:09
That was a bad idea not
having anything to eat.
• 40:09 - 40:15
And the other angle will be your
2 pi along this little circle.
• 40:15 - 40:18
So you still have two degrees
of freedom on a donut.
• 40:18 - 40:19
It's a surface.
• 40:19 - 40:20
You can write the
parameterization.
• 40:20 - 40:21
Yes?
• 40:21 - 40:22
STUDENT: Why is a
pie this way around.
• 40:22 - 40:26
Why is it like [INAUDIBLE].
• 40:26 - 40:28
PROFESSOR: It
doesn't have to be.
• 40:28 - 40:30
STUDENT: Or is it just
kind of like [INAUDIBLE]?
• 40:30 - 40:31
PROFESSOR: That's
what they call it.
• 40:31 - 40:32
Yeah.
• 40:32 - 40:35
So they are between 2 and 2 pi.
• 40:35 - 40:42
While I erase-- or should
I-- enough expectation
• 40:42 - 40:44
in terms of parameterization,
I have to night
• 40:44 - 40:47
• 40:47 - 40:53
If somebody would say I'm
giving you a patch of a surface,
• 40:53 - 40:57
but that patch of a
surface is in a frame--
• 40:57 - 41:02
it's a nice parameterization.
• 41:02 - 41:04
This is the P on the surface.
• 41:04 - 41:08
• 41:08 - 41:11
And you say, well,
the parameterization
• 41:11 - 41:14
is going to be R
of U and V equals
• 41:14 - 41:22
X of UVI plus Y of
UVJ plus Z of UVK.
• 41:22 - 41:25
• 41:25 - 41:28
And suppose that somebody says
this is you favorite test.
• 41:28 - 41:31
• 41:31 - 41:36
Find V. Well, that
would be absurd.
• 41:36 - 41:37
My god, how do we do that?
• 41:37 - 41:49
Find the flux
corresponding to-- do
• 41:49 - 41:52
we say restart--
just a second-- just
• 41:52 - 41:54
• 41:54 - 41:55
[INAUDIBLE]
• 41:55 - 41:57
• 41:57 - 42:01
We don't say what kind of
vector field that it is,
• 42:01 - 42:07
but we will say plus
corresponding to the vector
• 42:07 - 42:08
field.
• 42:08 - 42:09
F [INAUDIBLE].
• 42:09 - 42:14
• 42:14 - 42:16
And this vector
field, I'll tell you
• 42:16 - 42:22
in a second what's expected
from this to be a vector field.
• 42:22 - 42:35
Through, on the surface, we
find on the surface-- yes.
• 42:35 - 42:39
Mathematicians say
define normal surface S.
• 42:39 - 42:44
But a physicist will
say flux through,
• 42:44 - 42:55
the flux corresponding
to F through the surface.
• 42:55 - 42:59
• 42:59 - 43:01
Yes.
• 43:01 - 43:04
So you'll say why would that
be, and what is the flux?
• 43:04 - 43:12
By definition, how
should we denote it?
• 43:12 - 43:16
Let's make a beautiful script
F. That's crazy, right?
• 43:16 - 43:22
And then it goes doubling
over the surface F test.
• 43:22 - 43:25
Is anybody mechanical
engineering here?
• 43:25 - 43:28
Do you know the flux formula?
• 43:28 - 43:34
It's going to be [INAUDIBLE]
over S of F, this magic thing.
• 43:34 - 43:37
Not DN, DS.
• 43:37 - 43:39
Do you know what N means?
• 43:39 - 43:41
What it is N for
mechanical engineering,
• 43:41 - 43:43
[INAUDIBLE] engineers?
• 43:43 - 44:01
N to would be the unit normal
vector field to the surface S.
• 44:01 - 44:03
How would you want
to imagine that?
• 44:03 - 44:07
You would have a surface, and
you have this velocity vectors
• 44:07 - 44:13
here at the bottom that goes
to S. And this field goes up.
• 44:13 - 44:17
You'll have a force and
acceleration, velocity,
• 44:17 - 44:20
you have everything
going this way.
• 44:20 - 44:23
And you want to find
out what happens.
• 44:23 - 44:27
You introduce this notion
of flux through the surface.
• 44:27 - 44:29
Another way to have a
flux through the surface
• 44:29 - 44:31
maybe through the same
surface but associated
• 44:31 - 44:34
through another
kind of concept--
• 44:34 - 44:36
if there could be
something else.
• 44:36 - 44:40
In electromagnetism, F would be
something else, some other type
• 44:40 - 44:41
of vector field.
• 44:41 - 44:42
Yes, sir.
• 44:42 - 44:43
STUDENT: [INAUDIBLE].
• 44:43 - 44:46
PROFESSOR: So find out, by
the way until next time,
• 44:46 - 44:49
if you were an electrical
engineering major, what
• 44:49 - 44:52
would flux be for you guys?
• 44:52 - 44:56
Two surfaces, one would be the
meaning of the vector field
• 44:56 - 44:58
F for you, and
why would you care
• 44:58 - 45:01
flux or something like that.
• 45:01 - 45:03
I don't want to talk
• 45:03 - 45:07
It's for you to do the
search and find out.
• 45:07 - 45:10
So suppose that
somebody gives you
• 45:10 - 45:13
this notion that says you
have a parameteric surface.
• 45:13 - 45:19
Give an application
of that and find out
• 45:19 - 45:24
how you're going
be deal with it.
• 45:24 - 45:28
I'll give you an example
that shouldn't be too hard.
• 45:28 - 45:33
• 45:33 - 45:34
I'll make up my own example.
• 45:34 - 45:38
And looks like example 6, but
it's going to be different.
• 45:38 - 45:47
• 45:47 - 45:48
Example.
• 45:48 - 45:51
• 45:51 - 45:59
Find the flux F if F will
be a simple function.
• 45:59 - 46:05
Let's say something equals X, I
plus Y,J Z, K at every point X,
• 46:05 - 46:13
Y-- at every point
of the space XYZ.
• 46:13 - 46:16
That means you could have this
vector field defined everywhere
• 46:16 - 46:18
in space in [INAUDIBLE].
• 46:18 - 46:23
this acting on the surface.
• 46:23 - 46:25
So it's acting on the surface.
• 46:25 - 46:28
• 46:28 - 46:30
And then what will the flux be?
• 46:30 - 46:35
On the surface, which surface?
• 46:35 - 46:41
My favorite one, Z equals
X squared plus Y squared.
• 46:41 - 46:47
• 46:47 - 46:49
First of all, you say
wait, wait, Magdalena,
• 46:49 - 46:51
do you want to do it like that?
• 46:51 - 46:54
Do you want to say F
over XY to be a graph?
• 46:54 - 46:59
Or do you want to consider it
as a parameterized surface?
• 46:59 - 47:02
And that means it's the same
thing, equivalent to or if
• 47:02 - 47:09
and only if, who tells me again
what R was for such a surface?
• 47:09 - 47:10
STUDENT: XI.
• 47:10 - 47:11
PROFESSOR: X is
U. Y is V, so U--
• 47:11 - 47:12
STUDENT: [INAUDIBLE]
• 47:12 - 47:16
PROFESSOR: --I, that
would be J, then good.
• 47:16 - 47:20
U squared plus U squared UK.
• 47:20 - 47:23
Well, when you say that,
we have-- first of all,
• 47:23 - 47:27
we have no idea what
the heck we need to do,
• 47:27 - 47:32
because do we want to do it
in this form like a graph?
• 47:32 - 47:34
Or do we want to do
it parameterized?
• 47:34 - 47:37
We have to set up
formulas for the flats.
• 47:37 - 47:39
It's not so easy.
• 47:39 - 47:43
So assume that we are brave
enough and we start everything.
• 47:43 - 47:48
I want to understand what
flux really is as an integral.
• 47:48 - 47:56
And let me set it up for the
first case, the case of Z
• 47:56 - 47:58
equals F of X and Y.
And I'm happy with it
• 47:58 - 48:01
because that's
the simplest case.
• 48:01 - 48:04
Who's going to teach
me what I have to do?
• 48:04 - 48:05
You are confusing.
• 48:05 - 48:10
I have double integral over S
minus theory of F in general.
• 48:10 - 48:13
This is a general
vector value field.
• 48:13 - 48:16
• 48:16 - 48:17
It could be anything.
• 48:17 - 48:18
Could be anything.
• 48:18 - 48:23
But then I have to [INAUDIBLE],
because N corresponds
• 48:23 - 48:27
to the normal to the surface.
• 48:27 - 48:29
So I-- it's not so easy, right?
• 48:29 - 48:31
I have to be a little bit smart.
• 48:31 - 48:32
If I'm not smart--
• 48:32 - 48:33
STUDENT: [INAUDIBLE]
• 48:33 - 48:35
• 48:35 - 48:37
PROFESSOR: That-- you
are getting close.
• 48:37 - 48:41
So guys, the normal
two-way surface-- somebody
• 48:41 - 48:43
gave you a surface, OK?
• 48:43 - 48:46
And normal to a surface
is normal to the plane--
• 48:46 - 48:50
the tangent plane
of the surface.
• 48:50 - 48:52
So how did we get that?
• 48:52 - 48:54
There were many ways to do it.
• 48:54 - 48:56
Either you write
the tangent plane
• 48:56 - 49:03
and you know it by heart--
that was Z minus Z zero
• 49:03 - 49:07
equals-- what the heck was
that-- S of X times X minus X
• 49:07 - 49:11
equals-- plus X of Y
times Y minus Y zero.
• 49:11 - 49:14
And from here you collect--
what do you collect?
• 49:14 - 49:16
You move everybody--
it's a moving sale.
• 49:16 - 49:20
You move everybody to the
left hand side and that's it.
• 49:20 - 49:21
[INAUDIBLE] moving sale.
• 49:21 - 49:23
OK?
• 49:23 - 49:27
And everybody will be
giving you some components.
• 49:27 - 49:31
You're going to have minus S
of X-- S minus X zero-- minus S
• 49:31 - 49:37
of Y, Y minus Y zero, plus
1-- this is really funny.
• 49:37 - 49:40
1 times Z minus Z, Z.
• 49:40 - 49:43
be given by what?
• 49:43 - 49:46
The normal-- how do
you collect the normal?
• 49:46 - 49:47
STUDENT: [INAUDIBLE]
• 49:47 - 49:52
PROFESSOR: Pi is A, B, C. A,
B, and C will be the normal.
• 49:52 - 49:54
Except it's not unitary.
• 49:54 - 49:58
And the mechanical engineer
tells you, yeah, you're
• 49:58 - 50:01
stupid-- well, they
never say that.
• 50:01 - 50:06
They will stay look, you have
to be a little more careful.
• 50:06 - 50:08
Not say they are equal.
• 50:08 - 50:09
What do they mean?
• 50:09 - 50:11
They say for us,
in fluid mechanics,
• 50:11 - 50:15
solid mechanics, when we write
N, we mean you mean vector.
• 50:15 - 50:16
You are almost there.
• 50:16 - 50:17
What's missing?
• 50:17 - 50:18
STUDENT: Magnitude. [INAUDIBLE].
• 50:18 - 50:20
PROFESSOR: Very
good, the magnitude.
• 50:20 - 50:24
So they will say, go ahead and
you [INAUDIBLE] the magnitude.
• 50:24 - 50:29
And you are lucky now that
you know what N will be.
• 50:29 - 50:30
On the other hand--
• 50:30 - 50:31
STUDENT: [INAUDIBLE].
• 50:31 - 50:32
PROFESSOR: This is excellent.
• 50:32 - 50:36
The one on the bottom-- Alex
is thinking like in chess, two
• 50:36 - 50:38
• 50:38 - 50:41
You should get two extra
credit points with that.
• 50:41 - 50:41
STUDENT: All right.
• 50:41 - 50:43
• 50:43 - 50:46
DS is 1 plus S of X squared
plus F of X squared.
• 50:46 - 50:52
The 1 on the bottom and the
1 on the top will simplify.
• 50:52 - 50:53
So say it again, Magdalena.
• 50:53 - 50:55
Let me write it down here.
• 50:55 - 51:01
1 S of X, minus S of Y
1 over all this animal,
• 51:01 - 51:05
S of X squared plus S
of Y squared plus 1.
• 51:05 - 51:09
This is the thinking
like the early element
• 51:09 - 51:13
times the early element
will be the same thing.
• 51:13 - 51:18
I'll write it twice even if you
laugh at me because we are just
• 51:18 - 51:19
learning together,
and now you finally
• 51:19 - 51:21
see-- everybody can
see that desimplifies.
• 51:21 - 51:24
• 51:24 - 51:28
So it's going to be easy to
solve this integral in the end,
• 51:28 - 51:29
right?
• 51:29 - 51:32
So let's do the
problem, finally.
• 51:32 - 51:36
I'm going to erase it.
• 51:36 - 51:40
Let's do this problem
just for us, at any point.
• 51:40 - 51:43
I didn't say where.
• 51:43 - 51:47
Over the same thing.
• 51:47 - 51:50
The DS was over V01.
• 51:50 - 51:53
So the picture is
the same as before.
• 51:53 - 51:58
The S will be the
nutshell, the eggshell--
• 51:58 - 52:03
I don't know what it was--
over the domain D plane.
• 52:03 - 52:08
The domain D plane
was D of zero 1.
• 52:08 - 52:12
And I say that I need
to use another color.
• 52:12 - 52:17
This it's going to be my
shell, my surface S. Z
• 52:17 - 52:20
equals X squared
plus [INAUDIBLE].
• 52:20 - 52:23
How do you compute the flux?
• 52:23 - 52:26
Well, this is that.
• 52:26 - 52:28
So if we have to be a
little bit careful and smart
• 52:28 - 52:33
and say double integral over
S, and now without rushing,
• 52:33 - 52:36
we have to do a good job.
• 52:36 - 52:39
First of all, how do
you do the dot product?
• 52:39 - 52:41
The dot product--
• 52:41 - 52:44
STUDENT: [INAUDIBLE]
• 52:44 - 52:44
PROFESSOR: Right.
• 52:44 - 52:47
So first component
times first component,
• 52:47 - 52:50
a second component,
second component times
• 52:50 - 52:53
second component plus
that component times
• 52:53 - 52:54
third component.
• 52:54 - 53:01
So if 1 is X, F2 is 1.
• 53:01 - 53:02
Good.
• 53:02 - 53:04
Z, though, he's not free.
• 53:04 - 53:05
He's married.
• 53:05 - 53:07
Why is he married?
• 53:07 - 53:08
STUDENT: [INAUDIBLE].
• 53:08 - 53:11
PROFESSOR: Because he
depends on X and Y.
• 53:11 - 53:14
So Z was even here,
because I'm on the surface.
• 53:14 - 53:17
I don't care what F does
away from the surface,
• 53:17 - 53:20
but when he sticks
to the surface, when
• 53:20 - 53:24
he's origin is on
the surface, then he
• 53:24 - 53:28
has to listen to the surface.
• 53:28 - 53:30
And that Z is not independent.
• 53:30 - 53:33
The Z is X squared
by Y squared here.
• 53:33 - 53:37
In a bracket, we are
over the surface.
• 53:37 - 53:41
That product minus S
of X, minus S of Y. I
• 53:41 - 53:43
know you're going to
laugh at me because I
• 53:43 - 53:45
haven't written where they are.
• 53:45 - 53:46
But that's what I
• 53:46 - 53:47
DA.
• 53:47 - 53:49
• 53:49 - 53:52
Who are they?
• 53:52 - 53:54
Who is this guy?
• 53:54 - 53:55
STUDENT: The [INAUDIBLE].
• 53:55 - 53:55
PROFESSOR: What?
• 53:55 - 53:57
STUDENT: [INAUDIBLE].
• 53:57 - 53:58
PROFESSOR: Negative 2X.
• 53:58 - 53:59
Is it?
• 53:59 - 54:00
STUDENT: No.
• 54:00 - 54:01
• 54:01 - 54:02
STUDENT: [INAUDIBLE].
• 54:02 - 54:03
PROFESSOR: Negative 2Y.
• 54:03 - 54:06
• 54:06 - 54:07
I'm just kidding.
• 54:07 - 54:08
OK.
• 54:08 - 54:14
So finally we should be able
to compute this integral.
• 54:14 - 54:16
That looks awful.
• 54:16 - 54:18
Over D.
• 54:18 - 54:21
have the D, which
• 54:21 - 54:25
is the disk of
• 54:25 - 54:30
And we say, OK, I have,
oh my god, it's OK.
• 54:30 - 54:32
This times that is how much?
• 54:32 - 54:33
STUDENT: [INAUDIBLE].
• 54:33 - 54:34
PROFESSOR: Minus 2X squared.
• 54:34 - 54:35
Right?
• 54:35 - 54:36
There.
• 54:36 - 54:37
Take the green.
• 54:37 - 54:41
This times that is how much?
• 54:41 - 54:44
Minus the Y squared.
• 54:44 - 54:51
And this times that is finally
just X squared plus Y squared.
• 54:51 - 54:53
Very nice think.
• 54:53 - 54:56
I think that at
first, but now I see
• 54:56 - 54:59
that life is beautiful
again-- DX, DY--
• 54:59 - 55:03
that I can go ahead and do it.
• 55:03 - 55:05
I can get a hold of this.
• 55:05 - 55:11
And inside that, what do I--
what am I left with in the end?
• 55:11 - 55:12
STUDENT: [INAUDIBLE].
• 55:12 - 55:15
PROFESSOR: Minus 2
times this animal,
• 55:15 - 55:19
called X squared plus Y squared,
which is going to be R squared.
• 55:19 - 55:23
So the flux-- the flux for
this problem in the end
• 55:23 - 55:26
is going to be very
nice and sassy.
• 55:26 - 55:26
Look at that.
• 55:26 - 55:28
F would be--
• 55:28 - 55:30
STUDENT: There
would not be any--
• 55:30 - 55:30
STUDENT: [INAUDIBLE]
• 55:30 - 55:37
• 55:37 - 55:37
PROFESSOR: What?
• 55:37 - 55:41
STUDENT: You've got
minus 2 and the plus 1.
• 55:41 - 55:42
PROFESSOR: Oh, thank God.
• 55:42 - 55:44
Thank God you exist.
• 55:44 - 55:47
it before, but then I
• 55:47 - 55:49
said-- I don't know why.
• 55:49 - 55:50
I messed up.
• 55:50 - 55:53
So we have minus R squared.
• 55:53 - 55:54
Very good.
• 55:54 - 55:55
It's easy.
• 55:55 - 56:02
Times an R from the
Jacobian, DR is theta.
• 56:02 - 56:05
And theta is between 0 and 2 pi.
• 56:05 - 56:08
And R between 0 and 1.
• 56:08 - 56:11
And now I will need
a plumber to tell me
• 56:11 - 56:13
what I do the limits
of the integrals,
• 56:13 - 56:18
because I think I'm getting
• 56:18 - 56:21
• 56:21 - 56:22
STUDENT: [INAUDIBLE].
• 56:22 - 56:23
PROFESSOR: I'll do
it, and then you
• 56:23 - 56:27
tell me why I got what I got.
• 56:27 - 56:30
I have a minus
pulled out by nature.
• 56:30 - 56:33
And then I have integral--
• 56:33 - 56:34
STUDENT: R [INAUDIBLE].
• 56:34 - 56:36
PROFESSOR: R to the
fourth of a fourth.
• 56:36 - 56:37
Very good.
• 56:37 - 56:40
But you have your [INAUDIBLE] so
when I do between zero and 1--
• 56:40 - 56:42
STUDENT: It's [INAUDIBLE].
• 56:42 - 56:44
PROFESSOR: 1 over
4-- you are too
• 56:44 - 56:49
fast-- as 2 pi-- that's a
good thing-- minus pi over 2,
• 56:49 - 56:51
you said, Gus.
• 56:51 - 56:53
And I could see it
coming straight at me
• 56:53 - 56:55
and hit me between the eyes.
• 56:55 - 56:58
What is the problem.
• 56:58 - 57:00
Is there a problem?
• 57:00 - 57:03
Without an area as a flux, would
that say, what is the negative?
• 57:03 - 57:04
Yes.
• 57:04 - 57:06
How can I make it positive?
• 57:06 - 57:07
This is my question.
• 57:07 - 57:09
STUDENT: Change the direction.
• 57:09 - 57:11
PROFESSOR: Change
the direction of who?
• 57:11 - 57:12
STUDENT: The flux.
• 57:12 - 57:13
PROFESSOR: The flux.
• 57:13 - 57:15
I could change the direction.
• 57:15 - 57:18
So what is it that
doesn't match?
• 57:18 - 57:20
[INAUDIBLE]
• 57:20 - 57:23
If I want to keep-- the
flux will be the same.
• 57:23 - 57:25
When I can change the
orientation of the service.
• 57:25 - 57:28
And instead I get a minus then.
• 57:28 - 57:36
My N was it sticking
in-- oh, my god.
• 57:36 - 57:40
So is it sticking
in or sticking out?
• 57:40 - 57:41
Look at it.
• 57:41 - 57:42
• 57:42 - 57:48
I have minus the positive guy
minus another positive guy,
• 57:48 - 57:49
and 1 sticking out.
• 57:49 - 57:51
But it goes with
the holes inside.
• 57:51 - 57:54
This is the paraboloid
[INAUDIBLE].
• 57:54 - 57:57
If I have something I minus I
minus J, does it go out or in?
• 57:57 - 57:58
STUDENT: In.
• 57:58 - 58:00
PROFESSOR: It goes in.
• 58:00 - 58:01
It goes in, and it'll be up.
• 58:01 - 58:03
So it's going to be like
all these normals are
• 58:03 - 58:07
going to be like a vector
field like that, like amoebas.
• 58:07 - 58:10
But they are pointing
towards inside.
• 58:10 - 58:11
Do I like that?
• 58:11 - 58:13
Yes, because I'm a
crazy mathematician.
• 58:13 - 58:17
Does the engineer like that?
• 58:17 - 58:18
No.
• 58:18 - 58:19
Why?
• 58:19 - 58:22
The flux is pointing in or out?
• 58:22 - 58:23
The flux.
• 58:23 - 58:24
The flux.
• 58:24 - 58:26
The flux, the flux
is pointing out.
• 58:26 - 58:28
Are you guys with me?
• 58:28 - 58:31
X plus Y-- X plus I plus J.
It's like this pointing out.
• 58:31 - 58:34
So the flux get
out of the surface.
• 58:34 - 58:37
It's like to pour water
inside, and the water's
• 58:37 - 58:42
just a net-- not a net, but
like something that holds it in.
• 58:42 - 58:43
And like a--
• 58:43 - 58:44
STUDENT: Like a [INAUDIBLE]?
• 58:44 - 58:46
PROFESSOR: --pasta strainer.
• 58:46 - 58:48
And the water goes up
[SPRAYING NOISE], well,
• 58:48 - 58:49
like a jet.
• 58:49 - 58:49
Like that.
• 58:49 - 58:53
going through the surface.
• 58:53 - 58:56
Are you happy that I took
the normal pointing inside?
• 58:56 - 58:57
No.
• 58:57 - 58:58
That was crazy.
• 58:58 - 59:03
So here comes you, the
mechanical engineer majoring
• 59:03 - 59:06
in solid or [INAUDIBLE]
and say Magdalena,
• 59:06 - 59:09
you should have taken
the outer normal,
• 59:09 - 59:12
because look at the
flux pointing out.
• 59:12 - 59:14
Take the outer of normal,
and things are going
• 59:14 - 59:17
to looks right and nice again.
• 59:17 - 59:19
So if I were to
change the normal,
• 59:19 - 59:21
I would put the
plus, plus, minus.
• 59:21 - 59:24
I'll take the outer normal.
• 59:24 - 59:26
And in the end I
get plus 5 over 2.
• 59:26 - 59:29
So no remark.
• 59:29 - 59:36
If I change N to minus N, this
would become the outer normal.
• 59:36 - 59:41
Then the flux would
become pi over 2. solar
• 59:41 - 59:43
flux depends on the what?
• 59:43 - 59:45
The match between
the flux, the angles,
• 59:45 - 59:49
sort of between the flux if
function, vector [INAUDIBLE]
• 59:49 - 59:53
function, and the normal
that I take to the surface.
• 59:53 - 59:54
Right?
• 59:54 - 59:59
I can change the normal and
• 59:59 - 60:02
In absolute values,
the same flux.
• 60:02 - 60:06
So flux should be equal
[INAUDIBLE] the absolute value.
• 60:06 - 60:09
Unlike the area that should
be always a positive number.
• 60:09 - 60:11
Volume, that should always
be a positive number.
• 60:11 - 60:15
So if I get a limited area,
that means I messed up.
• 60:15 - 60:17
If I get a negative
on all of them,
• 60:17 - 60:21
it means messed up in my
computation somewhere.
• 60:21 - 60:23
But that doesn't mean
I messed up here.
• 60:23 - 60:25
I just chose the other normal.
• 60:25 - 60:26
It's possible.
• 60:26 - 60:31
So the flux can be taken as
is and put in absolute value.
• 60:31 - 60:31
All right.
• 60:31 - 60:34
OK.
• 60:34 - 60:37
We have to think of it like
the surface, and stuff that
• 60:37 - 60:40
goes through surface
in electric circuits.
• 60:40 - 60:43
Can you do some research
• 60:43 - 60:45
and electrical engineering?
• 60:45 - 60:50
And next time somebody
tells me a story about it.
• 60:50 - 60:53
Who is-- again-- who is
electrical engineering major
• 60:53 - 60:54
here?
• 60:54 - 60:56
Oh, so five people.
• 60:56 - 60:58
You're going to get four
extra credit points.
• 60:58 - 61:00
You guys are jealous.
• 61:00 - 61:04
I'm going to give you four extra
credit points if in 10 minutes
• 61:04 - 61:08
you can tell us a little bit
about where flux can be seen.
• 61:08 - 61:10
Well, you don't have
to come to the board.
• 61:10 - 61:13
You can just talk to us
from outside if you want,
• 61:13 - 61:15
or down inside the classroom.
• 61:15 - 61:17
Tell us where the
notion of flux appears
• 61:17 - 61:20
in the electric
circuits and why it
• 61:20 - 61:24
would be important for
Calculus 3 as well.
• 61:24 - 61:25
OK.
• 61:25 - 61:29
Now a big question
before I let you go.
• 61:29 - 61:35
Can I have a flux
that corresponds
• 61:35 - 61:36
to a parameterization?
• 61:36 - 61:42
That is my big worry, that
I have to do that as well.
• 61:42 - 61:45
Eventually, could I
have solved this problem
• 61:45 - 61:48
if the surface that
is parameterized
• 61:48 - 61:54
was my friend--
who was my friend?
• 61:54 - 61:55
I don't remember.
• 61:55 - 61:59
UI plus VJ plus U
squared plus-- you
• 61:59 - 62:03
gave it to me-- OK, that
was the previous example,
• 62:03 - 62:05
and that's the last
example on the board.
• 62:05 - 62:13
So you have double integral
of force field times NDS.
• 62:13 - 62:17
Now, what if I say I don't
want to do it like this-- Z
• 62:17 - 62:19
equals F of XY.
• 62:19 - 62:21
So I don't want to
do it like that.
• 62:21 - 62:25
I want to do it in
a different way.
• 62:25 - 62:34
That means you pulling out of
• 62:34 - 62:35
F was F, right?
• 62:35 - 62:37
You need to leave F alone,
poor fellow, because he
• 62:37 - 62:40
has no better way to do it.
• 62:40 - 62:44
This is becoming
complicated, the [INAUDIBLE]
• 62:44 - 62:45
mechanical engineering.
• 62:45 - 62:48
• 62:48 - 62:53
And what's given to you
before, but you don't remember?
• 62:53 - 62:56
R was given to you
as position vector.
• 62:56 - 63:00
R sub U and R sub V,
you may not remember--
• 63:00 - 63:03
that was a long time ago-- we
proved that R sub U and R sub
• 63:03 - 63:05
V were on the surface.
• 63:05 - 63:07
They are both tensions
of the surface.
• 63:07 - 63:09
It was a long time ago.
• 63:09 - 63:11
So the normal is
[INAUDIBLE], and that's
• 63:11 - 63:14
exactly what I wanted to
say the normal will be.
• 63:14 - 63:18
Not quite pressed product,
but just like before,
• 63:18 - 63:21
pressed product
divided by the norm,
• 63:21 - 63:28
because then the unit normal
vector has to be length 1.
• 63:28 - 63:31
So I have to divide
by the number.
• 63:31 - 63:31
[SNEEZE]
• 63:31 - 63:32
The DS--
• 63:32 - 63:33
STUDENT: Thank you.
• 63:33 - 63:36
PROFESSOR: --is going to--
OK, now it's up to you guys.
• 63:36 - 63:37
You're smart.
• 63:37 - 63:39
You know what I want to say.
• 63:39 - 63:44
So I'll pretend that you
know what DS is in terms
• 63:44 - 63:45
of the parameterization.
• 63:45 - 63:47
What's coming?
• 63:47 - 63:48
We said that.
• 63:48 - 63:49
It was a long time ago.
• 63:49 - 63:52
You can guess it by
just being smart--
• 63:52 - 63:52
STUDENT: [INAUDIBLE].
• 63:52 - 63:53
PROFESSOR: --or you can--
• 63:53 - 63:54
STUDENT: [INAUDIBLE].
• 63:54 - 63:55
PROFESSOR: Yes, exactly.
• 63:55 - 63:58
And you got another
one extra credit point.
• 63:58 - 64:02
• 64:02 - 64:05
STUDENT: [INAUDIBLE]
• 64:05 - 64:07
PROFESSOR: So since before,
they were simplified,
• 64:07 - 64:08
for god's sake.
• 64:08 - 64:11
Now we have the new kind
of writing area element DS.
• 64:11 - 64:13
They also have to simplify.
• 64:13 - 64:16
It wasn't hard to see.
• 64:16 - 64:18
So you could have
done it like that.
• 64:18 - 64:23
You could have done
it like that, how?
• 64:23 - 64:26
Somebody need to help me,
because I have no idea what
• 64:26 - 64:28
I'm going to do here.
• 64:28 - 64:29
Do we get the same thing or not?
• 64:29 - 64:30
This is the question.
• 64:30 - 64:32
And I'm going to
finish with that,
• 64:32 - 64:34
but I don't want to
go home-- I'm not
• 64:34 - 64:38
going to let you go home
until you finish this.
• 64:38 - 64:42
F was a simple,
beautiful vector field.
• 64:42 - 64:45
Given-- like that.
• 64:45 - 64:47
This is a force.
• 64:47 - 64:49
May the force be
with you like that.
• 64:49 - 64:54
But we changed it in U,V because
we are acting on the surface S,
• 64:54 - 64:56
what is the pressure
in V, right?
• 64:56 - 65:01
So you have UI plus VJ
plus-- you gave it to me--
• 65:01 - 65:04
U squared plus V squared.
• 65:04 - 65:05
Am I right, or am
I talking nonsense?
• 65:05 - 65:08
• 65:08 - 65:09
All right.
• 65:09 - 65:13
So now again I have
to be seeing them.
• 65:13 - 65:14
Am I getting the same thing?
• 65:14 - 65:16
If I'm not getting
the same thing,
• 65:16 - 65:19
I can just go home and
• 65:19 - 65:23
But I have to get
the same thing.
• 65:23 - 65:27
Otherwise, there is something
wrong with my setup.
• 65:27 - 65:33
So I have to have U, V.
U squared plus V squared.
• 65:33 - 65:34
Close.
• 65:34 - 65:37
Dot product.
• 65:37 - 65:42
This guy over on top-- say what?
• 65:42 - 65:48
Magdalena, this guy over on top
has to be-- has to be a what?
• 65:48 - 65:50
Well, I didn't say what it was.
• 65:50 - 65:53
I should do it now.
• 65:53 - 65:53
Right?
• 65:53 - 65:57
So how will we do that?
• 65:57 - 66:08
We were saying R of
UV will be UI plus VJ
• 66:08 - 66:10
plus U squared plus V squared.
• 66:10 - 66:11
OK.
• 66:11 - 66:15
So R sub U will be--
you teach me quickly,
• 66:15 - 66:18
and R sub [INAUDIBLE]
is-- voila.
• 66:18 - 66:20
STUDENT: [INAUDIBLE]
• 66:20 - 66:22
PROFESSOR: 1--
• 66:22 - 66:24
STUDENT: [INAUDIBLE]
• 66:24 - 66:28
PROFESSOR: Plus zero--
thank you-- plus 2U, OK.
• 66:28 - 66:35
0 plus 1J plus 2VK Am I done?
• 66:35 - 66:35
I'm done.
• 66:35 - 66:36
No, I'm not done.
• 66:36 - 66:37
What do I have to do?
• 66:37 - 66:39
Cross them.
• 66:39 - 66:41
• 66:41 - 66:44
Cross multiply IJK.
• 66:44 - 66:47
This looks nice.
• 66:47 - 66:49
Look, it's not so ugly.
• 66:49 - 66:51
I thought it would
be uglier, right?
• 66:51 - 66:52
OK.
• 66:52 - 66:55
What it is?
• 66:55 - 66:58
What it this thing?
• 66:58 - 66:58
STUDENT: [INAUDIBLE].
• 66:58 - 67:06
PROFESSOR: Minus the
U, I. Minus-- plus.
• 67:06 - 67:09
Minus, plus 1.
• 67:09 - 67:11
2V minus because it's--
• 67:11 - 67:12
STUDENT: Minus.
• 67:12 - 67:13
PROFESSOR: --minus in front.
• 67:13 - 67:15
Right.
• 67:15 - 67:18
So I'm alternating.
• 67:18 - 67:20
And 1K.
• 67:20 - 67:24
So again, I get minus
X of S minus XY and 1,
• 67:24 - 67:26
and again, I'm pointing
• 67:26 - 67:30
So my normal will point
inside the surface
• 67:30 - 67:34
like needles that are
perpendicular to the surface
• 67:34 - 67:35
pointing inside.
• 67:35 - 67:36
But that's OK.
• 67:36 - 67:39
In the end, I take
everything in absolute value.
• 67:39 - 67:40
Right?
• 67:40 - 67:48
• 67:48 - 67:51
So again, I do the same math.
• 67:51 - 67:55
So I get minus-- I don't
want to do it anymore.
• 67:55 - 67:59
Minus 2A squared, minus 2B
• 67:59 - 68:02
plus this squared,
then you save me
• 68:02 - 68:05
and you said minus 2
squared [INAUDIBLE] squared.
• 68:05 - 68:06
DUDV.
• 68:06 - 68:15
But DUDV means that UV is a
pair, a point in this, guys.
• 68:15 - 68:16
UV.
• 68:16 - 68:19
It's a pair in the
• 68:19 - 68:22
So I'm getting exactly,
what exactly the same thing
• 68:22 - 68:23
as before.
• 68:23 - 68:26
Because this is
minus R squared, so I
• 68:26 - 68:33
get integral, integral, minus
R squared times R. DR, D theta.
• 68:33 - 68:36
From zero to 1,
from zero to 2 pi,
• 68:36 - 68:38
and I get the same
• 68:38 - 68:39
STUDENT: [INAUDIBLE].
• 68:39 - 68:40
PROFESSOR: Minus what?
• 68:40 - 68:40
STUDENT: [INAUDIBLE]
• 68:40 - 68:41
PROFESSOR: Pi over--
• 68:41 - 68:42
STUDENT: [INAUDIBLE].
• 68:42 - 68:42
PROFESSOR: You see?
• 68:42 - 68:46
• 68:46 - 68:48
STUDENT: 2.
• 68:48 - 68:50
PROFESSOR: So what
matters is that we
• 68:50 - 68:52
take the flux in
absolute value because it
• 68:52 - 68:54
depends on the
orientation of the normal.
• 68:54 - 68:57
If we take the
normal [INAUDIBLE].
• 68:57 - 69:03
Please, one thing I want you
to do when you go home now,
• 69:03 - 69:06
open the book which
maybe you rarely do,
• 69:06 - 69:09
but now it's
really-- the material
• 69:09 - 69:10
became complicated enough.
• 69:10 - 69:14
We are not just doing math,
calculus, we are doing physics,
• 69:14 - 69:18
we are doing mechanics, we are
dealing with surface integrals
• 69:18 - 69:19
and flux.
• 69:19 - 69:27
I want you to open the book
at page-- I don't know.
• 69:27 - 69:33
At surface integrals
starts at page 1,063.
• 69:33 - 69:35
Section 13.5.
• 69:35 - 69:38
And it keeps going like that,
pretty pictures of surfaces
• 69:38 - 69:41
and fluxes and so on.
• 69:41 - 69:42
Vector fields.
• 69:42 - 69:44
And it keeps going like that.
• 69:44 - 69:49
But it doesn't cover anything
new except what I said today.
• 69:49 - 69:51
It's just that it shows
you examples that are not
• 69:51 - 69:55
as beautiful as the ones I
gave, but they are essentially
• 69:55 - 69:58
the same, only a little
bit nastier to complete.
• 69:58 - 70:02
So up to 1,072.
• 70:02 - 70:06
So that is what you're
going to do this weekend,
• 70:06 - 70:07
plus the homework.
• 70:07 - 70:08
Keep on the homework.
• 70:08 - 70:11
Now, if you get stuck
Saturday, Sunday,
• 70:11 - 70:14
you get stuck, what do you do?
• 70:14 - 70:15
STUDENT: [INAUDIBLE]
• 70:15 - 70:16
PROFESSOR: You email me.
• 70:16 - 70:18
So you say what in the world
is going on with this problem
• 70:18 - 70:25
because I tried it seven
times and-- 88 times.
• 70:25 - 70:27
And then you got
the brownie points.
• 70:27 - 70:28
STUDENT: [INAUDIBLE]
• 70:28 - 70:30
PROFESSOR: [INAUDIBLE] problem.
• 70:30 - 70:31
STUDENT: [INAUDIBLE] by 32.
• 70:31 - 70:33
PROFESSOR: There
was a problem, guys.
• 70:33 - 70:35
There are not so many problems.
• 70:35 - 70:39
But the only part, serious
part that we would catch,
• 70:39 - 70:42
he found it first, and
he tried it 88 times.
• 70:42 - 70:44
• 70:44 - 70:47
I'll never forget you, though,
because you are unique,
• 70:47 - 70:50
and that-- I appreciated
that very much.
• 70:50 - 70:55
So doing this weekend, do
not hesitate to pester.
• 70:55 - 70:58
I will answer all the web
work problems you have.
• 70:58 - 71:00
I want you to do well.
• 71:00 - 71:02
Next week is the last
week on new theory,
• 71:02 - 71:04
and then we start
working for the final,
• 71:04 - 71:09
so by the time of the final,
you'll be [INAUDIBLE].
• 71:09 - 71:10
STUDENT: [INAUDIBLE]?
• 71:10 - 71:11
PROFESSOR: Yes, sir.
• 71:11 - 71:13
Oh, I appreciated
that you did that.
• 71:13 - 71:14
STUDENT: [INAUDIBLE]
• 71:14 - 71:18
• 71:18 - 71:21
PROFESSOR: Again,
I forgot these.
• 71:21 - 71:25
• 71:25 - 71:28
With the extra points you
got, you shouldn't care.
• 71:28 - 71:29
Title:
TTU Math2450 Calculus3 Secs 13.4 -13.5
Description:

Green's Theorem, Surface Integrals and Flux Integrals

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Video Language:
English
 jackie.luft edited English subtitles for TTU Math2450 Calculus3 Secs 13.4 -13.5