0:00:00.000,0:00:00.500


0:00:00.500,0:00:04.050
PROFESSOR: I would like to[br]review just briefly what

0:00:04.050,0:00:06.689
we discussed last time.

0:00:06.689,0:00:15.530
We gave very important results,[br]and that was Green's Theorem.

0:00:15.530,0:00:19.110
And I would like to[br]know if you remember

0:00:19.110,0:00:25.280
when I said about the[br]settling for this problem.

0:00:25.280,0:00:30.660
So we'll assume we have[br]a domain without a hole,

0:00:30.660,0:00:34.790
D. D is a domain[br]without a hole inside,

0:00:34.790,0:00:39.760
without punctures or holes.

0:00:39.760,0:00:43.410


0:00:43.410,0:00:47.930
There is a scientific name in[br]mathematics for such a domain.

0:00:47.930,0:00:50.530
This is going to be[br]simply connected.

0:00:50.530,0:00:56.080


0:00:56.080,0:00:59.860
And this is a difficult[br]topological theorem,

0:00:59.860,0:01:02.040
but this is what we expect, OK?

0:01:02.040,0:01:03.600
And what does it mean?

0:01:03.600,0:01:04.720
What does it mean?

0:01:04.720,0:01:14.570
It means that in the C being[br]a Jordan curve was what?

0:01:14.570,0:01:15.470
How?

0:01:15.470,0:01:20.790
This was continuous,[br]no self intersections.

0:01:20.790,0:01:27.230


0:01:27.230,0:01:34.660
In such a case, we set up[br]M and N to be C1 functions.

0:01:34.660,0:01:39.440


0:01:39.440,0:01:43.480
And then we proceed through[br]the path integral of C.

0:01:43.480,0:01:46.690
Do you like this kind of C,[br]or you prefer a straight C?

0:01:46.690,0:01:57.200
The path integral of C of M[br]of xy dx class, N of xy, dy,

0:01:57.200,0:01:59.010
everything is in plane.

0:01:59.010,0:02:03.640
I'm sorry that I[br]cannot repeat that,

0:02:03.640,0:02:06.690
but we discussed that[br]time, is in the plane of 2.

0:02:06.690,0:02:10.139
And then what-- do[br]you remember in terms

0:02:10.139,0:02:16.120
of how this path integral,[br][INAUDIBLE] inside,

0:02:16.120,0:02:21.660
is connected to a double[br]integral over the whole domain.

0:02:21.660,0:02:26.400
In particular, do you remember--[br]this is easy to memorize--

0:02:26.400,0:02:28.270
but do you remember[br]what's inside?

0:02:28.270,0:02:36.800
Because for the final, you are[br]expected to know his result.

0:02:36.800,0:02:38.640
STUDENT: [INAUDIBLE]

0:02:38.640,0:02:43.170
PROFESSOR: N sub X.

0:02:43.170,0:02:44.540
STUDENT: Minus M sub Y.

0:02:44.540,0:02:47.120
PROFESSOR: Minus M[br]sub Y. [INAUDIBLE]

0:02:47.120,0:02:53.420
must M-- M and N-- M sub Y.[br]Here is the Y. Of course this

0:02:53.420,0:02:58.630
would be dA in plane,[br]and in the-- if you

0:02:58.630,0:03:02.490
want to represent this[br]in the general format,

0:03:02.490,0:03:05.630
the MdX minus the MdY.

0:03:05.630,0:03:08.300
Feel free to do that.

0:03:08.300,0:03:13.060
One was a correlary[br]or a consequence.

0:03:13.060,0:03:22.080
This theorem was that if I[br]were to take this big M to be

0:03:22.080,0:03:26.330
the minus Y as a function,[br]then this function N will

0:03:26.330,0:03:29.570
be plus X, what will I get?

0:03:29.570,0:03:37.932
I would get that minus[br]YdX plus NdY will be what?

0:03:37.932,0:03:40.127


0:03:40.127,0:03:40.960
STUDENT: [INAUDIBLE]

0:03:40.960,0:03:42.293
PROFESSOR: Two times, excellent.

0:03:42.293,0:03:43.590
You are very awake.

0:03:43.590,0:03:45.000
So I wanted to catch you.

0:03:45.000,0:03:46.210
I couldn't catch you.

0:03:46.210,0:03:48.320
I thought you would say[br]the A of the domain,

0:03:48.320,0:03:49.500
but you said it right.

0:03:49.500,0:03:52.780
You said Y is the[br]area of the domain.

0:03:52.780,0:03:55.700
You probably[br]already in your mind

0:03:55.700,0:04:01.820
did the math saying X sub X[br]is one, minus Y sub 1 is 1.

0:04:01.820,0:04:05.740
1 plus 1 is two, so the[br]two part [INAUDIBLE].

0:04:05.740,0:04:08.970
OK, so what did we do with it?

0:04:08.970,0:04:10.221
We just stared at it?

0:04:10.221,0:04:10.720
No.

0:04:10.720,0:04:11.845
We didn't just stare at it.

0:04:11.845,0:04:14.470
We did something nice[br]with it last time.

0:04:14.470,0:04:21.260
We proved that, finally,[br]that the area, this radius R

0:04:21.260,0:04:23.740
will be pi R[br]squared, and we also

0:04:23.740,0:04:28.100
proved that the area[br][INAUDIBLE] is what?

0:04:28.100,0:04:30.655
I'm testing you to[br]see if you remember.

0:04:30.655,0:04:31.280
STUDENT: AB pi.

0:04:31.280,0:04:31.620
PROFESSOR: AB pi.

0:04:31.620,0:04:32.340
Very good.

0:04:32.340,0:04:33.500
Or pi AB.

0:04:33.500,0:04:38.030
It's more, I like it the[br]way you said it, AB pi,

0:04:38.030,0:04:41.810
because pi is a transcendental[br]number, and you go around

0:04:41.810,0:04:44.590
and it's like partly[br]variable to put at the end.

0:04:44.590,0:04:47.490
And the real numbers[br]that could be anything,

0:04:47.490,0:04:54.630
so [INAUDIBLE] they are the[br]semi axes of the ellipse.

0:04:54.630,0:04:59.470
So we gain new knowledge and[br]we are ready to move forward.

0:04:59.470,0:05:02.220
And we're going to move[br]forward to something

0:05:02.220,0:05:10.590
called section 13.5, which[br]is the surface integral.

0:05:10.590,0:05:12.660
We will come back[br]to Green's Theorem

0:05:12.660,0:05:15.530
because there are[br]generalizations

0:05:15.530,0:05:18.680
of the Green's Theorem to[br]more complicate the case.

0:05:18.680,0:05:21.180
But in order to[br]see those, we have

0:05:21.180,0:05:23.650
to learn a little bit more.

0:05:23.650,0:05:27.610
In mathematics, you need to[br]know many things, many pieces

0:05:27.610,0:05:31.840
of the puzzle, and then[br]you put them together

0:05:31.840,0:05:34.280
to get the whole picture.

0:05:34.280,0:05:37.650
All right, so what[br]is 13.5 about?

0:05:37.650,0:05:39.220
This is just review.

0:05:39.220,0:05:46.900
13.5, if should be looking like[br]a friend, old friend, to you.

0:05:46.900,0:05:48.624
And I'll show you[br]in a minute why this

0:05:48.624,0:05:49.915
is called the surface integral.

0:05:49.915,0:05:54.440


0:05:54.440,0:05:57.950
I saw that US natives[br]don't pronounce integral,

0:05:57.950,0:05:59.640
they pronounce in-negral.

0:05:59.640,0:06:04.410
And everybody that I heard[br]in romance language-speaking

0:06:04.410,0:06:10.230
countries like Spanish,[br]Italian, Portuguese,

0:06:10.230,0:06:13.750
they put the T there[br]out, very visibly.

0:06:13.750,0:06:16.705
So it doesn't matter.

0:06:16.705,0:06:19.510
Even some accent difference[br]in different parts

0:06:19.510,0:06:22.046
of the United States[br]pronounce it differently.

0:06:22.046,0:06:25.040
So what is the surface[br]integral about?

0:06:25.040,0:06:32.250
It's about integrating a smooth[br]function, not a vector value,

0:06:32.250,0:06:34.530
but a real value function.

0:06:34.530,0:06:40.820
Let's say you have G or XY being[br]a nice interglobal function

0:06:40.820,0:06:46.350
over some surfaces.

0:06:46.350,0:06:50.230
And you say, I'm[br]going to take it,

0:06:50.230,0:07:03.360
double integral, over S of GDS,[br]where DS will be area level.

0:07:03.360,0:07:08.600


0:07:08.600,0:07:11.840
I had a student one time who[br]looked at two different books

0:07:11.840,0:07:16.210
and said, I have a problem[br]with this, [INAUDIBLE].

0:07:16.210,0:07:23.330
In one book it shows a[br]big, fat snake over S.

0:07:23.330,0:07:25.965
And in another book, a[br]double integral over it,

0:07:25.965,0:07:28.820
and I don't know which one it[br]is because I don't understand.

0:07:28.820,0:07:32.370
No matter how you denote it,[br]it's still a double integral.

0:07:32.370,0:07:33.260
You know why?

0:07:33.260,0:07:35.230
Because it's an[br]integral over a surface.

0:07:35.230,0:07:41.420
The same thing, integral over[br]a surface or a domain plane,

0:07:41.420,0:07:45.050
or anything two-dimensional[br]will be a double integral.

0:07:45.050,0:07:47.510
So it doesn't matter[br]how you denote it.

0:07:47.510,0:07:49.820
In the end, it's going[br]to be a double integral.

0:07:49.820,0:07:54.420
Now, what in the world[br]do we mean by that?

0:07:54.420,0:07:56.810
DS is an old friend of[br]yours, and I don't know

0:07:56.810,0:07:59.930
if you remember him at all.

0:07:59.930,0:08:08.300
He was infinitesimal element[br]on some curved or linear patch.

0:08:08.300,0:08:11.650
Imagine your favorite surface.

0:08:11.650,0:08:13.720
Let's assume it's a graph.

0:08:13.720,0:08:16.450
It doesn't have to be a graph,[br]but let's assume it's a graph.

0:08:16.450,0:08:19.700
And that's your[br]favorite surface S.

0:08:19.700,0:08:23.170
And then you draw[br]coordinate lines,

0:08:23.170,0:08:25.716
and you are looking at a patch.

0:08:25.716,0:08:28.580


0:08:28.580,0:08:32.080
And this patch looks small,[br]but it's not small enough.

0:08:32.080,0:08:35.030
I want this to be[br]infinitesimally small.

0:08:35.030,0:08:41.370
Imagine that these curvature[br]lines become closer and closer

0:08:41.370,0:08:42.400
to one another.

0:08:42.400,0:08:46.000
And then we look in the[br]directions of DX and DY,

0:08:46.000,0:08:48.980
and then you say, wait a[br]minute, I'm not in plane.

0:08:48.980,0:08:54.920
If I were in plane,[br]DA will be DX, DY.

0:08:54.920,0:08:59.240
If you work with [INAUDIBLE],[br]I will be DX with DY.

0:08:59.240,0:09:00.910
So we've matched[br]the orientation.

0:09:00.910,0:09:02.720
If you would change[br]DY, [INAUDIBLE]

0:09:02.720,0:09:04.760
put the minus in front.

0:09:04.760,0:09:08.310
But this happens[br]because-- thank God this

0:09:08.310,0:09:16.080
will be a rectangular 1 patch[br]in plane, in the plane of 2.

0:09:16.080,0:09:18.300
But what if you[br]were on the surface?

0:09:18.300,0:09:21.930
On the surface, you[br]don't have this animal.

0:09:21.930,0:09:25.570
You will have-- which animal--[br]I'm testing your knowledge.

0:09:25.570,0:09:27.430
I'm doing review with you.

0:09:27.430,0:09:29.140
For sure, you will[br]see something that

0:09:29.140,0:09:31.250
involves the S in the final.

0:09:31.250,0:09:33.840
Have you started browsing[br]through those finals

0:09:33.840,0:09:34.750
I sent you?

0:09:34.750,0:09:37.320
Just out of curiosity.

0:09:37.320,0:09:40.960
And do they look awful to you?

0:09:40.960,0:09:42.090
They look awful to you.

0:09:42.090,0:09:43.060
Come on.

0:09:43.060,0:09:45.070
I'm going to work with[br]you on some of those.

0:09:45.070,0:09:47.500
I don't want you to[br]have-- I don't want

0:09:47.500,0:09:49.190
you to be afraid of this final.

0:09:49.190,0:09:51.200
Because compared to[br]other exams that you'll

0:09:51.200,0:09:54.080
have in other courses,[br]where a lot of memorization

0:09:54.080,0:09:57.660
is emphasized, this[br]should not be a problem.

0:09:57.660,0:10:00.800
So you could go over[br]the types of problems

0:10:00.800,0:10:04.380
that are significant[br]in this course,

0:10:04.380,0:10:08.420
you will not have any-- you[br]shouldn't have any problem.

0:10:08.420,0:10:10.810
And I sent you three samples.

0:10:10.810,0:10:13.740
Didn't I send you three[br]samples with solutions?

0:10:13.740,0:10:17.312
Those are going to help[br]you once you read the exam

0:10:17.312,0:10:19.900
and you can go ahead[br]and try the exam

0:10:19.900,0:10:22.040
or go ahead, read the solutions.

0:10:22.040,0:10:26.780
If I give you more of that, then[br]you should be doctors in those,

0:10:26.780,0:10:30.320
and you would be able to[br]solve them yourselves.

0:10:30.320,0:10:33.240
What about this one?

0:10:33.240,0:10:34.570
This is not DA.

0:10:34.570,0:10:37.960
It's a DA times something.

0:10:37.960,0:10:44.070
There is some factor in front[br]of that, and why is that?

0:10:44.070,0:10:46.600
In case of Z equals[br]F of X and Y,

0:10:46.600,0:10:49.430
you should know that by heart,[br]and I know that some of you

0:10:49.430,0:10:50.400
know it.

0:10:50.400,0:10:52.495
You just have to ring[br]the bell, and I'll

0:10:52.495,0:10:53.850
start ringing the bell.

0:10:53.850,0:10:55.870
Look at my first step.

0:10:55.870,0:10:57.783
And now you know, right?

0:10:57.783,0:10:59.470
STUDENT: [INAUDIBLE] 1--

0:10:59.470,0:11:00.650
PROFESSOR: I start with 1.

0:11:00.650,0:11:01.700
You said it right.

0:11:01.700,0:11:02.830
1 plus--

0:11:02.830,0:11:03.590
STUDENT: F of X.

0:11:03.590,0:11:04.815
PROFESSOR: F of X--

0:11:04.815,0:11:06.180
STUDENT: F squared.

0:11:06.180,0:11:07.637
PROFESSOR: Squared plus--

0:11:07.637,0:11:08.470
STUDENT: [INAUDIBLE]

0:11:08.470,0:11:09.620
PROFESSOR: --SY squared.

0:11:09.620,0:11:11.270
So this what you're doing.

0:11:11.270,0:11:13.020
What are you going to do?

0:11:13.020,0:11:14.690
You're going to[br]do wait a minute.

0:11:14.690,0:11:18.130
This animal of mine,[br]that looks so scary,

0:11:18.130,0:11:21.431
this is nothing but what?

0:11:21.431,0:11:27.300
It's the same thing as, not[br]the picture, my picture.

0:11:27.300,0:11:32.150
It's going to be double integral[br]over a plane or domain D.

0:11:32.150,0:11:34.260
Well, I just said[br]goodbye to the picture,

0:11:34.260,0:11:37.250
but I find you are really smart.

0:11:37.250,0:11:41.150
I would have drawn the[br][INAUDIBLE] of a picture here.

0:11:41.150,0:11:43.760
This is S and this[br]is D. What is D?

0:11:43.760,0:11:46.810
It's the projection,[br]projects the shadow.

0:11:46.810,0:11:51.100
The projection of S[br]on the plane XY when

0:11:51.100,0:11:53.920
I have to deal with a graph.

0:11:53.920,0:11:57.380
So when I have to deal with a[br]graph, my life is really easy.

0:11:57.380,0:12:06.020
And I said I'd get double[br]integral over D of G of God

0:12:06.020,0:12:11.680
knows what in the end will[br]be a function of X and Y. OK?

0:12:11.680,0:12:15.900
And here I'm going to have[br]square root of this animal.

0:12:15.900,0:12:18.330
Let me change it,[br]F sub X squared

0:12:18.330,0:12:20.930
like-- because in this[br]one it is like that.

0:12:20.930,0:12:21.500
Plus 1.

0:12:21.500,0:12:24.280
It doesn't matter[br]where I put the 1.

0:12:24.280,0:12:25.120
DXDY.

0:12:25.120,0:12:31.060
DXDY will be like the area of an[br]infinitesimally small rectangle

0:12:31.060,0:12:34.250
based on displacement[br]DX and displacement DY

0:12:34.250,0:12:35.370
and disintegration.

0:12:35.370,0:12:37.350
So this is DA.

0:12:37.350,0:12:42.820
Make the distinction[br]between the DA and the DX.

0:12:42.820,0:12:46.950
Can I draw the two animals?

0:12:46.950,0:12:50.750
Let me try again.

0:12:50.750,0:12:55.490
So you have the[br]direction of X and Y.

0:12:55.490,0:12:59.300
You have to be imaginative and[br]see that some coordinate lines

0:12:59.300,0:13:02.990
are [INAUDIBLE] for fixing Y.

0:13:02.990,0:13:06.880
When I fix Y, I sliced a[br]lot like that very nicely.

0:13:06.880,0:13:11.710
That's the same piece of[br]cheese that I've been dreaming

0:13:11.710,0:13:13.150
because I didn't have lunch.

0:13:13.150,0:13:15.460
I was too busy not[br]to have any lunch.

0:13:15.460,0:13:17.400
So you slice it[br]up like that where

0:13:17.400,0:13:20.590
Y equals constant to slice[br]it up like that for X

0:13:20.590,0:13:21.700
equals constant.

0:13:21.700,0:13:24.950
What you get are so-called[br]coordinate lines.

0:13:24.950,0:13:27.440
So the coordinate[br]lines are [INAUDIBLE].

0:13:27.440,0:13:32.085
Y equals my zeros, and[br]X equals the zeros.

0:13:32.085,0:13:35.810
And when they get to be[br]many dense and refined,

0:13:35.810,0:13:42.310
your curvilinear element is[br]this-- between two curves

0:13:42.310,0:13:46.010
like this two curves like that.

0:13:46.010,0:13:48.190
Shrunk in the limit.

0:13:48.190,0:13:50.640
It's an infinitesimal element.

0:13:50.640,0:13:54.470
This shadow is going[br]to be a rectangle.

0:13:54.470,0:13:55.910
Say that again, Magdalena.

0:13:55.910,0:13:59.410
This is not just[br]delta X and delta Y.

0:13:59.410,0:14:03.650
This is DX and DY[br]because I shrink them

0:14:03.650,0:14:06.660
until it become[br]infinitesimally small.

0:14:06.660,0:14:11.160
So you can imagine,[br]which one is bigger?

0:14:11.160,0:14:13.984
DS is bigger, or DA is bigger?

0:14:13.984,0:14:14.900
STUDENT: DS is bigger.

0:14:14.900,0:14:16.080
PROFESSOR: DS is bigger.

0:14:16.080,0:14:16.920
DS is bigger.

0:14:16.920,0:14:18.640
And can I see it's true?

0:14:18.640,0:14:19.140
Yes.

0:14:19.140,0:14:22.875
Because for God's sake, this[br]is greater than 1, right?

0:14:22.875,0:14:28.110
And if I multiply the[br]little orange area, by that,

0:14:28.110,0:14:32.740
I'm going to get this,[br]which is greater than 1.

0:14:32.740,0:14:38.140
They could be equal when[br]both would be plainer, right?

0:14:38.140,0:14:41.320
If you have a plane or surface[br]on top of a plane or surface,

0:14:41.320,0:14:43.160
then you have two[br]tiny rectangles

0:14:43.160,0:14:48.110
and you have like a prism[br]between them, goes down.

0:14:48.110,0:14:50.030
But in general, the[br]curve in your [INAUDIBLE]

0:14:50.030,0:14:52.950
here-- let me make[br]him more curvilinear.

0:14:52.950,0:14:55.310
He looks so-- so square.

0:14:55.310,0:15:01.460
But he's between two lines,[br]but he's a curvilinear.

0:15:01.460,0:15:07.010
Dinah says that he belongs to a[br]curved surface, not a flat one.

0:15:07.010,0:15:08.020
All right.

0:15:08.020,0:15:11.240
When he could be flat,[br]these guys go away.

0:15:11.240,0:15:12.640
Zero and zero.

0:15:12.640,0:15:15.380
And that would be it.

0:15:15.380,0:15:23.750
If somebody else, they--[br]well, this is hard to imagine,

0:15:23.750,0:15:29.970
but what if it could[br]be a tiny-- this

0:15:29.970,0:15:32.030
would not be curvilinear, right?

0:15:32.030,0:15:37.290
But it would be something like[br]a rectangular patch of a plane.

0:15:37.290,0:15:40.170
You have a grid in that plane.

0:15:40.170,0:15:46.400
And then it's just-- DS[br]would be itself a rectangle.

0:15:46.400,0:15:48.860
When you project[br]that rectangle here,

0:15:48.860,0:15:51.840
it will still be a rectangle.

0:15:51.840,0:15:54.900
When we were little-- I mean,[br]little, we were in K-12,

0:15:54.900,0:15:59.230
we're smart in math better[br]than other people in class--

0:15:59.230,0:16:03.260
did you ever have to do[br]anything with the two areas?

0:16:03.260,0:16:04.680
I did.

0:16:04.680,0:16:05.670
This was the shadow.

0:16:05.670,0:16:08.550
The projection in this[br]was that [INAUDIBLE].

0:16:08.550,0:16:11.510
And do you know what[br]the relationship

0:16:11.510,0:16:14.190
would be if I have a plane.

0:16:14.190,0:16:17.520
I'm doing that for-- actually,[br]I'm doing that for Casey

0:16:17.520,0:16:20.030
because she has something[br]similar to that.

0:16:20.030,0:16:22.910
So imagine that[br]you have to project

0:16:22.910,0:16:27.650
a rectangle that's in plane to[br]a rectangle that is the shadow.

0:16:27.650,0:16:30.340
The rectangle is on the ground.

0:16:30.340,0:16:33.560
The flat ground.

0:16:33.560,0:16:35.970
What's the relationship[br]between the two ends?

0:16:35.970,0:16:40.470


0:16:40.470,0:16:42.879
STUDENT: [INAUDIBLE]

0:16:42.879,0:16:44.670
PROFESSOR: No matter[br]what it is, but assume

0:16:44.670,0:16:48.120
it's like a rectangle up[br]here and the shadow is also

0:16:48.120,0:16:50.205
a rectangle down here.

0:16:50.205,0:16:52.290
Obviously, the rectangle[br]down here, the shadow

0:16:52.290,0:16:55.810
will be much smaller than[br]this because this is oblique.

0:16:55.810,0:16:56.930
It's an oblique.

0:16:56.930,0:16:59.540
And assume that I[br]have this plane making

0:16:59.540,0:17:03.550
an angle, a fixed angle with[br]this laying on the table.

0:17:03.550,0:17:05.325
STUDENT: [INAUDIBLE]

0:17:05.325,0:17:06.200
PROFESSOR: Excellent.

0:17:06.200,0:17:06.991
STUDENT: --cosine--

0:17:06.991,0:17:08.720
PROFESSOR: Which one[br]is cosine of what?

0:17:08.720,0:17:12.069
So the S would be the[br]the equal sign of theta,

0:17:12.069,0:17:18.619
or the A will be the[br]S cosine of theta?

0:17:18.619,0:17:19.940
STUDENT: [INAUDIBLE] DA.

0:17:19.940,0:17:22.560
PROFESSOR: DA is the S[br]cosine of theta, a very smart

0:17:22.560,0:17:23.060
[INAUDIBLE].

0:17:23.060,0:17:24.680
How does she know [INAUDIBLE]?

0:17:24.680,0:17:25.868
STUDENT: Because it's[br]got to be less than one.

0:17:25.868,0:17:27.410
PROFESSOR: It's less[br]than one, right?

0:17:27.410,0:17:29.480
Cosine theta is[br]between zero and one,

0:17:29.480,0:17:31.585
so you think which one is less.

0:17:31.585,0:17:33.630
All right, very good.

0:17:33.630,0:17:36.700
So when you have a[br]simple example like that,

0:17:36.700,0:17:40.310
you were back to[br]your K-12, and you

0:17:40.310,0:17:42.130
were happy-- I[br]just meant we were

0:17:42.130,0:17:44.580
avoiding three years of exams.

0:17:44.580,0:17:47.580
We only have [INAUDIBLE].

0:17:47.580,0:17:49.990
But now exams became[br]serious, and look.

0:17:49.990,0:17:52.970
This is curvilinear[br]elemental variant.

0:17:52.970,0:17:58.920
So let me write it how[br]people call the S's then.

0:17:58.920,0:18:01.170
Some people call it[br]curvilinear elemental variant.

0:18:01.170,0:18:01.750
Yeah?

0:18:01.750,0:18:06.280
Many engineers I[br]talk to do that.

0:18:06.280,0:18:09.295


0:18:09.295,0:18:11.670
Now, I think we should just[br]call it surface area element.

0:18:11.670,0:18:14.700


0:18:14.700,0:18:18.213
[? I'm ?] a physicist, so you[br]also say surface area element.

0:18:18.213,0:18:22.550
So I think we should just[br]learn each other's language.

0:18:22.550,0:18:24.570
We are doing the same things.

0:18:24.570,0:18:27.930
We just-- we have a language[br]barrier between-- it's

0:18:27.930,0:18:31.100
not writing interdisciplinary,[br]so if we could establish

0:18:31.100,0:18:35.300
a little bit more work in[br]common, because there are so

0:18:35.300,0:18:38.380
many applications to[br]engineering of this thing,

0:18:38.380,0:18:41.150
you have no idea yet.

0:18:41.150,0:18:47.150
OK, let's pick a problem like[br]the ones we wrote in the book,

0:18:47.150,0:18:50.680
and see how hard it gets.

0:18:50.680,0:18:53.640
It shouldn't get very hard.

0:18:53.640,0:18:57.030
I'll start with one, the[br]only one, that is naturally

0:18:57.030,0:19:00.320
coming to your mind[br]right now, which would

0:19:00.320,0:19:03.690
be the one where G would be 1.

0:19:03.690,0:19:06.458
Somebody has to tell[br]me what that would be.

0:19:06.458,0:19:10.020


0:19:10.020,0:19:13.800
So guys, what if G would be 1?

0:19:13.800,0:19:14.915
STUDENT: [INAUDIBLE]

0:19:14.915,0:19:15.790
PROFESSOR: Very good.

0:19:15.790,0:19:17.520
It would be the[br]A of the surface.

0:19:17.520,0:19:21.740
I'm going to look for[br]some simple application.

0:19:21.740,0:19:22.550
Nothing is simple.

0:19:22.550,0:19:27.600
Why did we make this problem,[br]this book, so complicated?

0:19:27.600,0:19:28.480
OK, it' s good.

0:19:28.480,0:19:35.360
We can pick-- I can make[br]up a problem like this one.

0:19:35.360,0:19:39.440


0:19:39.440,0:19:41.420
But I can do a better job.

0:19:41.420,0:19:43.490
I can give you an[br]better example.

0:19:43.490,0:19:46.625
I'm looking at the[br]example 1 in section 13.5.

0:19:46.625,0:19:49.035
I'll give you[br]something like that

0:19:49.035,0:19:51.180
if I were to write an exam 1.

0:19:51.180,0:19:56.270
I put on it something[br]like Z equals

0:19:56.270,0:19:58.450
X squared plus 1 squared.

0:19:58.450,0:20:02.130
You know is my favorite[br]eggshell which is a [INAUDIBLE].

0:20:02.130,0:20:16.490


0:20:16.490,0:20:19.820
And somebody says,[br]I'm not interested

0:20:19.820,0:20:25.430
in the whole surface,[br]which is infinitely large.

0:20:25.430,0:20:29.880
I'm only interested in a[br]piece of a surface that

0:20:29.880,0:20:39.820
is above the disk D of[br]center O and radius 1.

0:20:39.820,0:20:41.210
So say, what, Magdalena?

0:20:41.210,0:20:48.080
Say that I want just[br]that part of the surface

0:20:48.080,0:20:54.030
that he's sitting above the[br]disk of center O and radius 1.

0:20:54.030,0:20:58.420
And I want to know how to[br]set up the surface integral.

0:20:58.420,0:21:02.194
Set up main surface[br]area integral.

0:21:02.194,0:21:07.420


0:21:07.420,0:21:10.000
And of course, when you[br]first see that you freak out

0:21:10.000,0:21:13.280
for a second, and then you say,[br]no, no, that's not a problem.

0:21:13.280,0:21:15.770
I know how to do that.

0:21:15.770,0:21:18.810
So example 1 out of[br]this section would

0:21:18.810,0:21:27.010
be a double integral over your[br]S. You have to call it names.

0:21:27.010,0:21:30.400
S. 1 instead of G and DS.

0:21:30.400,0:21:33.530


0:21:33.530,0:21:35.590
But then you say wait a minute.

0:21:35.590,0:21:40.380
I know that is true, but I[br]have to change it accordingly.

0:21:40.380,0:21:42.140
The same thing is here.

0:21:42.140,0:21:47.930
So I'm going to have it[br]over D. And D is the shadow,

0:21:47.930,0:21:52.770
DS is the plane of what?

0:21:52.770,0:21:55.440
Of 1 times.

0:21:55.440,0:21:59.790
I know I'm silly saying 1[br]times, but that's what it is.

0:21:59.790,0:22:07.400
Square root of-- S of X squared[br]plus S of Y squared plus 1.

0:22:07.400,0:22:13.930
DS, DY or DA as Rachel[br]said, somebody said.

0:22:13.930,0:22:15.457
Aaron said.

0:22:15.457,0:22:17.415
I don't know, you just[br]whispered, I should say.

0:22:17.415,0:22:20.330


0:22:20.330,0:22:21.550
All right.

0:22:21.550,0:22:28.050
So first of all, this[br]looks a little bit bad.

0:22:28.050,0:22:31.000
It makes me a[br]little bit nervous.

0:22:31.000,0:22:34.890
But in the end, with your[br]help, I'm going to do it.

0:22:34.890,0:22:37.941
And I'm going to do it by[br]using what kind of coordinates?

0:22:37.941,0:22:38.440
I'm--

0:22:38.440,0:22:39.560
STUDENT: [INAUDIBLE]

0:22:39.560,0:22:42.000
PROFESSOR: Former[br]coordinates of the Y and Z.

0:22:42.000,0:22:43.260
It would be a killer.

0:22:43.260,0:22:48.800
Double, double, square root[br]of 1 plus-- who's telling me

0:22:48.800,0:22:49.933
what's coming next?

0:22:49.933,0:22:50.880
STUDENT: 4X squared.

0:22:50.880,0:22:52.610
4X squared, excellent.

0:22:52.610,0:22:55.830
4R squared, you say.

0:22:55.830,0:22:56.947
STUDENT: [INAUDIBLE]

0:22:56.947,0:22:57.530
PROFESSOR: OK.

0:22:57.530,0:23:01.010
Let me write it with[br]X and Y, and then

0:23:01.010,0:23:02.670
realize that this is our square.

0:23:02.670,0:23:04.670
How about that?

0:23:04.670,0:23:08.140
And then I have DX,[br]DY over the domain D,

0:23:08.140,0:23:12.340
and now I finally become[br]smart and say I just

0:23:12.340,0:23:13.730
fooled around here.

0:23:13.730,0:23:17.450
I want to do it in four[br]coordinates finally.

0:23:17.450,0:23:22.210
And that means I'll say[br]zero to 2 pi for theta.

0:23:22.210,0:23:26.130
So that theta will be the[br]last of the [INAUDIBLE].

0:23:26.130,0:23:28.520
R will be from zero to 1.

0:23:28.520,0:23:33.110


0:23:33.110,0:23:35.030
And So what?

0:23:35.030,0:23:40.630
This is an ugly, fairly[br]ugly, I just [INAUDIBLE].

0:23:40.630,0:23:42.580
I don't know what[br]I'm going to do yet.

0:23:42.580,0:23:44.590
I reduced our confusion, right?

0:23:44.590,0:23:46.480
But I'm not done.

0:23:46.480,0:23:47.412
STUDENT: R.

0:23:47.412,0:23:51.890
PROFESSOR: R. Never forget it.

0:23:51.890,0:23:56.940
So if I didn't have this[br]R, I would be horrible.

0:23:56.940,0:23:58.580
Why would it be horrible?

0:23:58.580,0:24:01.250
Imagine you couldn't have the R.

0:24:01.250,0:24:02.550
STUDENT: [INAUDIBLE]

0:24:02.550,0:24:05.860
PROFESSOR: We have to look that[br]this thing in integral table

0:24:05.860,0:24:10.190
or some-- use the calculator,[br]which we are not allowed

0:24:10.190,0:24:12.110
to do in this kind of course.

0:24:12.110,0:24:14.350
So what do we do?

0:24:14.350,0:24:16.890
We say it's a new substitution.

0:24:16.890,0:24:19.070
I have an R. That's a blessing.

0:24:19.070,0:24:23.200
So U equals 4 squared plus 1.

0:24:23.200,0:24:29.890
DU equals 8R, DR. I[br]think R, DR is a block.

0:24:29.890,0:24:34.890
And I know that's what I'm[br]going to do is a U substitution.

0:24:34.890,0:24:36.260
And I'm almost there.

0:24:36.260,0:24:39.280


0:24:39.280,0:24:41.380
It's a pretty good[br]example, but the one

0:24:41.380,0:24:48.160
you have as a first example[br]in this section, 13.5,

0:24:48.160,0:24:50.070
it's a little bit[br]too computational.

0:24:50.070,0:24:52.760
It's not smart at all.

0:24:52.760,0:24:57.810
It has a similar function over a[br]rectangle, something like that.

0:24:57.810,0:24:59.560
But it's a little bit[br]too confrontational.

0:24:59.560,0:25:01.290
We are looking[br]for something that

0:25:01.290,0:25:04.920
is not going-- examples that are[br]going to be easy to do and not

0:25:04.920,0:25:09.380
involve too much heavy[br]competition by him, because you

0:25:09.380,0:25:10.640
do everything by him.

0:25:10.640,0:25:14.730
Not-- like you don't have[br]a calculator, et cetera.

0:25:14.730,0:25:19.590
And the exam is very[br]limited in time, DU over 8.

0:25:19.590,0:25:22.580
So you say OK, I'm[br]know what that is.

0:25:22.580,0:25:28.230
That's going to be the A of S.[br]And that is going to be 2 pi.

0:25:28.230,0:25:32.380
Why can't I be so confident[br]and pull 2 pi out?

0:25:32.380,0:25:33.340
STUDENT: [INAUDIBLE]

0:25:33.340,0:25:36.310
PROFESSOR: Because there[br]is no dependence on theta.

0:25:36.310,0:25:38.490
All right?

0:25:38.490,0:25:41.570
So I have that one.

0:25:41.570,0:25:46.340
And then you go all right,[br]integral, square of you

0:25:46.340,0:25:52.290
times the U over 8-- 1 over 8DU.

0:25:52.290,0:25:56.430
And I have to be careful[br]because when R is zero--

0:25:56.430,0:26:00.230
if I put zero and 1 here[br]like some of my students,

0:26:00.230,0:26:04.780
I'm dead meat, because I'm going[br]to lose a lot of credit, right?

0:26:04.780,0:26:06.570
So I have to pay attention.

0:26:06.570,0:26:09.100
R is 0, and U equals?

0:26:09.100,0:26:09.890
STUDENT: 1.

0:26:09.890,0:26:10.730
PROFESSOR: 1.

0:26:10.730,0:26:12.970
R equals 1.

0:26:12.970,0:26:16.760
U equals 5.

0:26:16.760,0:26:23.120
And I worked this out[br]and I should be done.

0:26:23.120,0:26:27.560
And that's-- you should[br]expect something like that.

0:26:27.560,0:26:33.630
Nice, not computational,[br]you kind of looking.

0:26:33.630,0:26:36.800
What is integral of square of U?

0:26:36.800,0:26:38.460
STUDENT: [INAUDIBLE]

0:26:38.460,0:26:41.690
PROFESSOR: So you have--[br]you do the three halves,

0:26:41.690,0:26:43.940
and you pull out the 2/3, right?

0:26:43.940,0:26:45.160
That's what you do.

0:26:45.160,0:26:49.110
And then you go between U equals[br]1 down, and U equals 5 up.

0:26:49.110,0:26:51.895
And it's like one of those[br]examples we worked before.

0:26:51.895,0:26:53.270
Remember, and more[br]important, you

0:26:53.270,0:26:58.460
had something like[br]that for surface area?

0:26:58.460,0:27:00.230
Oh, my god.

0:27:00.230,0:27:01.240
4 over 8.

0:27:01.240,0:27:03.108
How much is 4 over 8?

0:27:03.108,0:27:04.020
STUDENT: [INAUDIBLE]

0:27:04.020,0:27:04.853
PROFESSOR: One half.

0:27:04.853,0:27:08.760


0:27:08.760,0:27:09.260
Right?

0:27:09.260,0:27:15.330
So we will have 1 over[br]6, and write pi times 5

0:27:15.330,0:27:20.280
to the three halves minus 1.

0:27:20.280,0:27:21.600
So do I like it?

0:27:21.600,0:27:22.970
I would leave it like that.

0:27:22.970,0:27:23.470
I'm fine.

0:27:23.470,0:27:24.580
I'll forget about it.

0:27:24.580,0:27:26.640
I have people who care.

0:27:26.640,0:27:32.042
I don't care how some people[br]write it-- 5 with 5 minus 1

0:27:32.042,0:27:33.500
because they think[br]it looks better.

0:27:33.500,0:27:34.000
It doesn't.

0:27:34.000,0:27:37.150
That's the scientific[br]equation, and I'm fine with it.

0:27:37.150,0:27:38.416
Right?

0:27:38.416,0:27:39.260
OK.

0:27:39.260,0:27:43.320
So expect something like--[br]maybe I'm talking too much,

0:27:43.320,0:27:46.750
but maybe it's a good thing[br]to tell you what to expect

0:27:46.750,0:27:48.500
because we have to [INAUDIBLE].

0:27:48.500,0:27:50.500
At the same time, we're[br]teaching new things

0:27:50.500,0:27:54.970
as staff instructors doing[br]review of what's important.

0:27:54.970,0:28:03.050
I'm thinking if I'm doing things[br]right and at the same pace,

0:28:03.050,0:28:10.110
I should be finished[br]with chapter 13

0:28:10.110,0:28:12.230
at the end of next week.

0:28:12.230,0:28:15.460
Because after 13.5,[br]we have 13.6 which

0:28:15.460,0:28:18.125
is a generalization[br]of Green's Theorem.

0:28:18.125,0:28:20.810
13.6 as you recall is[br]called Stokes' Theorem.

0:28:20.810,0:28:25.090
13.7 is also a generalization[br]of Green's Theorem.

0:28:25.090,0:28:27.190
And they are all related.

0:28:27.190,0:28:31.500
It's like the trinity[br]on [INAUDIBLE].

0:28:31.500,0:28:33.170
That's the Divergence Theorem.

0:28:33.170,0:28:37.970
That is the last section,[br]13.7, Divergence Theorem.

0:28:37.970,0:28:41.580
So if I am going[br]at the right pace,

0:28:41.580,0:28:44.090
by-- what is next[br]wee on Thursday?

0:28:44.090,0:28:47.995
The-- 23rd?

0:28:47.995,0:28:50.840
I should be more or less[br]done with the chapter.

0:28:50.840,0:28:54.280
And I'm thinking I have[br]all the time in the world

0:28:54.280,0:28:57.410
to review with you[br]from that moment on.

0:28:57.410,0:28:59.490
In which sense are[br]we going to review?

0:28:59.490,0:29:04.740
We are going to review[br]by solving past finals.

0:29:04.740,0:29:05.610
Right?

0:29:05.610,0:29:08.881
That's what we are-- that's[br]what I'm planning to do.

0:29:08.881,0:29:12.380
I'm going to erase this[br]and move on to something

0:29:12.380,0:29:15.340
more spectacular.

0:29:15.340,0:29:16.250
Many-- OK.

0:29:16.250,0:29:18.310
This second part that[br]I want to teach you

0:29:18.310,0:29:23.030
now about, many instructors[br]in regular courses

0:29:23.030,0:29:28.590
just skip it because they do[br]not want to teach you-- not you,

0:29:28.590,0:29:29.610
you are honor students.

0:29:29.610,0:29:31.318
But they don't want[br]to teach the students

0:29:31.318,0:29:36.070
about some more general[br]ways to look at a surface.

0:29:36.070,0:29:40.235
Remember, guys, a surface[br]that is written like that

0:29:40.235,0:29:43.110
is called a graph.

0:29:43.110,0:29:47.560
But not all the[br]surfaces were graphs.

0:29:47.560,0:29:56.820
And actually for a surface[br]S, what the most general way

0:29:56.820,0:30:00.058
to represent the presentation[br]would be a parameterization.

0:30:00.058,0:30:05.560


0:30:05.560,0:30:10.450
And I'll do a little bit[br]of a review for those.

0:30:10.450,0:30:17.640
R-- little R or big R--[br]big R, because that's

0:30:17.640,0:30:20.540
the position vector the[br]way I serve it to you

0:30:20.540,0:30:24.350
on a plate, whether,[br]for curves in space.

0:30:24.350,0:30:29.113
I say that's R of P. And when[br]we moved on curves to surfaces,

0:30:29.113,0:30:34.430
I said you move your path[br]two directions of motion.

0:30:34.430,0:30:37.010
You have two-- what are[br]those called in mechanics?

0:30:37.010,0:30:38.440
Degrees of freedom.

0:30:38.440,0:30:40.570
So you have two[br]degrees of freedom

0:30:40.570,0:30:42.400
like latitude and longitude.

0:30:42.400,0:30:46.960
Then R belongs--[br]the position vector

0:30:46.960,0:30:52.710
is a function of two variables,[br]and it belongs to R3,

0:30:52.710,0:30:54.320
because it's a vector in R3.

0:30:54.320,0:30:58.220
And want to have-- imagine[br]that my hand is a surface.

0:30:58.220,0:30:59.550
Well, OK.

0:30:59.550,0:31:02.400
This is the position vector, I'm[br]just kind of sweeping my hand,

0:31:02.400,0:31:04.930
going this way, one[br]degree of freedom.

0:31:04.930,0:31:07.330
Or going that way, the[br]other degree of freedom.

0:31:07.330,0:31:10.600
This is what[br]parameterization is.

0:31:10.600,0:31:16.645
So for a sphere, if you want to[br]parameterize the whole sphere--

0:31:16.645,0:31:20.020
I'll be done in a second.

0:31:20.020,0:31:23.420
I need you to see[br]if you remember

0:31:23.420,0:31:25.400
how to parameterize a sphere.

0:31:25.400,0:31:26.450
I'm testing you.

0:31:26.450,0:31:28.400
I'm mean today.

0:31:28.400,0:31:29.750
So examples.

0:31:29.750,0:31:32.376
Example 1 is[br]parameterize a sphere.

0:31:32.376,0:31:36.660


0:31:36.660,0:31:38.310
Was it hard?

0:31:38.310,0:31:40.550
That was a long[br]time ago, my god.

0:31:40.550,0:31:45.570
X, Y, and Z are what?

0:31:45.570,0:31:47.880
Latitude from Santa Clause.

0:31:47.880,0:31:52.000
Always latitude from[br]the North Pole is 5.

0:31:52.000,0:31:54.390
Longitude is from zero to 5.

0:31:54.390,0:31:56.560
The meridian is zero to 5.

0:31:56.560,0:31:59.850
That was theta, the[br]parameter of theta.

0:31:59.850,0:32:04.940
R was the distance[br]from this to a point.

0:32:04.940,0:32:08.260
But R was allowed to be[br]from-- take many values.

0:32:08.260,0:32:11.870
Now if I'm moving on[br]a sphere of radius

0:32:11.870,0:32:15.634
A-- let me make[br]that radius a just

0:32:15.634,0:32:16.800
to make your life miserable.

0:32:16.800,0:32:19.840
Assume that A would[br]be a sample, A.

0:32:19.840,0:32:22.311
How am I going to write[br]that parameterization?

0:32:22.311,0:32:25.080
STUDENT: X equals[br]A plus [INAUDIBLE]?

0:32:25.080,0:32:28.960
PROFESSOR: A something,[br]A something, A something.

0:32:28.960,0:32:30.150
STUDENT: A [INAUDIBLE]

0:32:30.150,0:32:31.110
PROFESSOR: He is right.

0:32:31.110,0:32:32.067
I have to move on.

0:32:32.067,0:32:32.900
STUDENT: [INAUDIBLE]

0:32:32.900,0:32:34.510
PROFESSOR: Go slow.

0:32:34.510,0:32:37.370
So I have-- the last[br]one-- you were right,

0:32:37.370,0:32:40.250
Buddy, you have the[br]memory of a medical doctor

0:32:40.250,0:32:42.340
and some day you will[br]be a medical doctor.

0:32:42.340,0:32:44.510
Not everybody has a good memory.

0:32:44.510,0:32:49.980
So the way you can do that[br]is, wait a minute, this is pi,

0:32:49.980,0:32:50.670
right?

0:32:50.670,0:32:52.250
This [INAUDIBLE].

0:32:52.250,0:32:55.330
If you want the Z, you[br]start with that first.

0:32:55.330,0:33:00.588
And since Z is adjacent, you[br]go R, cosine, sine, phi equals

0:33:00.588,0:33:01.800
sine phi.

0:33:01.800,0:33:05.079
Now we started with X[br]because he's worked on this

0:33:05.079,0:33:06.120
and remembers everything.

0:33:06.120,0:33:07.590
He has it memorized.

0:33:07.590,0:33:10.430
Sine phi for both.

0:33:10.430,0:33:13.520
And times what in both cases?

0:33:13.520,0:33:15.310
He's just the guy[br]who's not here.

0:33:15.310,0:33:17.020
So sine phi.

0:33:17.020,0:33:20.820
It helps to memorize N[br]cosine theta, and sine theta.

0:33:20.820,0:33:22.600
Is that really easy to memorize?

0:33:22.600,0:33:26.270
So where phi was the[br]latitude from the North

0:33:26.270,0:33:30.080
Pole between zero[br]and phi, it theta

0:33:30.080,0:33:35.540
was the longitude-- excuse[br]me, guys-- longitude from zero

0:33:35.540,0:33:41.410
to 2 pi, all around one more.

0:33:41.410,0:33:43.920
So you say wait a[br]minute, Magdalena,

0:33:43.920,0:33:45.190
these are Euler's angle.

0:33:45.190,0:33:47.280
What do they call in mechanics?

0:33:47.280,0:33:49.670
I think they call[br]them Euler angles.

0:33:49.670,0:33:53.410
But anyway, for[br]phi theta, we call

0:33:53.410,0:33:55.620
them latitude and longitude.

0:33:55.620,0:33:59.990
I'll replace them, because look,[br]I want R to be in terms of U,V.

0:33:59.990,0:34:02.390
So in mathematics, it's[br]not about location.

0:34:02.390,0:34:05.470
We can call them[br]whatever we want.

0:34:05.470,0:34:10.409
Mathematics is about the freedom[br]to call people names-- no--

0:34:10.409,0:34:13.760
to call things names[br]and people names--

0:34:13.760,0:34:15.909
STUDENT: Could U not equal zero?

0:34:15.909,0:34:16.540
PROFESSOR: Who?

0:34:16.540,0:34:17.840
STUDENT: U.

0:34:17.840,0:34:18.530
PROFESSOR: Yes.

0:34:18.530,0:34:19.076
So U can--

0:34:19.076,0:34:19.909
STUDENT: [INAUDIBLE]

0:34:19.909,0:34:24.520
PROFESSOR: --yeah, but[br]why didn't I write zero?

0:34:24.520,0:34:25.469
Well--

0:34:25.469,0:34:26.929
STUDENT: [INAUDIBLE][br]makes sense.

0:34:26.929,0:34:30.330
PROFESSOR: --because,[br]yeah, you can take both.

0:34:30.330,0:34:33.040
If I want to study[br]differentiability,

0:34:33.040,0:34:36.500
I usually have to take it less[br]than and less than and less

0:34:36.500,0:34:39.050
than and less than because we[br]studied differentiability on

0:34:39.050,0:34:40.190
[INAUDIBLE].

0:34:40.190,0:34:43.310
But right now, I can take them[br]from the North Pole itself

0:34:43.310,0:34:47.090
to the South Pole itself-- so.

0:34:47.090,0:34:50.610
I'm not deleting any meridian.

0:34:50.610,0:34:53.739
If I were-- suppose[br]I were to delete it.

0:34:53.739,0:34:55.960
By the way, what does this mean?

0:34:55.960,0:34:57.030
I'm just kidding.

0:34:57.030,0:34:58.030
I'll put it back.

0:34:58.030,0:35:00.910
But Alex had a smart[br]question over there,

0:35:00.910,0:35:04.030
and he made me thinking.

0:35:04.030,0:35:06.600
It's a dangerous thing[br]when people make you think.

0:35:06.600,0:35:10.260
So it goes from zero to 2 pi.

0:35:10.260,0:35:12.050
Why would that be?

0:35:12.050,0:35:15.100
Imagine you have all the[br]meridians in the world

0:35:15.100,0:35:17.680
except for one.

0:35:17.680,0:35:22.470
From the sphere, you cut it and[br]remove the Greenwich meridian,

0:35:22.470,0:35:25.560
the one that passes[br]through Greenwich Village.

0:35:25.560,0:35:29.960
The one-- not the one in New[br]York, the one next to London,

0:35:29.960,0:35:31.400
right?

0:35:31.400,0:35:32.500
So put it back.

0:35:32.500,0:35:34.800
Put that meridian back.

0:35:34.800,0:35:38.660
It's like you take an[br]orange, and you make a slice.

0:35:38.660,0:35:40.240
I am-- OK.

0:35:40.240,0:35:43.050
Stop with the fruit[br]because I'm hungry.

0:35:43.050,0:35:47.180
Now, example two.

0:35:47.180,0:35:52.730
Now, imagine another surface[br]area you're used to, the what?

0:35:52.730,0:35:58.490
The paraboloid is one of our[br]favorite guys this semester.

0:35:58.490,0:36:00.170
X squared plus Y squared.

0:36:00.170,0:36:02.127
What is the[br]parameterization of that?

0:36:02.127,0:36:06.950


0:36:06.950,0:36:09.680
Well, if I write it[br]like that, it's a graph.

0:36:09.680,0:36:11.600
But if I don't want to[br]write it as a graph,

0:36:11.600,0:36:14.300
I have to write[br]it as a parameter.

0:36:14.300,0:36:16.070
What am I going to do?

0:36:16.070,0:36:19.380
I really know X to be U, right?

0:36:19.380,0:36:21.380
That's the simplest[br]choice possible.

0:36:21.380,0:36:25.300
Y could be V. And then Z will[br]be U squared plus V squared.

0:36:25.300,0:36:26.360
And there I am.

0:36:26.360,0:36:26.860
[SNEEZE]

0:36:26.860,0:36:29.110
So I'm going to write-- bless[br]your heart, [INAUDIBLE].

0:36:29.110,0:36:31.530


0:36:31.530,0:36:38.320
V plus J plus U squared[br]plus V squared, K. So this

0:36:38.320,0:36:44.410
is the parameterization[br]of a paraboloid.

0:36:44.410,0:36:46.734
That one of them--[br]there are infinitely

0:36:46.734,0:36:48.650
many-- the one that comes[br]to mind because it's

0:36:48.650,0:36:52.600
the easiest one to think about.

0:36:52.600,0:36:53.560
STUDENT: [INAUDIBLE].

0:36:53.560,0:36:54.520
PROFESSOR: Good.

0:36:54.520,0:36:59.060
For a minute, guys,[br]you didn't need me.

0:36:59.060,0:37:02.390
You didn't need me at all[br]to come up with those.

0:37:02.390,0:37:05.870
But maybe you would need me[br]to remember, or maybe not--

0:37:05.870,0:37:08.480
to remind you of the helicoid.

0:37:08.480,0:37:09.810
Helicoid.

0:37:09.810,0:37:13.250
Did you go to the,[br]as I told you to go

0:37:13.250,0:37:15.710
to the [INAUDIBLE] spectrum--[br]what was that called?

0:37:15.710,0:37:15.860
The--

0:37:15.860,0:37:16.530
STUDENT: Science spectrum.

0:37:16.530,0:37:17.760
PROFESSOR: Science spectrum.

0:37:17.760,0:37:24.620
And dip into soap solution[br]the thingy was-- a metal

0:37:24.620,0:37:27.950
rod with a-- with a what?

0:37:27.950,0:37:34.210
With; a helix made[br]of metal so the soap

0:37:34.210,0:37:37.040
film would take which shape?

0:37:37.040,0:37:42.230
The shape of this spiral that's[br]going to go inside here, right?

0:37:42.230,0:37:45.630
That's called a helicoid.

0:37:45.630,0:37:46.150
OK.

0:37:46.150,0:37:46.720
All right.

0:37:46.720,0:37:47.595
You're not mad at me.

0:37:47.595,0:37:48.094
STUDENT: No.

0:37:48.094,0:37:49.010
PROFESSOR: OK, good.

0:37:49.010,0:37:54.410
So in this case, R[br]of UV will be what?

0:37:54.410,0:37:57.690
It was a long time ago, once[br]upon a time I gave it to you.

0:37:57.690,0:38:00.290
It's extremely hard to[br]memorize if you don't work

0:38:00.290,0:38:03.560
with it on a regular basis.

0:38:03.560,0:38:06.940
If it were a helix,[br]what would it be?

0:38:06.940,0:38:09.970
If it were a helix, it[br]would be R of T right?

0:38:09.970,0:38:14.320
It would be like equal[br]sign T, A sine T, BT.

0:38:14.320,0:38:17.190
Say it again, Magdalena,[br]that was a long time ago,

0:38:17.190,0:38:17.970
chapter 10.

0:38:17.970,0:38:19.690
Chapter 10.

0:38:19.690,0:38:23.960
Equal sign, T, A sine[br]T, MBT, standard helix.

0:38:23.960,0:38:25.350
This is not going to be that.

0:38:25.350,0:38:33.310
It's going to be-- U cosine B.[br]U sine B. Look at the picture.

0:38:33.310,0:38:37.691
And imagine that these guys[br]are extended to infinity.

0:38:37.691,0:38:39.190
It's not just the[br]stairs themselves,

0:38:39.190,0:38:41.460
or whatever they are.

0:38:41.460,0:38:46.350
There are infinite lines,[br]straight lines, and busy.

0:38:46.350,0:38:47.760
This is done.

0:38:47.760,0:38:49.140
NB is a positive constant.

0:38:49.140,0:38:51.760


0:38:51.760,0:38:56.520
But your parameters are[br]U and V. Any other guy

0:38:56.520,0:39:00.400
that comes to mind, I'm out[br]of imagination right now.

0:39:00.400,0:39:04.520
You can do a torus on the[br]fold that looks like a donut.

0:39:04.520,0:39:05.740
You will have two parameters.

0:39:05.740,0:39:08.450
Imagine a donut.

0:39:08.450,0:39:10.970
How do you-- I'm not[br]going to write that.

0:39:10.970,0:39:13.480
Eventually I could give you[br]that as an extra credit thing.

0:39:13.480,0:39:18.660
What are the two degrees of[br]freedom of moving on the donut,

0:39:18.660,0:39:21.422
assuming that you would[br]like to move in circles?

0:39:21.422,0:39:25.417


0:39:25.417,0:39:26.250
STUDENT: [INAUDIBLE]

0:39:26.250,0:39:29.750


0:39:29.750,0:39:32.010
PROFESSOR: Let me draw a[br]donut, because I'm hungry,

0:39:32.010,0:39:34.720
and I really-- I cannot help it.

0:39:34.720,0:39:38.450
I just have to-- this is[br]called a torus in mathematics.

0:39:38.450,0:39:42.744
And you'll have-- one degree[br]of freedom will be like this,

0:39:42.744,0:39:44.660
the other degree of[br]freedom will be like that.

0:39:44.660,0:39:46.880
This is U and B.[br]Instead of U and B,

0:39:46.880,0:39:50.110
mathematicians,[br]apologists, geometers,

0:39:50.110,0:39:53.370
they call those angles phi[br]and theta because they really

0:39:53.370,0:39:56.100
are between zero and 2 pi.

0:39:56.100,0:39:59.800
It has a rotation like[br]that along the donut.

0:39:59.800,0:40:02.180
You can cut, slice[br]the donut, or if they

0:40:02.180,0:40:04.450
don't put cheese filling in it.

0:40:04.450,0:40:08.790
That was a bad idea not[br]having anything to eat.

0:40:08.790,0:40:15.300
And the other angle will be your[br]2 pi along this little circle.

0:40:15.300,0:40:18.380
So you still have two degrees[br]of freedom on a donut.

0:40:18.380,0:40:19.230
It's a surface.

0:40:19.230,0:40:20.210
You can write the[br]parameterization.

0:40:20.210,0:40:20.860
Yes?

0:40:20.860,0:40:22.443
STUDENT: Why is a[br]pie this way around.

0:40:22.443,0:40:25.870
Why is it like [INAUDIBLE].

0:40:25.870,0:40:27.710
PROFESSOR: It[br]doesn't have to be.

0:40:27.710,0:40:29.710
STUDENT: Or is it just[br]kind of like [INAUDIBLE]?

0:40:29.710,0:40:31.420
PROFESSOR: That's[br]what they call it.

0:40:31.420,0:40:32.480
Yeah.

0:40:32.480,0:40:35.230
So they are between 2 and 2 pi.

0:40:35.230,0:40:41.780
While I erase-- or should[br]I-- enough expectation

0:40:41.780,0:40:44.380
in terms of parameterization,[br]I have to night

0:40:44.380,0:40:47.280
teach you something about that.

0:40:47.280,0:40:53.480
If somebody would say I'm[br]giving you a patch of a surface,

0:40:53.480,0:40:57.440
but that patch of a[br]surface is in a frame--

0:40:57.440,0:41:01.700
it's a nice parameterization.

0:41:01.700,0:41:04.110
This is the P on the surface.

0:41:04.110,0:41:07.520


0:41:07.520,0:41:10.510
And you say, well,[br]the parameterization

0:41:10.510,0:41:14.190
is going to be R[br]of U and V equals

0:41:14.190,0:41:22.110
X of UVI plus Y of[br]UVJ plus Z of UVK.

0:41:22.110,0:41:25.090


0:41:25.090,0:41:28.006
And suppose that somebody says[br]this is you favorite test.

0:41:28.006,0:41:30.880


0:41:30.880,0:41:35.990
Find V. Well, that[br]would be absurd.

0:41:35.990,0:41:37.420
My god, how do we do that?

0:41:37.420,0:41:48.670
Find the flux[br]corresponding to-- do

0:41:48.670,0:41:51.840
we say restart--[br]just a second-- just

0:41:51.840,0:41:54.130
to restart with applications.

0:41:54.130,0:41:54.630
[INAUDIBLE]

0:41:54.630,0:41:57.270


0:41:57.270,0:42:01.490
We don't say what kind of[br]vector field that it is,

0:42:01.490,0:42:06.820
but we will say plus[br]corresponding to the vector

0:42:06.820,0:42:08.417
field.

0:42:08.417,0:42:09.000
F [INAUDIBLE].

0:42:09.000,0:42:14.380


0:42:14.380,0:42:16.300
And this vector[br]field, I'll tell you

0:42:16.300,0:42:22.450
in a second what's expected[br]from this to be a vector field.

0:42:22.450,0:42:35.130
Through, on the surface, we[br]find on the surface-- yes.

0:42:35.130,0:42:39.150
Mathematicians say[br]define normal surface S.

0:42:39.150,0:42:44.460
But a physicist will[br]say flux through,

0:42:44.460,0:42:55.350
the flux corresponding[br]to F through the surface.

0:42:55.350,0:42:58.702


0:42:58.702,0:43:01.170
Yes.

0:43:01.170,0:43:03.940
So you'll say why would that[br]be, and what is the flux?

0:43:03.940,0:43:11.955
By definition, how[br]should we denote it?

0:43:11.955,0:43:16.370
Let's make a beautiful script[br]F. That's crazy, right?

0:43:16.370,0:43:21.590
And then it goes doubling[br]over the surface F test.

0:43:21.590,0:43:24.910
Is anybody mechanical[br]engineering here?

0:43:24.910,0:43:28.380
Do you know the flux formula?

0:43:28.380,0:43:33.980
It's going to be [INAUDIBLE][br]over S of F, this magic thing.

0:43:33.980,0:43:36.890
Not DN, DS.

0:43:36.890,0:43:38.960
Do you know what N means?

0:43:38.960,0:43:41.250
What it is N for[br]mechanical engineering,

0:43:41.250,0:43:42.940
[INAUDIBLE] engineers?

0:43:42.940,0:44:01.070
N to would be the unit normal[br]vector field to the surface S.

0:44:01.070,0:44:03.050
How would you want[br]to imagine that?

0:44:03.050,0:44:07.110
You would have a surface, and[br]you have this velocity vectors

0:44:07.110,0:44:13.350
here at the bottom that goes[br]to S. And this field goes up.

0:44:13.350,0:44:17.460
You'll have a force and[br]acceleration, velocity,

0:44:17.460,0:44:19.850
you have everything[br]going this way.

0:44:19.850,0:44:22.800
And you want to find[br]out what happens.

0:44:22.800,0:44:26.720
You introduce this notion[br]of flux through the surface.

0:44:26.720,0:44:29.400
Another way to have a[br]flux through the surface

0:44:29.400,0:44:31.460
maybe through the same[br]surface but associated

0:44:31.460,0:44:34.040
through another[br]kind of concept--

0:44:34.040,0:44:36.180
if there could be[br]something else.

0:44:36.180,0:44:39.960
In electromagnetism, F would be[br]something else, some other type

0:44:39.960,0:44:41.075
of vector field.

0:44:41.075,0:44:42.145
Yes, sir.

0:44:42.145,0:44:43.020
STUDENT: [INAUDIBLE].

0:44:43.020,0:44:45.550
PROFESSOR: So find out, by[br]the way until next time,

0:44:45.550,0:44:49.390
if you were an electrical[br]engineering major, what

0:44:49.390,0:44:51.930
would flux be for you guys?

0:44:51.930,0:44:55.910
Two surfaces, one would be the[br]meaning of the vector field

0:44:55.910,0:44:58.280
F for you, and[br]why would you care

0:44:58.280,0:45:01.240
about the electromagnetic[br]flux or something like that.

0:45:01.240,0:45:02.920
I don't want to talk[br]too much about it.

0:45:02.920,0:45:07.150
It's for you to do the[br]search and find out.

0:45:07.150,0:45:09.880
So suppose that[br]somebody gives you

0:45:09.880,0:45:13.300
this notion that says you[br]have a parameteric surface.

0:45:13.300,0:45:19.050
Give an application[br]of that and find out

0:45:19.050,0:45:24.166
how you're going[br]be deal with it.

0:45:24.166,0:45:27.840
I'll give you an example[br]that shouldn't be too hard.

0:45:27.840,0:45:32.700


0:45:32.700,0:45:34.440
I'll make up my own example.

0:45:34.440,0:45:38.469
And looks like example 6, but[br]it's going to be different.

0:45:38.469,0:45:47.391


0:45:47.391,0:45:47.891
Example.

0:45:47.891,0:45:51.330


0:45:51.330,0:45:58.600
Find the flux F if F will[br]be a simple function.

0:45:58.600,0:46:05.235
Let's say something equals X, I[br]plus Y,J Z, K at every point X,

0:46:05.235,0:46:13.030
Y-- at every point[br]of the space XYZ.

0:46:13.030,0:46:16.380
That means you could have this[br]vector field defined everywhere

0:46:16.380,0:46:18.360
in space in [INAUDIBLE].

0:46:18.360,0:46:22.660
But you only care about[br]this acting on the surface.

0:46:22.660,0:46:25.064
So it's acting on the surface.

0:46:25.064,0:46:28.000


0:46:28.000,0:46:30.330
And then what will the flux be?

0:46:30.330,0:46:34.570
On the surface, which surface?

0:46:34.570,0:46:41.012
My favorite one, Z equals[br]X squared plus Y squared.

0:46:41.012,0:46:47.290


0:46:47.290,0:46:49.260
First of all, you say[br]wait, wait, Magdalena,

0:46:49.260,0:46:50.990
do you want to do it like that?

0:46:50.990,0:46:54.330
Do you want to say F[br]over XY to be a graph?

0:46:54.330,0:46:58.800
Or do you want to consider it[br]as a parameterized surface?

0:46:58.800,0:47:01.915
And that means it's the same[br]thing, equivalent to or if

0:47:01.915,0:47:09.000
and only if, who tells me again[br]what R was for such a surface?

0:47:09.000,0:47:09.500
STUDENT: XI.

0:47:09.500,0:47:11.380
PROFESSOR: X is[br]U. Y is V, so U--

0:47:11.380,0:47:12.213
STUDENT: [INAUDIBLE]

0:47:12.213,0:47:15.750
PROFESSOR: --I, that[br]would be J, then good.

0:47:15.750,0:47:19.720
U squared plus U squared UK.

0:47:19.720,0:47:23.300
Well, when you say that,[br]we have-- first of all,

0:47:23.300,0:47:26.920
we have no idea what[br]the heck we need to do,

0:47:26.920,0:47:32.450
because do we want to do it[br]in this form like a graph?

0:47:32.450,0:47:34.060
Or do we want to do[br]it parameterized?

0:47:34.060,0:47:37.340
We have to set up[br]formulas for the flats.

0:47:37.340,0:47:38.650
It's not so easy.

0:47:38.650,0:47:43.250
So assume that we are brave[br]enough and we start everything.

0:47:43.250,0:47:48.500
I want to understand what[br]flux really is as an integral.

0:47:48.500,0:47:55.510
And let me set it up for the[br]first case, the case of Z

0:47:55.510,0:47:58.390
equals F of X and Y.[br]And I'm happy with it

0:47:58.390,0:48:01.300
because that's[br]the simplest case.

0:48:01.300,0:48:03.930
Who's going to teach[br]me what I have to do?

0:48:03.930,0:48:05.240
You are confusing.

0:48:05.240,0:48:09.920
I have double integral over S[br]minus theory of F in general.

0:48:09.920,0:48:12.810
This is a general[br]vector value field.

0:48:12.810,0:48:15.810


0:48:15.810,0:48:16.940
It could be anything.

0:48:16.940,0:48:18.220
Could be anything.

0:48:18.220,0:48:23.210
But then I have to [INAUDIBLE],[br]because N corresponds

0:48:23.210,0:48:27.180
to the normal to the surface.

0:48:27.180,0:48:29.270
So I-- it's not so easy, right?

0:48:29.270,0:48:30.935
I have to be a little bit smart.

0:48:30.935,0:48:31.987
If I'm not smart--

0:48:31.987,0:48:32.820
STUDENT: [INAUDIBLE]

0:48:32.820,0:48:35.124


0:48:35.124,0:48:36.790
PROFESSOR: That-- you[br]are getting close.

0:48:36.790,0:48:40.870
So guys, the normal[br]two-way surface-- somebody

0:48:40.870,0:48:42.550
gave you a surface, OK?

0:48:42.550,0:48:46.100
And normal to a surface[br]is normal to the plane--

0:48:46.100,0:48:49.590
the tangent plane[br]of the surface.

0:48:49.590,0:48:51.600
So how did we get that?

0:48:51.600,0:48:53.720
There were many ways to do it.

0:48:53.720,0:48:56.190
Either you write[br]the tangent plane

0:48:56.190,0:49:02.700
and you know it by heart--[br]that was Z minus Z zero

0:49:02.700,0:49:07.460
equals-- what the heck was[br]that-- S of X times X minus X

0:49:07.460,0:49:10.710
equals-- plus X of Y[br]times Y minus Y zero.

0:49:10.710,0:49:14.110
And from here you collect--[br]what do you collect?

0:49:14.110,0:49:16.140
You move everybody--[br]it's a moving sale.

0:49:16.140,0:49:19.550
You move everybody to the[br]left hand side and that's it.

0:49:19.550,0:49:21.200
[INAUDIBLE] moving sale.

0:49:21.200,0:49:22.850
OK?

0:49:22.850,0:49:26.900
And everybody will be[br]giving you some components.

0:49:26.900,0:49:31.030
You're going to have minus S[br]of X-- S minus X zero-- minus S

0:49:31.030,0:49:36.620
of Y, Y minus Y zero, plus[br]1-- this is really funny.

0:49:36.620,0:49:39.700
1 times Z minus Z, Z.

0:49:39.700,0:49:43.370
Your normal will[br]be given by what?

0:49:43.370,0:49:46.297
The normal-- how do[br]you collect the normal?

0:49:46.297,0:49:47.130
STUDENT: [INAUDIBLE]

0:49:47.130,0:49:52.300
PROFESSOR: Pi is A, B, C. A,[br]B, and C will be the normal.

0:49:52.300,0:49:54.300
Except it's not unitary.

0:49:54.300,0:49:57.820
And the mechanical engineer[br]tells you, yeah, you're

0:49:57.820,0:50:00.990
stupid-- well, they[br]never say that.

0:50:00.990,0:50:06.080
They will stay look, you have[br]to be a little more careful.

0:50:06.080,0:50:07.680
Not say they are equal.

0:50:07.680,0:50:08.940
What do they mean?

0:50:08.940,0:50:10.540
They say for us,[br]in fluid mechanics,

0:50:10.540,0:50:14.980
solid mechanics, when we write[br]N, we mean you mean vector.

0:50:14.980,0:50:16.150
You are almost there.

0:50:16.150,0:50:16.907
What's missing?

0:50:16.907,0:50:18.240
STUDENT: Magnitude. [INAUDIBLE].

0:50:18.240,0:50:19.739
PROFESSOR: Very[br]good, the magnitude.

0:50:19.739,0:50:24.460
So they will say, go ahead and[br]you [INAUDIBLE] the magnitude.

0:50:24.460,0:50:28.990
And you are lucky now that[br]you know what N will be.

0:50:28.990,0:50:29.980
On the other hand--

0:50:29.980,0:50:31.130
STUDENT: [INAUDIBLE].

0:50:31.130,0:50:32.350
PROFESSOR: This is excellent.

0:50:32.350,0:50:35.600
The one on the bottom-- Alex[br]is thinking like in chess, two

0:50:35.600,0:50:37.900
or three moves ahead.

0:50:37.900,0:50:40.689
You should get two extra[br]credit points with that.

0:50:40.689,0:50:41.480
STUDENT: All right.

0:50:41.480,0:50:42.730
PROFESSOR: You already got it.

0:50:42.730,0:50:46.360
DS is 1 plus S of X squared[br]plus F of X squared.

0:50:46.360,0:50:51.750
The 1 on the bottom and the[br]1 on the top will simplify.

0:50:51.750,0:50:53.000
So say it again, Magdalena.

0:50:53.000,0:50:54.980
Let me write it down here.

0:50:54.980,0:51:01.440
1 S of X, minus S of Y[br]1 over all this animal,

0:51:01.440,0:51:04.940
S of X squared plus S[br]of Y squared plus 1.

0:51:04.940,0:51:08.850
This is the thinking[br]like the early element

0:51:08.850,0:51:13.470
times the early element[br]will be the same thing.

0:51:13.470,0:51:17.590
I'll write it twice even if you[br]laugh at me because we are just

0:51:17.590,0:51:19.430
learning together,[br]and now you finally

0:51:19.430,0:51:21.350
see-- everybody can[br]see that desimplifies.

0:51:21.350,0:51:23.970


0:51:23.970,0:51:27.850
So it's going to be easy to[br]solve this integral in the end,

0:51:27.850,0:51:29.130
right?

0:51:29.130,0:51:32.120
So let's do the[br]problem, finally.

0:51:32.120,0:51:35.590
I'm going to erase it.

0:51:35.590,0:51:40.050
Let's do this problem[br]just for us, at any point.

0:51:40.050,0:51:42.710
I didn't say where.

0:51:42.710,0:51:46.780
Over the same thing.

0:51:46.780,0:51:49.940
The DS was over V01.

0:51:49.940,0:51:53.025
So the picture is[br]the same as before.

0:51:53.025,0:51:57.560
The S will be the[br]nutshell, the eggshell--

0:51:57.560,0:52:02.780
I don't know what it was--[br]over the domain D plane.

0:52:02.780,0:52:07.620
The domain D plane[br]was D of zero 1.

0:52:07.620,0:52:12.160
And I say that I need[br]to use another color.

0:52:12.160,0:52:17.430
This it's going to be my[br]shell, my surface S. Z

0:52:17.430,0:52:20.470
equals X squared[br]plus [INAUDIBLE].

0:52:20.470,0:52:23.140
How do you compute the flux?

0:52:23.140,0:52:25.510
Well, this is that.

0:52:25.510,0:52:28.270
So if we have to be a[br]little bit careful and smart

0:52:28.270,0:52:33.160
and say double integral over[br]S, and now without rushing,

0:52:33.160,0:52:35.730
we have to do a good job.

0:52:35.730,0:52:39.210
First of all, how do[br]you do the dot product?

0:52:39.210,0:52:40.856
The dot product--

0:52:40.856,0:52:43.512
STUDENT: [INAUDIBLE]

0:52:43.512,0:52:44.220
PROFESSOR: Right.

0:52:44.220,0:52:46.530
So first component[br]times first component,

0:52:46.530,0:52:50.290
a second component,[br]second component times

0:52:50.290,0:52:52.530
second component plus[br]that component times

0:52:52.530,0:52:53.710
third component.

0:52:53.710,0:53:01.270
So if 1 is X, F2 is 1.

0:53:01.270,0:53:02.170
Good.

0:53:02.170,0:53:04.290
Z, though, he's not free.

0:53:04.290,0:53:05.270
He's married.

0:53:05.270,0:53:07.045
Why is he married?

0:53:07.045,0:53:07.920
STUDENT: [INAUDIBLE].

0:53:07.920,0:53:10.580
PROFESSOR: Because he[br]depends on X and Y.

0:53:10.580,0:53:14.230
So Z was even here,[br]because I'm on the surface.

0:53:14.230,0:53:16.690
I don't care what F does[br]away from the surface,

0:53:16.690,0:53:20.050
but when he sticks[br]to the surface, when

0:53:20.050,0:53:24.030
he's origin is on[br]the surface, then he

0:53:24.030,0:53:27.860
has to listen to the surface.

0:53:27.860,0:53:29.880
And that Z is not independent.

0:53:29.880,0:53:33.110
The Z is X squared[br]by Y squared here.

0:53:33.110,0:53:36.740
In a bracket, we are[br]over the surface.

0:53:36.740,0:53:41.110
That product minus S[br]of X, minus S of Y. I

0:53:41.110,0:53:42.960
know you're going to[br]laugh at me because I

0:53:42.960,0:53:44.670
haven't written where they are.

0:53:44.670,0:53:46.370
But that's what I[br]need your help for.

0:53:46.370,0:53:46.870
DA.

0:53:46.870,0:53:49.390


0:53:49.390,0:53:51.810
Who are they?

0:53:51.810,0:53:53.759
Who is this guy?

0:53:53.759,0:53:54.800
STUDENT: The [INAUDIBLE].

0:53:54.800,0:53:55.466
PROFESSOR: What?

0:53:55.466,0:53:56.580
STUDENT: [INAUDIBLE].

0:53:56.580,0:53:58.000
PROFESSOR: Negative 2X.

0:53:58.000,0:53:58.856
Is it?

0:53:58.856,0:53:59.850
STUDENT: No.

0:53:59.850,0:54:01.350
PROFESSOR: How about this guy?

0:54:01.350,0:54:02.430
STUDENT: [INAUDIBLE].

0:54:02.430,0:54:03.388
PROFESSOR: Negative 2Y.

0:54:03.388,0:54:05.750
How about this guy?

0:54:05.750,0:54:06.710
I'm just kidding.

0:54:06.710,0:54:07.680
OK.

0:54:07.680,0:54:14.230
So finally we should be able[br]to compute this integral.

0:54:14.230,0:54:15.650
That looks awful.

0:54:15.650,0:54:17.560
Over D.

0:54:17.560,0:54:20.575
So instead of S, we[br]have the D, which

0:54:20.575,0:54:25.170
is the disk of[br]radius one in plane.

0:54:25.170,0:54:29.740
And we say, OK, I have,[br]oh my god, it's OK.

0:54:29.740,0:54:31.716
This times that is how much?

0:54:31.716,0:54:32.590
STUDENT: [INAUDIBLE].

0:54:32.590,0:54:33.890
PROFESSOR: Minus 2X squared.

0:54:33.890,0:54:35.190
Right?

0:54:35.190,0:54:35.880
There.

0:54:35.880,0:54:37.120
Take the green.

0:54:37.120,0:54:40.870
This times that is how much?

0:54:40.870,0:54:44.250
Minus the Y squared.

0:54:44.250,0:54:50.780
And this times that is finally[br]just X squared plus Y squared.

0:54:50.780,0:54:53.440
Very nice think.

0:54:53.440,0:54:55.560
I think that at[br]first, but now I see

0:54:55.560,0:54:59.110
that life is beautiful[br]again-- DX, DY--

0:54:59.110,0:55:02.770
that I can go ahead and do it.

0:55:02.770,0:55:05.053
I can get a hold of this.

0:55:05.053,0:55:11.134
And inside that, what do I--[br]what am I left with in the end?

0:55:11.134,0:55:12.050
STUDENT: [INAUDIBLE].

0:55:12.050,0:55:14.650
PROFESSOR: Minus 2[br]times this animal,

0:55:14.650,0:55:18.550
called X squared plus Y squared,[br]which is going to be R squared.

0:55:18.550,0:55:23.230
So the flux-- the flux for[br]this problem in the end

0:55:23.230,0:55:25.640
is going to be very[br]nice and sassy.

0:55:25.640,0:55:26.310
Look at that.

0:55:26.310,0:55:28.145
F would be--

0:55:28.145,0:55:29.520
STUDENT: There[br]would not be any--

0:55:29.520,0:55:30.353
STUDENT: [INAUDIBLE]

0:55:30.353,0:55:36.580


0:55:36.580,0:55:37.390
PROFESSOR: What?

0:55:37.390,0:55:40.989
STUDENT: You've got[br]minus 2 and the plus 1.

0:55:40.989,0:55:42.030
PROFESSOR: Oh, thank God.

0:55:42.030,0:55:43.730
Thank God you exist.

0:55:43.730,0:55:47.250
So I thought about[br]it before, but then I

0:55:47.250,0:55:49.040
said-- I don't know why.

0:55:49.040,0:55:50.120
I messed up.

0:55:50.120,0:55:52.540
So we have minus R squared.

0:55:52.540,0:55:53.530
Very good.

0:55:53.530,0:55:54.810
It's easy.

0:55:54.810,0:56:01.950
Times an R from the[br]Jacobian, DR is theta.

0:56:01.950,0:56:04.550
And theta is between 0 and 2 pi.

0:56:04.550,0:56:08.020
And R between 0 and 1.

0:56:08.020,0:56:11.050
And now I will need[br]a plumber to tell me

0:56:11.050,0:56:12.790
what I do the limits[br]of the integrals,

0:56:12.790,0:56:17.752
because I think I'm getting[br]a negative answer, so.

0:56:17.752,0:56:20.527


0:56:20.527,0:56:21.692
STUDENT: [INAUDIBLE].

0:56:21.692,0:56:23.150
PROFESSOR: I'll do[br]it, and then you

0:56:23.150,0:56:26.700
tell me why I got what I got.

0:56:26.700,0:56:29.710
I have a minus[br]pulled out by nature.

0:56:29.710,0:56:32.850
And then I have integral--

0:56:32.850,0:56:34.000
STUDENT: R [INAUDIBLE].

0:56:34.000,0:56:35.820
PROFESSOR: R to the[br]fourth of a fourth.

0:56:35.820,0:56:36.590
Very good.

0:56:36.590,0:56:40.447
But you have your [INAUDIBLE] so[br]when I do between zero and 1--

0:56:40.447,0:56:41.530
STUDENT: It's [INAUDIBLE].

0:56:41.530,0:56:43.830
PROFESSOR: 1 over[br]4-- you are too

0:56:43.830,0:56:49.400
fast-- as 2 pi-- that's a[br]good thing-- minus pi over 2,

0:56:49.400,0:56:50.890
you said, Gus.

0:56:50.890,0:56:53.160
And I could see it[br]coming straight at me

0:56:53.160,0:56:55.320
and hit me between the eyes.

0:56:55.320,0:56:57.720
What is the problem.

0:56:57.720,0:56:59.750
Is there a problem?

0:56:59.750,0:57:02.820
Without an area as a flux, would[br]that say, what is the negative?

0:57:02.820,0:57:03.810
Yes.

0:57:03.810,0:57:05.630
How can I make it positive?

0:57:05.630,0:57:07.480
This is my question.

0:57:07.480,0:57:08.730
STUDENT: Change the direction.

0:57:08.730,0:57:10.624
PROFESSOR: Change[br]the direction of who?

0:57:10.624,0:57:12.220
STUDENT: The flux.

0:57:12.220,0:57:13.360
PROFESSOR: The flux.

0:57:13.360,0:57:14.850
I could change the direction.

0:57:14.850,0:57:17.640
So what is it that[br]doesn't match?

0:57:17.640,0:57:19.980
[INAUDIBLE]

0:57:19.980,0:57:22.645
If I want to keep-- the[br]flux will be the same.

0:57:22.645,0:57:25.270
When I can change the[br]orientation of the service.

0:57:25.270,0:57:27.820
And instead I get a minus then.

0:57:27.820,0:57:36.410
My N was it sticking[br]in-- oh, my god.

0:57:36.410,0:57:40.340
So is it sticking[br]in or sticking out?

0:57:40.340,0:57:40.990
Look at it.

0:57:40.990,0:57:41.900
Think about it.

0:57:41.900,0:57:47.660
I have minus the positive guy[br]minus another positive guy,

0:57:47.660,0:57:48.980
and 1 sticking out.

0:57:48.980,0:57:51.010
But it goes with[br]the holes inside.

0:57:51.010,0:57:54.020
This is the paraboloid[br][INAUDIBLE].

0:57:54.020,0:57:57.470
If I have something I minus I[br]minus J, does it go out or in?

0:57:57.470,0:57:57.970
STUDENT: In.

0:57:57.970,0:57:59.540
PROFESSOR: It goes in.

0:57:59.540,0:58:00.804
It goes in, and it'll be up.

0:58:00.804,0:58:02.720
So it's going to be like[br]all these normals are

0:58:02.720,0:58:07.170
going to be like a vector[br]field like that, like amoebas.

0:58:07.170,0:58:09.740
But they are pointing[br]towards inside.

0:58:09.740,0:58:11.090
Do I like that?

0:58:11.090,0:58:13.160
Yes, because I'm a[br]crazy mathematician.

0:58:13.160,0:58:16.740
Does the engineer like that?

0:58:16.740,0:58:17.710
No.

0:58:17.710,0:58:18.660
Why?

0:58:18.660,0:58:21.790
The flux is pointing in or out?

0:58:21.790,0:58:23.150
The flux.

0:58:23.150,0:58:23.650
The flux.

0:58:23.650,0:58:25.900
The flux, the flux[br]is pointing out.

0:58:25.900,0:58:27.590
Are you guys with me?

0:58:27.590,0:58:31.370
X plus Y-- X plus I plus J.[br]It's like this pointing out.

0:58:31.370,0:58:33.540
So the flux get[br]out of the surface.

0:58:33.540,0:58:36.790
It's like to pour water[br]inside, and the water's

0:58:36.790,0:58:41.580
just a net-- not a net, but[br]like something that holds it in.

0:58:41.580,0:58:42.954
And like a--

0:58:42.954,0:58:44.120
STUDENT: Like a [INAUDIBLE]?

0:58:44.120,0:58:45.765
PROFESSOR: --pasta strainer.

0:58:45.765,0:58:47.640
And the water goes up[br][SPRAYING NOISE], well,

0:58:47.640,0:58:48.670
like a jet.

0:58:48.670,0:58:49.320
Like that.

0:58:49.320,0:58:52.580
So that is your flux[br]going through the surface.

0:58:52.580,0:58:56.190
Are you happy that I took[br]the normal pointing inside?

0:58:56.190,0:58:56.790
No.

0:58:56.790,0:58:57.780
That was crazy.

0:58:57.780,0:59:02.820
So here comes you, the[br]mechanical engineer majoring

0:59:02.820,0:59:06.100
in solid or [INAUDIBLE][br]and say Magdalena,

0:59:06.100,0:59:09.250
you should have taken[br]the outer normal,

0:59:09.250,0:59:12.294
because look at the[br]flux pointing out.

0:59:12.294,0:59:14.210
Take the outer of normal,[br]and things are going

0:59:14.210,0:59:16.590
to looks right and nice again.

0:59:16.590,0:59:18.610
So if I were to[br]change the normal,

0:59:18.610,0:59:21.400
I would put the[br]plus, plus, minus.

0:59:21.400,0:59:23.500
I'll take the outer normal.

0:59:23.500,0:59:26.010
And in the end I[br]get plus 5 over 2.

0:59:26.010,0:59:29.100
So no remark.

0:59:29.100,0:59:36.280
If I change N to minus N, this[br]would become the outer normal.

0:59:36.280,0:59:40.680
Then the flux would[br]become pi over 2. solar

0:59:40.680,0:59:42.730
flux depends on the what?

0:59:42.730,0:59:44.980
The match between[br]the flux, the angles,

0:59:44.980,0:59:48.990
sort of between the flux if[br]function, vector [INAUDIBLE]

0:59:48.990,0:59:53.340
function, and the normal[br]that I take to the surface.

0:59:53.340,0:59:54.000
Right?

0:59:54.000,0:59:59.200
I can change the normal and[br]I get the opposite answer.

0:59:59.200,1:00:01.505
In absolute values,[br]the same flux.

1:00:01.505,1:00:05.770
So flux should be equal[br][INAUDIBLE] the absolute value.

1:00:05.770,1:00:09.060
Unlike the area that should[br]be always a positive number.

1:00:09.060,1:00:11.430
Volume, that should always[br]be a positive number.

1:00:11.430,1:00:14.940
So if I get a limited area,[br]that means I messed up.

1:00:14.940,1:00:17.040
If I get a negative[br]on all of them,

1:00:17.040,1:00:20.920
it means messed up in my[br]computation somewhere.

1:00:20.920,1:00:23.070
But that doesn't mean[br]I messed up here.

1:00:23.070,1:00:24.810
I just chose the other normal.

1:00:24.810,1:00:26.200
It's possible.

1:00:26.200,1:00:30.720
So the flux can be taken as[br]is and put in absolute value.

1:00:30.720,1:00:31.440
All right.

1:00:31.440,1:00:33.700
OK.

1:00:33.700,1:00:36.980
We have to think of it like[br]the surface, and stuff that

1:00:36.980,1:00:40.320
goes through surface[br]in electric circuits.

1:00:40.320,1:00:42.700
Can you do some research[br]for you about flux

1:00:42.700,1:00:45.120
and electrical engineering?

1:00:45.120,1:00:50.280
And next time somebody[br]tells me a story about it.

1:00:50.280,1:00:52.710
Who is-- again-- who is[br]electrical engineering major

1:00:52.710,1:00:54.280
here?

1:00:54.280,1:00:56.160
Oh, so five people.

1:00:56.160,1:00:58.220
You're going to get four[br]extra credit points.

1:00:58.220,1:00:59.920
You guys are jealous.

1:00:59.920,1:01:03.510
I'm going to give you four extra[br]credit points if in 10 minutes

1:01:03.510,1:01:08.490
you can tell us a little bit[br]about where flux can be seen.

1:01:08.490,1:01:10.470
Well, you don't have[br]to come to the board.

1:01:10.470,1:01:13.070
You can just talk to us[br]from outside if you want,

1:01:13.070,1:01:14.800
or down inside the classroom.

1:01:14.800,1:01:16.950
Tell us where the[br]notion of flux appears

1:01:16.950,1:01:19.910
in the electric[br]circuits and why it

1:01:19.910,1:01:24.290
would be important for[br]Calculus 3 as well.

1:01:24.290,1:01:25.190
OK.

1:01:25.190,1:01:28.710
Now a big question[br]before I let you go.

1:01:28.710,1:01:34.550
Can I have a flux[br]that corresponds

1:01:34.550,1:01:36.120
to a parameterization?

1:01:36.120,1:01:41.560
That is my big worry, that[br]I have to do that as well.

1:01:41.560,1:01:45.140
Eventually, could I[br]have solved this problem

1:01:45.140,1:01:47.950
if the surface that[br]is parameterized

1:01:47.950,1:01:54.080
was my friend--[br]who was my friend?

1:01:54.080,1:01:54.980
I don't remember.

1:01:54.980,1:01:59.350
UI plus VJ plus U[br]squared plus-- you

1:01:59.350,1:02:03.490
gave it to me-- OK, that[br]was the previous example,

1:02:03.490,1:02:05.450
and that's the last[br]example on the board.

1:02:05.450,1:02:12.610
So you have double integral[br]of force field times NDS.

1:02:12.610,1:02:16.930
Now, what if I say I don't[br]want to do it like this-- Z

1:02:16.930,1:02:19.460
equals F of XY.

1:02:19.460,1:02:21.406
So I don't want to[br]do it like that.

1:02:21.406,1:02:25.070
I want to do it in[br]a different way.

1:02:25.070,1:02:33.720
That means you pulling out of[br]your brain some old memories.

1:02:33.720,1:02:35.260
F was F, right?

1:02:35.260,1:02:37.480
You need to leave F alone,[br]poor fellow, because he

1:02:37.480,1:02:39.900
has no better way to do it.

1:02:39.900,1:02:43.902
This is becoming[br]complicated, the [INAUDIBLE]

1:02:43.902,1:02:44.860
mechanical engineering.

1:02:44.860,1:02:47.862


1:02:47.862,1:02:52.530
And what's given to you[br]before, but you don't remember?

1:02:52.530,1:02:55.810
R was given to you[br]as position vector.

1:02:55.810,1:02:59.710
R sub U and R sub V,[br]you may not remember--

1:02:59.710,1:03:03.475
that was a long time ago-- we[br]proved that R sub U and R sub

1:03:03.475,1:03:05.450
V were on the surface.

1:03:05.450,1:03:07.190
They are both tensions[br]of the surface.

1:03:07.190,1:03:09.360
It was a long time ago.

1:03:09.360,1:03:11.280
So the normal is[br][INAUDIBLE], and that's

1:03:11.280,1:03:14.050
exactly what I wanted to[br]say the normal will be.

1:03:14.050,1:03:17.820
Not quite pressed product,[br]but just like before,

1:03:17.820,1:03:21.240
pressed product[br]divided by the norm,

1:03:21.240,1:03:28.470
because then the unit normal[br]vector has to be length 1.

1:03:28.470,1:03:30.660
So I have to divide[br]by the number.

1:03:30.660,1:03:31.160
[SNEEZE]

1:03:31.160,1:03:32.019
The DS--

1:03:32.019,1:03:32.810
STUDENT: Thank you.

1:03:32.810,1:03:36.020
PROFESSOR: --is going to--[br]OK, now it's up to you guys.

1:03:36.020,1:03:36.780
You're smart.

1:03:36.780,1:03:39.360
You know what I want to say.

1:03:39.360,1:03:43.770
So I'll pretend that you[br]know what DS is in terms

1:03:43.770,1:03:45.030
of the parameterization.

1:03:45.030,1:03:46.515
What's coming?

1:03:46.515,1:03:47.790
We said that.

1:03:47.790,1:03:49.340
It was a long time ago.

1:03:49.340,1:03:51.500
You can guess it by[br]just being smart--

1:03:51.500,1:03:51.740
STUDENT: [INAUDIBLE].

1:03:51.740,1:03:52.781
PROFESSOR: --or you can--

1:03:52.781,1:03:53.830
STUDENT: [INAUDIBLE].

1:03:53.830,1:03:55.290
PROFESSOR: Yes, exactly.

1:03:55.290,1:03:57.800
And you got another[br]one extra credit point.

1:03:57.800,1:04:01.990


1:04:01.990,1:04:04.579
STUDENT: [INAUDIBLE]

1:04:04.579,1:04:06.620
PROFESSOR: So since before,[br]they were simplified,

1:04:06.620,1:04:07.630
for god's sake.

1:04:07.630,1:04:10.640
Now we have the new kind[br]of writing area element DS.

1:04:10.640,1:04:13.250
They also have to simplify.

1:04:13.250,1:04:15.510
It wasn't hard to see.

1:04:15.510,1:04:18.330
So you could have[br]done it like that.

1:04:18.330,1:04:22.600
You could have done[br]it like that, how?

1:04:22.600,1:04:25.510
Somebody need to help me,[br]because I have no idea what

1:04:25.510,1:04:28.050
I'm going to do here.

1:04:28.050,1:04:29.384
Do we get the same thing or not?

1:04:29.384,1:04:30.258
This is the question.

1:04:30.258,1:04:31.800
And I'm going to[br]finish with that,

1:04:31.800,1:04:33.850
but I don't want to[br]go home-- I'm not

1:04:33.850,1:04:37.890
going to let you go home[br]until you finish this.

1:04:37.890,1:04:42.270
F was a simple,[br]beautiful vector field.

1:04:42.270,1:04:45.490
Given-- like that.

1:04:45.490,1:04:46.760
This is a force.

1:04:46.760,1:04:48.720
May the force be[br]with you like that.

1:04:48.720,1:04:54.010
But we changed it in U,V because[br]we are acting on the surface S,

1:04:54.010,1:04:56.050
what is the pressure[br]in V, right?

1:04:56.050,1:05:00.700
So you have UI plus VJ[br]plus-- you gave it to me--

1:05:00.700,1:05:03.680
U squared plus V squared.

1:05:03.680,1:05:05.490
Am I right, or am[br]I talking nonsense?

1:05:05.490,1:05:08.010


1:05:08.010,1:05:08.650
All right.

1:05:08.650,1:05:12.850
So now again I have[br]to be seeing them.

1:05:12.850,1:05:14.140
Am I getting the same thing?

1:05:14.140,1:05:16.280
If I'm not getting[br]the same thing,

1:05:16.280,1:05:19.280
I can just go home and[br]get drunk and be sad.

1:05:19.280,1:05:22.710
But I have to get[br]the same thing.

1:05:22.710,1:05:27.020
Otherwise, there is something[br]wrong with my setup.

1:05:27.020,1:05:32.620
So I have to have U, V.[br]U squared plus V squared.

1:05:32.620,1:05:34.500
Close.

1:05:34.500,1:05:37.000
Dot product.

1:05:37.000,1:05:41.840
This guy over on top-- say what?

1:05:41.840,1:05:48.120
Magdalena, this guy over on top[br]has to be-- has to be a what?

1:05:48.120,1:05:50.040
Well, I didn't say what it was.

1:05:50.040,1:05:52.720
I should do it now.

1:05:52.720,1:05:53.460
Right?

1:05:53.460,1:05:57.160
So how will we do that?

1:05:57.160,1:06:07.960
We were saying R of[br]UV will be UI plus VJ

1:06:07.960,1:06:10.500
plus U squared plus V squared.

1:06:10.500,1:06:11.100
OK.

1:06:11.100,1:06:14.920
So R sub U will be--[br]you teach me quickly,

1:06:14.920,1:06:18.146
and R sub [INAUDIBLE][br]is-- voila.

1:06:18.146,1:06:19.770
STUDENT: [INAUDIBLE]

1:06:19.770,1:06:21.564
PROFESSOR: 1--

1:06:21.564,1:06:23.530
STUDENT: [INAUDIBLE]

1:06:23.530,1:06:28.120
PROFESSOR: Plus zero--[br]thank you-- plus 2U, OK.

1:06:28.120,1:06:34.830
0 plus 1J plus 2VK Am I done?

1:06:34.830,1:06:35.330
I'm done.

1:06:35.330,1:06:35.900
No, I'm not done.

1:06:35.900,1:06:36.820
What do I have to do?

1:06:36.820,1:06:38.590
Cross them.

1:06:38.590,1:06:41.470


1:06:41.470,1:06:44.500
Cross multiply IJK.

1:06:44.500,1:06:47.355
This looks nice.

1:06:47.355,1:06:49.240
Look, it's not so ugly.

1:06:49.240,1:06:51.481
I thought it would[br]be uglier, right?

1:06:51.481,1:06:51.980
OK.

1:06:51.980,1:06:54.690
What it is?

1:06:54.690,1:06:57.515
What it this thing?

1:06:57.515,1:06:58.390
STUDENT: [INAUDIBLE].

1:06:58.390,1:07:06.180
PROFESSOR: Minus the[br]U, I. Minus-- plus.

1:07:06.180,1:07:09.130
Minus, plus 1.

1:07:09.130,1:07:11.455
2V minus because it's--

1:07:11.455,1:07:12.080
STUDENT: Minus.

1:07:12.080,1:07:13.450
PROFESSOR: --minus in front.

1:07:13.450,1:07:15.360
Right.

1:07:15.360,1:07:17.510
So I'm alternating.

1:07:17.510,1:07:19.980
And 1K.

1:07:19.980,1:07:24.040
So again, I get minus[br]X of S minus XY and 1,

1:07:24.040,1:07:26.420
and again, I'm pointing[br]in, and that's bad.

1:07:26.420,1:07:29.520
So my normal will point[br]inside the surface

1:07:29.520,1:07:33.670
like needles that are[br]perpendicular to the surface

1:07:33.670,1:07:34.840
pointing inside.

1:07:34.840,1:07:36.000
But that's OK.

1:07:36.000,1:07:39.050
In the end, I take[br]everything in absolute value.

1:07:39.050,1:07:39.550
Right?

1:07:39.550,1:07:47.640


1:07:47.640,1:07:51.100
So again, I do the same math.

1:07:51.100,1:07:54.760
So I get minus-- I don't[br]want to do it anymore.

1:07:54.760,1:07:59.410
Minus 2A squared, minus 2B[br]squared, plus your squared,

1:07:59.410,1:08:01.580
plus this squared,[br]then you save me

1:08:01.580,1:08:04.870
and you said minus 2[br]squared [INAUDIBLE] squared.

1:08:04.870,1:08:06.260
DUDV.

1:08:06.260,1:08:15.020
But DUDV means that UV is a[br]pair, a point in this, guys.

1:08:15.020,1:08:16.120
UV.

1:08:16.120,1:08:18.960
It's a pair in the[br]disk of radius one.

1:08:18.960,1:08:21.990
So I'm getting exactly,[br]what exactly the same thing

1:08:21.990,1:08:22.939
as before.

1:08:22.939,1:08:25.640
Because this is[br]minus R squared, so I

1:08:25.640,1:08:32.910
get integral, integral, minus[br]R squared times R. DR, D theta.

1:08:32.910,1:08:35.660
From zero to 1,[br]from zero to 2 pi,

1:08:35.660,1:08:38.158
and I get the same[br]answer, which was?

1:08:38.158,1:08:39.490
STUDENT: [INAUDIBLE].

1:08:39.490,1:08:39.800
PROFESSOR: Minus what?

1:08:39.800,1:08:40.300
STUDENT: [INAUDIBLE]

1:08:40.300,1:08:40.590
PROFESSOR: Pi over--

1:08:40.590,1:08:41.679
STUDENT: [INAUDIBLE].

1:08:41.679,1:08:42.470
PROFESSOR: You see?

1:08:42.470,1:08:46.024
I already forgot.

1:08:46.024,1:08:48.359
STUDENT: 2.

1:08:48.359,1:08:49.910
PROFESSOR: So what[br]matters is that we

1:08:49.910,1:08:52.290
take the flux in[br]absolute value because it

1:08:52.290,1:08:54.420
depends on the[br]orientation of the normal.

1:08:54.420,1:08:57.189
If we take the[br]normal [INAUDIBLE].

1:08:57.189,1:09:02.950
Please, one thing I want you[br]to do when you go home now,

1:09:02.950,1:09:06.029
open the book which[br]maybe you rarely do,

1:09:06.029,1:09:08.529
but now it's[br]really-- the material

1:09:08.529,1:09:10.130
became complicated enough.

1:09:10.130,1:09:14.170
We are not just doing math,[br]calculus, we are doing physics,

1:09:14.170,1:09:17.640
we are doing mechanics, we are[br]dealing with surface integrals

1:09:17.640,1:09:19.069
and flux.

1:09:19.069,1:09:27.120
I want you to open the book[br]at page-- I don't know.

1:09:27.120,1:09:32.790
At surface integrals[br]starts at page 1,063.

1:09:32.790,1:09:34.960
Section 13.5.

1:09:34.960,1:09:38.250
And it keeps going like that,[br]pretty pictures of surfaces

1:09:38.250,1:09:40.680
and fluxes and so on.

1:09:40.680,1:09:41.840
Vector fields.

1:09:41.840,1:09:43.660
And it keeps going like that.

1:09:43.660,1:09:48.890
But it doesn't cover anything[br]new except what I said today.

1:09:48.890,1:09:51.310
It's just that it shows[br]you examples that are not

1:09:51.310,1:09:54.810
as beautiful as the ones I[br]gave, but they are essentially

1:09:54.810,1:09:58.190
the same, only a little[br]bit nastier to complete.

1:09:58.190,1:10:02.480
So up to 1,072.

1:10:02.480,1:10:05.740
So that is what you're[br]going to do this weekend,

1:10:05.740,1:10:06.760
plus the homework.

1:10:06.760,1:10:07.800
Keep on the homework.

1:10:07.800,1:10:11.070
Now, if you get stuck[br]Saturday, Sunday,

1:10:11.070,1:10:13.977
whenever you try your homework[br]you get stuck, what do you do?

1:10:13.977,1:10:14.810
STUDENT: [INAUDIBLE]

1:10:14.810,1:10:15.810
PROFESSOR: You email me.

1:10:15.810,1:10:18.460
So you say what in the world[br]is going on with this problem

1:10:18.460,1:10:25.045
because I tried it seven[br]times and-- 88 times.

1:10:25.045,1:10:27.177
And then you got[br]the brownie points.

1:10:27.177,1:10:28.010
STUDENT: [INAUDIBLE]

1:10:28.010,1:10:30.200
PROFESSOR: [INAUDIBLE] problem.

1:10:30.200,1:10:31.470
STUDENT: [INAUDIBLE] by 32.

1:10:31.470,1:10:33.240
PROFESSOR: There[br]was a problem, guys.

1:10:33.240,1:10:35.110
There are not so many problems.

1:10:35.110,1:10:38.930
But the only part, serious[br]part that we would catch,

1:10:38.930,1:10:41.670
he found it first, and[br]he tried it 88 times.

1:10:41.670,1:10:44.400


1:10:44.400,1:10:47.155
I'll never forget you, though,[br]because you are unique,

1:10:47.155,1:10:49.970
and that-- I appreciated[br]that very much.

1:10:49.970,1:10:55.380
So doing this weekend, do[br]not hesitate to pester.

1:10:55.380,1:10:58.400
I will answer all the web[br]work problems you have.

1:10:58.400,1:11:00.100
I want you to do well.

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Next week is the last[br]week on new theory,

1:11:01.980,1:11:04.380
and then we start[br]working for the final,

1:11:04.380,1:11:09.060
so by the time of the final,[br]you'll be [INAUDIBLE].

1:11:09.060,1:11:10.317
STUDENT: [INAUDIBLE]?

1:11:10.317,1:11:11.150
PROFESSOR: Yes, sir.

1:11:11.150,1:11:13.377
Oh, I appreciated[br]that you did that.

1:11:13.377,1:11:14.210
STUDENT: [INAUDIBLE]

1:11:14.210,1:11:18.200


1:11:18.200,1:11:20.592
PROFESSOR: Again,[br]I forgot these.

1:11:20.592,1:11:24.920


1:11:24.920,1:11:28.456
With the extra points you[br]got, you shouldn't care.

1:11:28.456,1:11:28.956