0:00:00.000,0:00:02.916
PROFESSOR: Do you have[br]any kind of questions?

0:00:02.916,0:00:05.346
There were a few questions[br]about the homework.

0:00:05.346,0:00:11.081
Casey, you have that problem[br]we need to the minus D?

0:00:11.081,0:00:11.664
STUDENT: Yeah.

0:00:11.664,0:00:12.622
PROFESSOR: The minus D?

0:00:12.622,0:00:15.714
Let's do that in class, because[br]there were several people who

0:00:15.714,0:00:17.520
faced that problem.

0:00:17.520,0:00:22.120
You said you faced it, and[br]you got it and can I cheat?

0:00:22.120,0:00:25.870
Can I take your work so I[br]can present it at the board?

0:00:25.870,0:00:27.214
I'm serious about it.

0:00:27.214,0:00:27.880
STUDENT: OK, um.

0:00:27.880,0:00:31.364
PROFESSOR: So I know[br]we've done this together.

0:00:31.364,0:00:33.690
I don't even[br]remember the problem.

0:00:33.690,0:00:35.410
How was it?

0:00:35.410,0:00:37.060
Homework problem.

0:00:37.060,0:00:39.530
STUDENT: She knows it.

0:00:39.530,0:00:40.030
[INAUDIBLE]

0:00:40.030,0:00:43.502


0:00:43.502,0:00:44.990
STUDENT: But she knows it.

0:00:44.990,0:00:47.800
PROFESSOR: They'll work with you[br]to be minus M. Can you tell me,

0:00:47.800,0:00:48.462
Casey?

0:00:48.462,0:00:50.446
Can I tell the statement?

0:00:50.446,0:00:52.926
STUDENT: Well, a black[br]guy's [INAUDIBLE].

0:00:52.926,0:00:55.902
PROFESSOR: If you find it, give[br]it to me and I'll give you $2.

0:00:55.902,0:00:57.390
STUDENT: It's your problem.

0:00:57.390,0:00:58.382
[INAUDIBLE]

0:00:58.382,0:00:59.870
PROFESSOR: How is it's this one.

0:00:59.870,0:01:00.862
STUDENT: No.

0:01:00.862,0:01:01.612
PROFESSOR: Oh, no.

0:01:01.612,0:01:02.505
It's not this one.

0:01:02.505,0:01:03.838
STUDENT: Can I have it from you?

0:01:03.838,0:01:05.822
I won't give you anything.

0:01:05.822,0:01:07.310
STUDENT: Um, it's doable.

0:01:07.310,0:01:09.790
[INAUDIBLE]

0:01:09.790,0:01:13.338
PROFESSOR: So can somebody with[br]me now, that's my handwriting.

0:01:13.338,0:01:14.254
STUDENT: Yeah, I know.

0:01:14.254,0:01:14.750
It is weird.

0:01:14.750,0:01:14.998
PROFESSOR: OK.

0:01:14.998,0:01:15.742
All right.

0:01:15.742,0:01:17.726
So the problem says--

0:01:17.726,0:01:18.222
STUDENT: Do you have[br]a problem with me?

0:01:18.222,0:01:18.718
PROFESSOR: Any--

0:01:18.718,0:01:20.968
STUDENT: And then we're all[br]here for her. [INAUDIBLE].

0:01:20.968,0:01:23.182
Doesn't it feel like[br][INAUDIBLE] kind of a bit?

0:01:23.182,0:01:25.390
PROFESSOR: It's the one[br]that has X of D equals

0:01:25.390,0:01:28.582
into the minus the cosign D.

0:01:28.582,0:01:30.046
STUDENT: Oh, [INAUDIBLE].

0:01:30.046,0:01:32.486
PROFESSOR: Y of T, and you[br]go by your exclamation.

0:01:32.486,0:01:37.370
I understand that you[br]love this problem.

0:01:37.370,0:01:43.645
And so you've had this type[br]of pathing to grow compute.

0:01:43.645,0:01:48.796
The pathing to grow with respect[br]to the [INAUDIBLE] fellow man

0:01:48.796,0:01:51.400
well meant that[br]in life is slowly

0:01:51.400,0:01:53.780
because nobody[br][INAUDIBLE] with you.

0:01:53.780,0:01:58.190


0:01:58.190,0:02:04.070
And to go over C[br]of the integer will

0:02:04.070,0:02:07.520
be a very nice friend of[br]yours, [INAUDIBLE] explain it,

0:02:07.520,0:02:11.210
but of course, they are both[br]functions of T in general,

0:02:11.210,0:02:13.910
and you will have[br]the DS element,

0:02:13.910,0:02:15.300
and what does this mean?

0:02:15.300,0:02:17.462
S is [INAUDIBLE].

0:02:17.462,0:02:22.260
It means that you are[br]archic element should

0:02:22.260,0:02:25.512
be expressed in terms of what?

0:02:25.512,0:02:27.630
Who in the world is[br]the archeling infinite

0:02:27.630,0:02:29.850
decimal element.

0:02:29.850,0:02:32.040
It's the speed times the t.

0:02:32.040,0:02:33.050
STUDENT: Say it again?

0:02:33.050,0:02:34.730
PROFESSOR: It's the speed.

0:02:34.730,0:02:36.400
STUDENT: And what was the speed?

0:02:36.400,0:02:41.790
R in front of T. [INAUDIBLE].

0:02:41.790,0:02:42.760
Right?

0:02:42.760,0:02:46.090
So you will have to[br]transform this path integral

0:02:46.090,0:02:49.590
into an integral, respected[br]T, where T takes values

0:02:49.590,0:02:52.610
from a T0 to a T1.

0:02:52.610,0:02:59.212
And I don't want to give[br]you your notebook back.

0:02:59.212,0:02:59.920
STUDENT: It's OK.

0:02:59.920,0:03:02.295
PROFESSOR: OK, now I'll do[br]the same thing all over again,

0:03:02.295,0:03:06.590
and you control me and then[br]if I do something wrong,

0:03:06.590,0:03:07.860
you've done me.

0:03:07.860,0:03:12.740
And what were the--[br]what was the path?

0:03:12.740,0:03:15.076
Specified as what?

0:03:15.076,0:03:18.004
STUDENT: XYZ? [INAUDIBLE].

0:03:18.004,0:03:19.956
PROFESSOR: Yeah, the path was--

0:03:19.956,0:03:22.400
STUDENT: [INAUDIBLE].

0:03:22.400,0:03:24.910
PROFESSOR: T equals[br]from zero to pi over 2.

0:03:24.910,0:03:26.750
I have to write it down.

0:03:26.750,0:03:29.690


0:03:29.690,0:03:34.750
So let us write the--[br][INAUDIBLE] are from the T.

0:03:34.750,0:03:44.100
The speed square root of is[br]from the T squared plus Y

0:03:44.100,0:03:49.620
prime of T squared, because[br]the sampling occurred.

0:03:49.620,0:03:53.650
Before we do that, we have[br]to go ahead and compute

0:03:53.650,0:03:54.795
X prime and Y prime.

0:03:54.795,0:03:58.662


0:03:58.662,0:04:01.092
And of course[br]that's product rule,

0:04:01.092,0:04:03.036
and I need a better marker.

0:04:03.036,0:04:04.008
STUDENT: [INAUDIBLE].

0:04:04.008,0:04:04.980
PROFESSOR: Yes, sir?

0:04:04.980,0:04:06.771
STUDENT: Do you think[br]the arc too is really

0:04:06.771,0:04:08.880
taken as an arc [INAUDIBLE]?

0:04:08.880,0:04:09.380
This

0:04:09.380,0:04:10.380
PROFESSOR: This is the--

0:04:10.380,0:04:13.070
STUDENT: Because we take-- I[br]would consider it as a path

0:04:13.070,0:04:15.530
function that looks like[br]an arc, or, like, thinking

0:04:15.530,0:04:18.728
that it's missing one[br]rule, and that's about it.

0:04:18.728,0:04:19.720
That's fine.

0:04:19.720,0:04:21.620
PROFESSOR: No, no, no[br]no, no, no, no, no.

0:04:21.620,0:04:22.120
no.

0:04:22.120,0:04:23.090
OK, let me explain.

0:04:23.090,0:04:28.090
So suppose you are[br][INAUDIBLE] arc in plane

0:04:28.090,0:04:29.798
and this is your r of t.

0:04:29.798,0:04:30.629
STUDENT: Oh, OK.

0:04:30.629,0:04:32.670
PROFESSOR: And that's[br]called the position vector,

0:04:32.670,0:04:35.661
and that's x of t, y of t.

0:04:35.661,0:04:36.160
OK.

0:04:36.160,0:04:38.014
What is your velocity vector?

0:04:38.014,0:04:41.730
Velocity vector would be[br]in tangent to the curve.

0:04:41.730,0:04:44.974
Suppose you go in this[br]direction, counterclockwise,

0:04:44.974,0:04:48.310
and then our prime of[br]t will be this guy.

0:04:48.310,0:04:50.705
And it's gonna be[br]x prime, y prime.

0:04:50.705,0:04:53.350
And we have to[br]find its magnitude.

0:04:53.350,0:04:55.700
And its magnitude[br]will be this animal.

0:04:55.700,0:04:59.012
So the only thing here[br]is tricky because you

0:04:59.012,0:05:03.480
will have to do this[br]carefully, and there

0:05:03.480,0:05:05.710
will be a simplification[br]coming from the plus

0:05:05.710,0:05:07.810
and minus of the binomial.

0:05:07.810,0:05:11.250
So a few people missed it[br]because of that reason.

0:05:11.250,0:05:13.780
So let's see what[br]we have-- minus

0:05:13.780,0:05:17.410
e to the minus t, first[br]prime, times second one

0:05:17.410,0:05:24.844
prime plus the first one[br]prime times the second prime.

0:05:24.844,0:05:25.770
Good.

0:05:25.770,0:05:28.066
We are done with this first guy.

0:05:28.066,0:05:34.443
The second guy will be minus[br]e to the minus t sin t.

0:05:34.443,0:05:36.910
Why do I do this?

0:05:36.910,0:05:40.730
Because I'm afraid that[br]this being on the final.

0:05:40.730,0:05:43.410
Well, it's good practice.

0:05:43.410,0:05:47.228
You may expect[br]something a little bit

0:05:47.228,0:05:53.014
similar to that, so why don't we[br]do this as part of our review,

0:05:53.014,0:05:55.324
which will be a very good idea.

0:05:55.324,0:05:58.290
We are gonna do lots of[br]review this week and next

0:05:58.290,0:06:01.710
week already, because[br]the final is coming close

0:06:01.710,0:06:09.832
and you have to go over[br]everything that you've covered.

0:06:09.832,0:06:16.428
Let's square them,[br]and add them together.

0:06:16.428,0:06:22.720


0:06:22.720,0:06:23.740
OK.

0:06:23.740,0:06:31.740
When we add them together,[br]this guy, the speed [INAUDIBLE]

0:06:31.740,0:06:33.880
is going to-- bless you.

0:06:33.880,0:06:36.990
It is going-- it's not going[br]to bless, it's going-- OK,

0:06:36.990,0:06:41.050
you are being blessed, and[br]now let's look at that.

0:06:41.050,0:06:46.120
You have e to the[br]minus 2t cosine squared

0:06:46.120,0:06:48.170
and e to the minus[br]2t sine squared,

0:06:48.170,0:06:50.672
and when you add[br]those parts, the sine

0:06:50.672,0:06:53.120
squared plus cosine[br]squared stick together.

0:06:53.120,0:06:56.808
They form a block called 1.

0:06:56.808,0:06:58.580
Do you guys agree with me?

0:06:58.580,0:07:02.960
So what we have as the[br]first result of that

0:07:02.960,0:07:04.882
would be this guy.

0:07:04.882,0:07:09.550
But then, when you[br]take twice the product

0:07:09.550,0:07:12.810
of these guys in the[br]binomial formula,

0:07:12.810,0:07:15.560
and twice the product of these[br]guys, what do you notice?

0:07:15.560,0:07:19.820
We have exactly the[br]same individuals inside,

0:07:19.820,0:07:23.250
but when you do twice the[br]product of these two red ones,

0:07:23.250,0:07:26.800
you have minus, minus, plus.

0:07:26.800,0:07:29.195
But when you do twice the[br]product of these guys,

0:07:29.195,0:07:31.870
you have minus, plus, minus.

0:07:31.870,0:07:33.410
So they will cancel out.

0:07:33.410,0:07:37.058
The part in the middle[br]will cancel out.

0:07:37.058,0:07:41.454
And finally, when I square[br]this part and that part,

0:07:41.454,0:07:42.620
what's going to happen them?

0:07:42.620,0:07:47.460
And I'm gonna shut up because I[br]want you to give me the answer.

0:07:47.460,0:07:50.570
Square this animal, square[br]this animal, add them together,

0:07:50.570,0:07:51.950
what do you have?

0:07:51.950,0:07:54.345
STUDENT: Squared, squared[br]total-- [INTERPOSING VOICES]

0:07:54.345,0:07:58.770
PROFESSOR: T to the minus 2t,[br]so exactly the same as this guy.

0:07:58.770,0:08:02.033
So all I know under[br]the square root,

0:08:02.033,0:08:12.230
I'm gonna get square root of[br]2 times e to the minus 2t.

0:08:12.230,0:08:15.052
Which is e to the minus[br]t square root of 2.

0:08:15.052,0:08:17.966
Am I right, [INAUDIBLE][br]that's what we got last time?

0:08:17.966,0:08:18.942
All right.

0:08:18.942,0:08:25.680
So I know who this will be.

0:08:25.680,0:08:30.450
I don't know who this will be,[br]but I'm gonna need your help.

0:08:30.450,0:08:34.710
Here I write it, x squared[br]of t plus y squared of t

0:08:34.710,0:08:38.740
in terms of t, squaring them[br]and adding them together.

0:08:38.740,0:08:42.700
It's gonna be again a piece of[br]cake, because you've got it.

0:08:42.700,0:08:45.670
How much is it?

0:08:45.670,0:08:47.650
I'm waiting for you to tell me.

0:08:47.650,0:08:50.620
This is this one.

0:08:50.620,0:08:52.120
[INAUDIBLE]

0:08:52.120,0:08:52.840
E to the?

0:08:52.840,0:08:54.060
STUDENT: Minus t

0:08:54.060,0:08:56.135
PROFESSOR: And anything else?

0:08:56.135,0:08:59.100
STUDENT: Was it 2?

0:08:59.100,0:08:59.600
[INAUDIBLE]

0:08:59.600,0:09:04.060


0:09:04.060,0:09:05.914
PROFESSOR: Why it times 2?

0:09:05.914,0:09:07.375
STUDENT: Times 2[br]in the last one.

0:09:07.375,0:09:09.958
Because we had an e to the minus[br]2t plus an e to the minus 2t.

0:09:09.958,0:09:13.706
PROFESSOR: So I took[br]this guy and squared it,

0:09:13.706,0:09:15.505
and I took this guy[br]and squared it--

0:09:15.505,0:09:17.130
STUDENT: No, we don't[br]have [INAUDIBLE].

0:09:17.130,0:09:19.432
PROFESSOR: And I sum them up.

0:09:19.432,0:09:21.520
And I close the issue.

0:09:21.520,0:09:28.397
Unless I have sine squared plus[br]cosine squared, which is 1,

0:09:28.397,0:09:30.782
so we adjust it to the minus 2t.

0:09:30.782,0:09:32.230
Agree with me?

0:09:32.230,0:09:34.090
All right, now we have[br]all the ingredients.

0:09:34.090,0:09:35.715
Do we have all the[br]ingredients we need?

0:09:35.715,0:09:37.696
We have this, we have[br]that, we have that.

0:09:37.696,0:09:40.730
And we should just go ahead[br]and solve the problem.

0:09:40.730,0:09:48.106
So, integral from 0 to pi over[br]2, this friend of yours, e

0:09:48.106,0:09:52.490
to the minus 2t[br]plus [INAUDIBLE].

0:09:52.490,0:09:59.026
The speed was over there, e to[br]the minus t times square root

0:09:59.026,0:10:00.002
of 2.

0:10:00.002,0:10:01.260
That was the speed.

0:10:01.260,0:10:05.384
[INAUDIBLE] magnitude, dt.

0:10:05.384,0:10:07.376
Is this what we got?

0:10:07.376,0:10:08.372
All right.

0:10:08.372,0:10:10.364
Now, we are almost[br]done, in the sense

0:10:10.364,0:10:14.348
that we should wrap things up.

0:10:14.348,0:10:16.860
Square root 2 gets out.

0:10:16.860,0:10:22.889
And then integral of it to the[br]minus 3t from zero to pi over 2

0:10:22.889,0:10:24.308
is our friend.

0:10:24.308,0:10:26.673
We know how to deal with him.

0:10:26.673,0:10:28.570
We have dt.

0:10:28.570,0:10:35.240
So when you integrate[br]that, what do you have?

0:10:35.240,0:10:36.680
Let me erase--

0:10:36.680,0:10:40.302
STUDENT: Negative square[br]root 2 over 3 [INAUDIBLE] 3t.

0:10:40.302,0:10:41.010
PROFESSOR: Right.

0:10:41.010,0:10:42.960
So let me erase this part.

0:10:42.960,0:10:47.600


0:10:47.600,0:10:50.610
So we have-- first we[br]have to copy this guy.

0:10:50.610,0:10:54.728
Then we have e to the[br]minus 3t divided by minus 3

0:10:54.728,0:10:58.150
because that is[br]the antiderivative.

0:10:58.150,0:11:04.950
And we take that into t equals[br]zero and t equals pi over 2.

0:11:04.950,0:11:10.650
Square root of 2 says I'm[br]going out, and actually minus 3

0:11:10.650,0:11:12.450
says also I'm going out.

0:11:12.450,0:11:16.474
So he doesn't want to be[br]involved in this discussion.

0:11:16.474,0:11:17.788
[INAUDIBLE]

0:11:17.788,0:11:23.650
Now, e to the minus 3 pi over[br]2 is the first thing we got.

0:11:23.650,0:11:26.520
And then minus e to the 0.

0:11:26.520,0:11:27.803
What's e to the 0?

0:11:27.803,0:11:28.610
STUDENT: 1.

0:11:28.610,0:11:30.310
PROFESSOR: 1.

0:11:30.310,0:11:35.050
So in the end, you have[br]to change the sign.

0:11:35.050,0:11:40.540
You have root 2 over 3[br]times bracket notation when

0:11:40.540,0:11:44.020
you type this in[br]WeBWorK because based

0:11:44.020,0:11:49.150
on your syntax, if your syntax[br]is bad, you are-- for example,

0:11:49.150,0:11:53.480
here we have to put[br]^ minus 3 pi over 2.

0:11:53.480,0:11:54.906
Are you guys with me?

0:11:54.906,0:11:57.030
Do you understand the words[br]coming out of my mouth?

0:11:57.030,0:12:01.840
So here you have to[br]type the right syntax,

0:12:01.840,0:12:03.743
and you did, and you got--

0:12:03.743,0:12:05.162
STUDENT: And I[br]didn't [INAUDIBLE]

0:12:05.162,0:12:08.000
but I need to type[br]the decimal answer.

0:12:08.000,0:12:10.332
In terms of decimal places.

0:12:10.332,0:12:11.540
PROFESSOR: This is a problem.

0:12:11.540,0:12:13.700
It shouldn't be like that.

0:12:13.700,0:12:18.040
Sometimes unfortunately--[br]well, fortunately it rarely

0:12:18.040,0:12:27.420
happens that WeBWorK program[br]does not take your answer

0:12:27.420,0:12:28.480
in a certain format.

0:12:28.480,0:12:30.620
Maybe the pi screws[br]everything up.

0:12:30.620,0:12:31.640
I don't know.

0:12:31.640,0:12:34.630
But if you do this[br]with your calculator,

0:12:34.630,0:12:38.460
eventually, you can What was[br]the approximate answer you got,

0:12:38.460,0:12:39.740
[INAUDIBLE]?

0:12:39.740,0:12:42.447
STUDENT: 0.467

0:12:42.447,0:12:43.780
PROFESSOR: And blah, blah, blah.

0:12:43.780,0:12:44.321
I don't know.

0:12:44.321,0:12:48.180
I think WeBWorK only cares[br]for the first two decimals

0:12:48.180,0:12:49.375
to be correct.

0:12:49.375,0:12:50.325
As I remember.

0:12:50.325,0:12:51.275
I don't know.

0:12:51.275,0:12:53.790
Now I have to ask[br]the programmer.

0:12:53.790,0:12:56.020
So this would be[br]approximately-- you

0:12:56.020,0:12:57.580
plug in the approximate answer.

0:12:57.580,0:12:59.499
I solved the problem,[br]so I should give myself

0:12:59.499,0:13:02.520
the credit, plus[br]a piece of candy,

0:13:02.520,0:13:07.300
but I hope I was able to[br]save you from some grief

0:13:07.300,0:13:10.910
because you have so much review[br]going on that you shouldn't

0:13:10.910,0:13:14.290
spend time on problems[br]that you headache

0:13:14.290,0:13:16.260
for computational reasons.

0:13:16.260,0:13:18.520
Actually, I have[br]computational reasons,

0:13:18.520,0:13:22.010
because we are not androids[br]and we are not computers.

0:13:22.010,0:13:27.190
What we can do is[br]think of a problem

0:13:27.190,0:13:31.230
and let the software[br]solve the problem for us.

0:13:31.230,0:13:36.020
So our strength does not consist[br]in how fast we can compute,

0:13:36.020,0:13:42.306
but on how well we can solve a[br]problem so that the calculator

0:13:42.306,0:13:45.880
or computer can carry on.

0:13:45.880,0:13:46.880
All right.

0:13:46.880,0:13:49.880


0:13:49.880,0:13:55.610
I know I covered[br]up to 13.6, and let

0:13:55.610,0:13:57.900
me remind you what we covered.

0:13:57.900,0:14:02.634
We covered some beautiful[br]sections that were called 13.4.

0:14:02.634,0:14:04.410
This was Green's theorem.

0:14:04.410,0:14:07.875
And now, I'm really proud[br]of you that all of you

0:14:07.875,0:14:10.350
know Green's theorem very well.

0:14:10.350,0:14:17.440
And the surface[br]integral, which was 13.5.

0:14:17.440,0:14:21.940
And then I promised[br]you that today we'd

0:14:21.940,0:14:25.874
move on to 13.6, which[br]is Stokes' theorem,

0:14:25.874,0:14:29.250
and I'm gonna do that.

0:14:29.250,0:14:34.380
But before I do that, I want[br]to attract your attention

0:14:34.380,0:14:49.920
to a fact that this is a bigger[br]result that incorporates 13.4.

0:14:49.920,0:14:53.470
So Stokes' theorem[br]is a more general

0:14:53.470,0:14:58.410
result. So let me make a[br]diagram, like a Venn diagram.

0:14:58.410,0:15:01.600
This is all the cases[br]of Stokes' theorem,

0:15:01.600,0:15:03.501
and Green's is one of them.

0:15:03.501,0:15:07.110


0:15:07.110,0:15:11.490
And this is something you've[br]learned, and you did very well,

0:15:11.490,0:15:14.650
and we only considered[br]this theorem

0:15:14.650,0:15:17.560
on a domain that's[br]interconnected.

0:15:17.560,0:15:19.520
It has no holes in it.

0:15:19.520,0:15:21.920
Green's theorem[br]can be also taught

0:15:21.920,0:15:25.080
on something like a[br]doughnut, but that's not

0:15:25.080,0:15:26.970
the purpose of this course.

0:15:26.970,0:15:28.540
You have it in the book.

0:15:28.540,0:15:30.110
It's very sophisticated.

0:15:30.110,0:15:37.470
13.6 starts at-- oh, my[br]God, I don't know the pages.

0:15:37.470,0:15:39.770
And being a co-author[br]of this book

0:15:39.770,0:15:42.640
means that I should[br]remember the pages.

0:15:42.640,0:15:44.200
All right, there it is.

0:15:44.200,0:15:50.760
13.6 is that page 1075.

0:15:50.760,0:15:56.695
OK, and let's see what[br]this theorem is about.

0:15:56.695,0:16:01.227


0:16:01.227,0:16:05.400
I'm gonna state it as[br]first Stokes' theorem,

0:16:05.400,0:16:08.850
and then I will see why Green's[br]theorem is a particular case.

0:16:08.850,0:16:11.110
We don't know yet why that is.

0:16:11.110,0:16:13.620
Well, assume you[br]have a force field,

0:16:13.620,0:16:15.250
may the force be with you.

0:16:15.250,0:16:30.260
This is a big vector-valued[br]function over a domain in R 3

0:16:30.260,0:16:39.150
that includes a surface s.

0:16:39.150,0:16:41.975
We don't say much[br]about the surface S

0:16:41.975,0:16:45.650
because we try to[br]avoid the terminology,

0:16:45.650,0:16:48.080
but you guys should[br]assume that this

0:16:48.080,0:17:00.190
is a simply connected[br]surface patch with a boundary

0:17:00.190,0:17:10.760
c, such that c is[br]a Jordan curve.

0:17:10.760,0:17:13.960


0:17:13.960,0:17:18.720
We use the word Jordan curve[br]as a boundary of the surface,

0:17:18.720,0:17:21.060
but we don't say[br]simply connected.

0:17:21.060,0:17:23.431
And I'm going to ask[br]you, what in the world

0:17:23.431,0:17:25.955
do we mean when we[br]simply connected?

0:17:25.955,0:17:27.920
I've used this before.

0:17:27.920,0:17:30.880
I just want to test your[br]memory and attention.

0:17:30.880,0:17:33.170
Do you remember what that meant?

0:17:33.170,0:17:39.170
I have some sort of little[br]hill, or s could be a flat disc,

0:17:39.170,0:17:41.330
or it could be a[br]patch of the plane,

0:17:41.330,0:17:46.170
or it could be just any[br]kind of surface that

0:17:46.170,0:17:49.142
is bounded by Jordan curve c.

0:17:49.142,0:17:50.100
What is a Jordan curve?

0:17:50.100,0:17:52.480
But what can we say about c?

0:17:52.480,0:17:56.140


0:17:56.140,0:18:00.500
So c would be nice[br]piecewise continuous-- we

0:18:00.500,0:18:02.380
assumed it continuous actually.

0:18:02.380,0:18:03.320
Most cases--

0:18:03.320,0:18:05.361
STUDENT: It has to connect[br]to itself, doesn't it?

0:18:05.361,0:18:07.660
PROFESSOR: No[br]self-intersections.

0:18:07.660,0:18:10.570
So we knew that from before,[br]but what does it mean,

0:18:10.570,0:18:15.440
simply connected for us?

0:18:15.440,0:18:16.395
I said it before.

0:18:16.395,0:18:19.930
I don't know how[br]attentive you were.

0:18:19.930,0:18:22.600
Connectedness makes[br]you think of something.

0:18:22.600,0:18:23.850
No holes in it.

0:18:23.850,0:18:26.290
So that means no holes.

0:18:26.290,0:18:27.260
No punctures.

0:18:27.260,0:18:31.200
No holes, no punctures.

0:18:31.200,0:18:33.070
So why-- don't draw it.

0:18:33.070,0:18:35.250
I will draw it so you can laugh.

0:18:35.250,0:18:37.320
Assume that the dog[br]came here and took

0:18:37.320,0:18:39.420
a bite of this surface.

0:18:39.420,0:18:41.720
And now you have a hole in it.

0:18:41.720,0:18:43.710
Well, you're not supposed[br]to have a hole in it,

0:18:43.710,0:18:45.960
so tell the dog to go away.

0:18:45.960,0:18:48.820
So you're not gonna have[br]any problems, any puncture,

0:18:48.820,0:18:52.600
any hole, any problem with this.

0:18:52.600,0:18:57.780
Now the surface is assumed[br]to be a regular surface,

0:18:57.780,0:18:59.830
and we've seen that before.

0:18:59.830,0:19:03.000
And since it's a regular[br]surface, that means

0:19:03.000,0:19:05.150
it's immersed in[br]the ambient space,

0:19:05.150,0:19:09.100
and you have an N orientation.

0:19:09.100,0:19:16.860
Orientation which is the[br]unit normal to the surface.

0:19:16.860,0:19:19.630


0:19:19.630,0:19:20.880
Can you draw it, Magdalena?

0:19:20.880,0:19:24.180
Yes, in a minute,[br]I will draw it.

0:19:24.180,0:19:29.560
At every point you[br]have an N unit normal.

0:19:29.560,0:19:32.560
What was the unit[br]normal for you when

0:19:32.560,0:19:35.540
you parametrize the surface?

0:19:35.540,0:19:40.210
That was the stick that[br]has length 1 perpendicular

0:19:40.210,0:19:41.860
to the tangent length, right?

0:19:41.860,0:19:44.840
So if you wanted to[br]do it for general R,

0:19:44.840,0:19:48.860
you would take those R sub u or[br]sub v the partial [INAUDIBLE].

0:19:48.860,0:19:53.220
And draw the cross[br]product, and this

0:19:53.220,0:19:55.060
is what I'm trying to do now.

0:19:55.060,0:19:57.570
And just make the length be 1.

0:19:57.570,0:20:02.170
So if the surface is[br]regular, I can parametrize it

0:20:02.170,0:20:06.650
as [INAUDIBLE] will exist[br]in that orientation.

0:20:06.650,0:20:09.000
I want something more.

0:20:09.000,0:20:17.360
I want N orientation to be[br]compatible to the direction

0:20:17.360,0:20:26.450
of travel on c-- along c.

0:20:26.450,0:20:29.540
Along, not on, but[br]on is not bad either.

0:20:29.540,0:20:32.460
So assume that this[br]is a hill, and I'm

0:20:32.460,0:20:34.170
running around the boundary.

0:20:34.170,0:20:37.340
Look, I'm just running around[br]the boundary, which is c.

0:20:37.340,0:20:40.270
Am I running in a[br]particular direction that

0:20:40.270,0:20:42.870
tells you I'm a mathematician?

0:20:42.870,0:20:44.660
It tells you that I'm a weirdo.

0:20:44.660,0:20:45.520
Yes.

0:20:45.520,0:20:49.240
So in what kind of[br]direction am I running?

0:20:49.240,0:20:50.610
Counterclockwise.

0:20:50.610,0:20:51.110
Why?

0:20:51.110,0:20:52.740
Because I'm a nerd.

0:20:52.740,0:20:54.560
Like Sheldon or something.

0:20:54.560,0:20:59.410
So let's go around,[br]and so what does

0:20:59.410,0:21:05.294
it mean I am compatible[br]with the orientation?

0:21:05.294,0:21:07.010
Think of the right hand rule.

0:21:07.010,0:21:10.210
Or forget about right hand[br]rule, I hate that word.

0:21:10.210,0:21:12.940
Let's think faucet.

0:21:12.940,0:21:17.340
So if your motion[br]is along the c,

0:21:17.340,0:21:20.950
so that it's like you are[br]unscrewing the faucet,

0:21:20.950,0:21:22.650
it's going up.

0:21:22.650,0:21:25.050
That should mean[br]that your orientation

0:21:25.050,0:21:28.660
n should go up, or, not[br]down, in the other direction.

0:21:28.660,0:21:32.750
So if I take c to be my[br]orientation around the curve,

0:21:32.750,0:21:36.000
then the orientation of[br]the surface should go up.

0:21:36.000,0:21:41.150
Am I allowed to go around the[br]opposite direction on the c.

0:21:41.150,0:21:42.880
Yes I am.

0:21:42.880,0:21:46.020
That's, how it this called,[br]inverse trigonometric,

0:21:46.020,0:21:48.035
or how do we call such a thing.

0:21:48.035,0:21:49.398
STUDENT: Clockwise?

0:21:49.398,0:21:52.146
PROFESSOR: Clockwise,[br]you guessed it.

0:21:52.146,0:21:53.520
OK, clockwise.

0:21:53.520,0:21:57.310
If I would go[br]clockwise in plane,

0:21:57.310,0:22:00.980
then the N should[br]be pointing down.

0:22:00.980,0:22:04.090
So it should be oriented[br]just the opposite way

0:22:04.090,0:22:06.410
on the surface S.

0:22:06.410,0:22:10.590
All right, that's sort[br]of easy to understand now

0:22:10.590,0:22:12.794
because most of[br]you are engineers

0:22:12.794,0:22:17.240
and you deal with this[br]kind of stuff every day.

0:22:17.240,0:22:18.960
What is Stokes' theorem?

0:22:18.960,0:22:23.490
Stokes' theorem says well, in[br]that case, the path integral

0:22:23.490,0:22:30.211
over c of FdR, F dot dR.

0:22:30.211,0:22:32.040
What the heck is this?

0:22:32.040,0:22:35.250
I'm not gonna finish the[br]sentence, because I'm mean.

0:22:35.250,0:22:37.930
There is a sentence there,[br]an equation, but I'm mean.

0:22:37.930,0:22:43.400
So I'm asking you first,[br]what in the world is this?

0:22:43.400,0:22:45.590
F is the may the[br]force be with you.

0:22:45.590,0:22:47.880
R is the vector position.

0:22:47.880,0:22:49.770
What is this animal?

0:22:49.770,0:22:51.912
The book doesn't tell you.

0:22:51.912,0:22:54.210
This is the work that[br]you know so well.

0:22:54.210,0:22:55.750
All right.

0:22:55.750,0:22:59.910
So you may hear math majors[br]saying they don't care.

0:22:59.910,0:23:02.420
They don't care because they're[br]not engineers or physicists,

0:23:02.420,0:23:04.710
but work is very important.

0:23:04.710,0:23:10.588
The work along the curve[br]will be equal to-- now

0:23:10.588,0:23:14.004
comes the beauty--[br]the beautiful part.

0:23:14.004,0:23:17.270
This is a double integral[br]over the surface [INAUDIBLE]

0:23:17.270,0:23:22.240
with respect to the[br]area element dS.

0:23:22.240,0:23:26.580
Oh, guess what, you wouldn't[br]know unless somebody taught you

0:23:26.580,0:23:29.810
before coming to[br]class, this is going

0:23:29.810,0:23:34.220
to be curl F. What is curl?

0:23:34.220,0:23:35.140
It's a vector.

0:23:35.140,0:23:39.075
So I have to do dot product[br]with another vector.

0:23:39.075,0:23:40.780
And that vector is N.

0:23:40.780,0:23:45.410
Some people read the book[br]ahead of time, which is great.

0:23:45.410,0:23:52.370
I would say 0.5% or less[br]of the students read ahead

0:23:52.370,0:23:53.170
in a textbook.

0:23:53.170,0:23:56.650
I used to do that[br]when I was young.

0:23:56.650,0:24:00.296
I didn't always have[br]the time to do it,

0:24:00.296,0:24:04.828
but whenever I had the[br]possibility I did it.

0:24:04.828,0:24:11.950
Now a quiz for you.

0:24:11.950,0:24:15.830
No, don't take any sheets[br]out, but a quiz for you.

0:24:15.830,0:24:18.930
Could you prove[br]to me, just based

0:24:18.930,0:24:24.710
on this thing that looks[br]weird, that Green's theorem is

0:24:24.710,0:24:28.190
a particular case of this?

0:24:28.190,0:24:32.160
So prove-- where[br]should I put it?

0:24:32.160,0:24:35.155
That was Stokes' theorem.

0:24:35.155,0:24:36.940
Stokes' theorem.

0:24:36.940,0:24:39.830
And I'll say[br]exercise number one,

0:24:39.830,0:24:42.812
sometimes I put this[br]in the final exam,

0:24:42.812,0:24:46.040
so I consider this[br]to be important.

0:24:46.040,0:25:06.560
Prove that Green's theorem is[br]nothing but a particular case

0:25:06.560,0:25:08.036
of Stokes' theorem.

0:25:08.036,0:25:12.464


0:25:12.464,0:25:17.384
And I make a face in the sense[br]that I'm trying to build trust.

0:25:17.384,0:25:18.860
Maybe you don't trust me.

0:25:18.860,0:25:21.500
But I-- let's do this together.

0:25:21.500,0:25:23.820
Let's prove together[br]that this is what it is.

0:25:23.820,0:25:31.160
Now, the thing is, if I were[br]to give you a test right now

0:25:31.160,0:25:33.580
on Green's theorem,[br]how many of you would

0:25:33.580,0:25:35.630
know what Green's theorem said?

0:25:35.630,0:25:39.900
So I'll put it here[br]in an-- open an icon.

0:25:39.900,0:25:44.170
Imagine this would be[br]an icon-- or a window,

0:25:44.170,0:25:48.090
a window on the computer screen.

0:25:48.090,0:25:52.898
Like a tutorial reminding[br]you what Green's theorem was.

0:25:52.898,0:25:54.854
So Green's theorem said what?

0:25:54.854,0:25:57.790


0:25:57.790,0:26:00.910
We have to-- bless you.

0:26:00.910,0:26:05.770
So Zander started[br]the theorem by-- we

0:26:05.770,0:26:12.660
have a domain D that was[br]also simply connected.

0:26:12.660,0:26:13.695
What does it mean?

0:26:13.695,0:26:15.515
No punctures, no holes.

0:26:15.515,0:26:16.425
No holes.

0:26:16.425,0:26:20.500
Even if you have a[br]puncture that's a point,

0:26:20.500,0:26:22.090
that's still a hole.

0:26:22.090,0:26:27.670
You may not see it, but if[br]anybody punctured the portion

0:26:27.670,0:26:30.230
of a plane, you are in trouble.

0:26:30.230,0:26:32.190
So there are no such things.

0:26:32.190,0:26:34.040
And c is a Jordan curve.

0:26:34.040,0:26:41.470


0:26:41.470,0:26:44.460
And then you say, OK,[br]how is it, how was F?

0:26:44.460,0:26:46.420
F was a c 1.

0:26:46.420,0:26:48.690
What does it mean[br]that if is a c 1?

0:26:48.690,0:26:52.550
F is a vector-valued function[br]that's differentiable,

0:26:52.550,0:26:56.020
and its derivatives[br]are continuous,

0:26:56.020,0:26:57.520
partial derivatives.

0:26:57.520,0:27:07.930
And so you think F of xy will[br]be M of xy I plus n of xyj

0:27:07.930,0:27:14.850
is a vector field, so[br]it's a multivariable,

0:27:14.850,0:27:16.550
so I have two variables.

0:27:16.550,0:27:18.400
OK?

0:27:18.400,0:27:21.130
So you think, OK, I[br]know what this is.

0:27:21.130,0:27:23.100
Like, this would be a force.

0:27:23.100,0:27:25.590
If this were a[br]force, I would get

0:27:25.590,0:27:33.740
the vector-- the work--[br]how can I write this again?

0:27:33.740,0:27:35.210
We didn't write it like that.

0:27:35.210,0:27:38.180
We wrote it as Mdx[br]plus Ndy, which

0:27:38.180,0:27:40.921
is the same thing as before.

0:27:40.921,0:27:41.420
Why?

0:27:41.420,0:27:45.780
Because Mr. F is MI plus NJ.

0:27:45.780,0:27:47.195
Not k, Magdalena.

0:27:47.195,0:27:50.420
You were too nice, but you[br]didn't want to shout at me.

0:27:50.420,0:27:56.610
And dR was what? dR was[br]dxI plus dyJ, right?

0:27:56.610,0:28:00.560
So when you do this, the[br]product which is called work,

0:28:00.560,0:28:04.420
the integral will[br]read Mdx plus Ndy,

0:28:04.420,0:28:08.315
and this is what it[br]was in Green's theorem.

0:28:08.315,0:28:10.580
And what did we claim it was?

0:28:10.580,0:28:13.450
Now, you know it, because[br]you've done a lot of homework.

0:28:13.450,0:28:16.570
You're probably sick and tired[br]of Green's theorem and you say,

0:28:16.570,0:28:20.540
I understand that work-- a[br]path integral can be expressed

0:28:20.540,0:28:23.930
as a double integral some way.

0:28:23.930,0:28:25.600
Do you know this by heart?

0:28:25.600,0:28:28.920
You proved this to me last[br]time you know it by heart.

0:28:28.920,0:28:33.020
That was N sub x minus M sub y.

0:28:33.020,0:28:40.080
And we memorized it-- dA[br]over this is a planar domain.

0:28:40.080,0:28:43.750
It's a domain in[br]plane d, [INAUDIBLE].

0:28:43.750,0:28:45.760
I said it, but I[br]didn't write it down.

0:28:45.760,0:28:50.654
So double integral over d,[br]N sub x minus M sub y dA.

0:28:50.654,0:28:51.320
We've done that.

0:28:51.320,0:28:55.330
That was section-- what[br]section was that, guys?

0:28:55.330,0:28:56.820
13.4.

0:28:56.820,0:29:00.830
Yeah, for sure, you will have a[br]problem on the final like that.

0:29:00.830,0:29:03.800
Do not expect lots of problems.

0:29:03.800,0:29:05.320
Do not expect 25 problems.

0:29:05.320,0:29:06.700
You will not have the time.

0:29:06.700,0:29:09.198
So you will have[br]some 15, 16 problems.

0:29:09.198,0:29:12.110
This will be one of them.

0:29:12.110,0:29:14.560
You mastered this[br]Green's theorem.

0:29:14.560,0:29:18.640
When you sent me[br]questions from WeBWorK

0:29:18.640,0:29:23.070
I realized that you were able[br]to solve the problems where

0:29:23.070,0:29:26.025
these would be[br]easy to manipulate,

0:29:26.025,0:29:27.140
like constants and so on.

0:29:27.140,0:29:29.410
That's a beautiful case.

0:29:29.410,0:29:33.190
There was one that gave[br][INAUDIBLE] a headache,

0:29:33.190,0:29:35.318
and then I decided--[br]number 22, right?

0:29:35.318,0:29:40.410
Where this was more complicated[br]as an integrant in y, and guys,

0:29:40.410,0:29:43.130
your domain was like that.

0:29:43.130,0:29:46.680
And then normally to integrate[br]with respect to y and then x,

0:29:46.680,0:29:50.770
you would have had to split this[br]integral into two integrals--

0:29:50.770,0:29:53.002
one over a part of the[br]triangle, the other one

0:29:53.002,0:29:54.770
over part of the triangle.

0:29:54.770,0:30:00.060
So the easier way[br]was to do it how?

0:30:00.060,0:30:04.320
To do it like that, with[br]horizontal integrals.

0:30:04.320,0:30:05.360
And we've done that.

0:30:05.360,0:30:07.920
I told you-- I gave[br]you too much, actually,

0:30:07.920,0:30:12.780
I served it to you on a[br]plate, the proof-- solution

0:30:12.780,0:30:13.752
of that problem.

0:30:13.752,0:30:15.210
But you have many others.

0:30:15.210,0:30:20.745
Now, how do we prove that[br]this individual equation that

0:30:20.745,0:30:23.360
looks so sophisticated[br]is nothing

0:30:23.360,0:30:29.520
but that for the case[br]when S is a planar patch?

0:30:29.520,0:30:33.920
If S is like a hill,[br]yeah, then we believe it.

0:30:33.920,0:30:37.560
But what if S is the[br]domain d in plane Well,

0:30:37.560,0:30:43.080
then this S is exactly this d.

0:30:43.080,0:30:44.850
So it reduces to d.

0:30:44.850,0:30:46.630
So you say, wait a[br]minute, doesn't it

0:30:46.630,0:30:48.170
have to be curvilinear?

0:30:48.170,0:30:49.710
Nope.

0:30:49.710,0:30:56.290
Any surface that is bounded[br]by c verifies Stokes' theorem.

0:30:56.290,0:30:57.570
Say it again, Magdalena.

0:30:57.570,0:31:01.770
Any surface S that[br]is regular, so I'm

0:31:01.770,0:31:04.560
within the conditions[br]of the theorem, that

0:31:04.560,0:31:07.300
is bounded by a Jordan curve,[br]will satisfy the theorem.

0:31:07.300,0:31:09.340
So let's see what I've become.

0:31:09.340,0:31:11.520
That should became[br]a friend of yours,

0:31:11.520,0:31:14.540
and we already know[br]who this guy is.

0:31:14.540,0:31:24.660
So the integral FdR is your[br]friend integral Mdx plus Ndy

0:31:24.660,0:31:26.790
that's staring at you over c.

0:31:26.790,0:31:31.410
It's an integral over one[br]form, and it says that's work.

0:31:31.410,0:31:35.120
And the right-hand side, it's[br]a little bit more complicated.

0:31:35.120,0:31:36.950
So we have to think.

0:31:36.950,0:31:38.300
We have to think.

0:31:38.300,0:31:41.740
It's not about computation,[br]it's about how good we

0:31:41.740,0:31:45.790
are at identifying everybody.

0:31:45.790,0:31:50.130
If I go, for this particular[br]case, S is d, right?

0:31:50.130,0:31:51.560
Right.

0:31:51.560,0:31:56.965
So I have a double integral[br]over D. Sometimes you ask me,

0:31:56.965,0:32:02.000
but I saw that over a domain[br]that's a two-dimensional domain

0:32:02.000,0:32:05.420
people wrote only one[br]snake, and it looks fat,

0:32:05.420,0:32:08.030
like somebody fed[br]the snake too much.

0:32:08.030,0:32:10.730
Mathematicians are lazy people.

0:32:10.730,0:32:14.070
They don't want to write always[br]double snake, triple snake.

0:32:14.070,0:32:16.570
So sometimes they say,[br]I have an integral

0:32:16.570,0:32:18.030
over an n-dimensional domain.

0:32:18.030,0:32:20.136
I'll make it a fat snake.

0:32:20.136,0:32:22.860
And that should be enough.

0:32:22.860,0:32:25.680
Curl F N-- we have[br]to do this together.

0:32:25.680,0:32:26.460
Is it hard?

0:32:26.460,0:32:27.390
I don't know.

0:32:27.390,0:32:30.880
You have to help me.

0:32:30.880,0:32:33.221
So what in the world was that?

0:32:33.221,0:32:34.470
I pretend I forgot everything.

0:32:34.470,0:32:35.207
I have amnesia.

0:32:35.207,0:32:36.082
STUDENT: [INAUDIBLE].

0:32:36.082,0:32:39.040


0:32:39.040,0:32:41.190
PROFESSOR: Yeah, so[br]actually some of you

0:32:41.190,0:32:45.790
told me by email[br]that you prefer that.

0:32:45.790,0:32:49.240
I really like it[br]that you-- maybe I

0:32:49.240,0:32:52.106
should have started a[br]Facebook group or something.

0:32:52.106,0:32:55.640
Because instead of the[br]personal email interaction

0:32:55.640,0:32:59.660
between me and you,[br]everybody could see this.

0:32:59.660,0:33:03.810
So some of you tell me, I[br]like better this notation,

0:33:03.810,0:33:05.950
because I use it[br]in my engineering

0:33:05.950,0:33:12.160
course, curl F. OK, good, it's[br]up to you what you want to use.

0:33:12.160,0:33:15.520
d/dx, d/dy-- I mean it.

0:33:15.520,0:33:18.820
In principle, in r3,[br]but I'm really lucky.

0:33:18.820,0:33:23.134
Because in this case, F[br]is in r2, value in r2.

0:33:23.134,0:33:24.115
STUDENT: You mean d/dz?

0:33:24.115,0:33:24.740
PROFESSOR: Huh?

0:33:24.740,0:33:26.140
STUDENT: d/dx, d/dy, d/dz.

0:33:26.140,0:33:27.050
PROFESSOR: I'm sorry.

0:33:27.050,0:33:29.782
You are so on the ball.

0:33:29.782,0:33:30.859
Thank you, Alexander.

0:33:30.859,0:33:33.150
STUDENT: No, I thought I had[br]completely misunderstood--

0:33:33.150,0:33:36.280
PROFESSOR: No, no, no,[br]no, I wrote it twice.

0:33:36.280,0:33:41.860
So M and N and 0, M is a[br]function of x and y only.

0:33:41.860,0:33:44.420
N of course-- do I[br]have to write that?

0:33:44.420,0:33:46.550
No, I'm just being silly.

0:33:46.550,0:33:49.789
And what do I get in this case?

0:33:49.789,0:33:50.664
STUDENT: [INAUDIBLE].

0:33:50.664,0:33:56.080


0:33:56.080,0:33:59.146
PROFESSOR: I times this[br]guy-- how much is this guy?

0:33:59.146,0:33:59.869
STUDENT: 0.

0:33:59.869,0:34:00.410
PROFESSOR: 0.

0:34:00.410,0:34:01.700
Why is that 0?

0:34:01.700,0:34:05.150
Because this contains no z,[br]and I prime with respect to z.

0:34:05.150,0:34:11.500
So that is nonsense,[br]0i minus 0j.

0:34:11.500,0:34:12.190
Why is that?

0:34:12.190,0:34:16.550
Because 0 minus something[br]that doesn't depend on z.

0:34:16.550,0:34:22.520
So plus, finally-- the[br]only guy that matters there

0:34:22.520,0:34:28.409
is [INAUDIBLE], which[br]is this, which is that.

0:34:28.409,0:34:31.859
So because I have[br]derivative of N

0:34:31.859,0:34:34.520
with respect to h[br]minus derivative of M

0:34:34.520,0:34:35.563
with respect to y.

0:34:35.563,0:34:39.260


0:34:39.260,0:34:42.320
And now I stare at it,[br]and I say, times k.

0:34:42.320,0:34:45.945
That's the only guy that's[br]not 0, the only component.

0:34:45.945,0:34:49.270
Now I'm going to go[br]ahead and multiply this

0:34:49.270,0:34:51.090
in the top product with him.

0:34:51.090,0:34:54.614
But we have to be smart[br]and think, N is what?

0:34:54.614,0:34:56.117
STUDENT: [INAUDIBLE].

0:34:56.117,0:34:57.700
PROFESSOR: It's[br]normal to the surface.

0:34:57.700,0:35:01.930
But the surface is[br]a patch of a plane.

0:35:01.930,0:35:03.460
The normal would be trivial.

0:35:03.460,0:35:05.376
What will the normal be?

0:35:05.376,0:35:10.890
The vector field of all[br]pencils that are k-- k.

0:35:10.890,0:35:12.820
It's all k, k everywhere.

0:35:12.820,0:35:14.550
All over the domain is k.

0:35:14.550,0:35:16.270
So N becomes k.

0:35:16.270,0:35:17.623
Where is it?

0:35:17.623,0:35:21.080
There, N becomes k.

0:35:21.080,0:35:24.200
So when you multiply[br]in the dot product

0:35:24.200,0:35:26.624
this guy with this[br]guy, what do you have?

0:35:26.624,0:35:27.620
STUDENT: [INAUDIBLE].

0:35:27.620,0:35:35.010
PROFESSOR: N sub x[br]minus M sub y dA.

0:35:35.010,0:35:37.890


0:35:37.890,0:35:41.680
QED-- what does it mean, QED?

0:35:41.680,0:35:44.100
$1, which I don't[br]have, for the person

0:35:44.100,0:35:46.740
who will tell me what that is.

0:35:46.740,0:35:50.961


0:35:50.961,0:36:05.310
Latin-- quod erat demonstrandum,[br]which was to be proved, yes?

0:36:05.310,0:36:06.390
So I'm done.

0:36:06.390,0:36:10.155
When people put QED, that means[br]they are done with the proof.

0:36:10.155,0:36:13.810
But now since mathematicians[br]are a little bit illiterate,

0:36:13.810,0:36:17.340
they don't know much about[br]philosophy or linguistics.

0:36:17.340,0:36:19.440
Now many of them,[br]instead of QED,

0:36:19.440,0:36:22.540
they put a little square box.

0:36:22.540,0:36:24.610
And we do the same in our books.

0:36:24.610,0:36:28.230
So that means I'm[br]done with the proof.

0:36:28.230,0:36:30.520
Let's go home, but not that.

0:36:30.520,0:36:39.590
So we proved that for the[br]particular case of the planar

0:36:39.590,0:36:43.550
domains, Stokes' theorem[br]becomes Green's theorem.

0:36:43.550,0:36:46.480
And actually this is the curl.

0:36:46.480,0:36:49.580
And this-- well, not the curl.

0:36:49.580,0:36:55.140
But you have the curl of F[br]multiplied with dot product

0:36:55.140,0:36:58.180
with k and this green[br]fellow is exactly

0:36:58.180,0:37:02.215
the same as N sub[br]x minus N sub y

0:37:02.215,0:37:06.750
smooth function,[br]real value function.

0:37:06.750,0:37:10.140
All right, am I done?

0:37:10.140,0:37:13.450
Yes, with this Exercise 1,[br]which is a proof, I'm done.

0:37:13.450,0:37:17.700
You haven't seen many[br]proofs in calculus.

0:37:17.700,0:37:21.060
You've seen some from[br]me that we never cover.

0:37:21.060,0:37:24.850
We don't do epsilon delta in[br]regular classes of calculus,

0:37:24.850,0:37:25.980
only in honors.

0:37:25.980,0:37:28.510
And not in all the honors[br]you've seen some proofs

0:37:28.510,0:37:29.980
with epsilon delta.

0:37:29.980,0:37:35.650
You've seen one or two[br]proofs from me occasionally.

0:37:35.650,0:37:40.461
And this was one simple proof[br]that I wanted to work with you.

0:37:40.461,0:37:42.996
Now, do you know if[br]you're ever going

0:37:42.996,0:37:47.260
to see proofs in math[br]classes, out of curiosity?

0:37:47.260,0:37:51.170
US It depends how much[br]math you want to take.

0:37:51.170,0:37:55.950
If you're a math major, you[br]take a course called 3310.

0:37:55.950,0:37:58.720
That's called[br]Introduction to Proofs.

0:37:58.720,0:38:02.090
If you are not a math[br]major, but assume

0:38:02.090,0:38:06.200
you are in this[br]dual program-- we

0:38:06.200,0:38:11.290
have a beautiful and tough dual[br]major, mathematics and computer

0:38:11.290,0:38:14.860
science, 162 hours.

0:38:14.860,0:38:17.710
Then you see everything[br]you would normally

0:38:17.710,0:38:19.760
see for an engineering major.

0:38:19.760,0:38:22.400
But in addition, you[br]see a few more courses

0:38:22.400,0:38:24.560
that have excellent proofs.

0:38:24.560,0:38:28.740
And one of them is linear[br]algebra, Linear Algebra 2360.

0:38:28.740,0:38:32.690
We do a few proofs--[br]depends who teaches that.

0:38:32.690,0:38:35.020
And in 3310 also[br]you see some proofs

0:38:35.020,0:38:37.720
like, by way of contradiction,[br]let's prove this and that.

0:38:37.720,0:38:41.220
OK, so it's sort of fun.

0:38:41.220,0:38:45.930
But we don't attempt long[br]and nasty, complicated proofs

0:38:45.930,0:38:49.035
until you are in graduate[br]school, normally.

0:38:49.035,0:38:52.590
Some of you will do[br]graduate studies.

0:38:52.590,0:38:54.320
Some of you-- I[br]know four of you--

0:38:54.320,0:38:57.510
want to go to medical school.

0:38:57.510,0:39:02.424
And then many of you hopefully[br]will get a graduate program

0:39:02.424,0:39:06.400
in engineering.

0:39:06.400,0:39:11.210
OK, let's see another[br]example for this section.

0:39:11.210,0:39:15.260
I don't particularly[br]like all the examples

0:39:15.260,0:39:17.680
we have in the book.

0:39:17.680,0:39:21.672
But I have my favorites.

0:39:21.672,0:39:31.096
And I'm going to go[br]ahead and choose one.

0:39:31.096,0:39:34.600


0:39:34.600,0:39:38.500
There is one that's a[br]little bit complicated.

0:39:38.500,0:39:40.650
And you asked me about it.

0:39:40.650,0:39:45.740
And I wanted to[br]talk about this one.

0:39:45.740,0:39:51.005
Because it gave several[br]of you a headache.

0:39:51.005,0:40:00.060
There is Example 1, which[br]says-- what does it say?

0:40:00.060,0:40:07.460
Evaluate fat integral[br]over C of 1 over 2 i

0:40:07.460,0:40:23.340
squared dx plus zdy plus xdz[br]where C is the intersection

0:40:23.340,0:40:47.210
curve between the plane x plus[br]z equals 1 and the ellipsoid x

0:40:47.210,0:40:50.410
squared plus 2y[br]squared plus z squared

0:40:50.410,0:41:01.306
equals 1 that's oriented[br]counterclockwise as viewed

0:41:01.306,0:41:03.348
from the above picture.

0:41:03.348,0:41:06.040
And I need to draw the picture.

0:41:06.040,0:41:09.630
The picture looks really ugly.

0:41:09.630,0:41:11.590
You have this ellipsoid.

0:41:11.590,0:41:17.470


0:41:17.470,0:41:23.920
And when you draw[br]this intersection

0:41:23.920,0:41:29.780
between this plane and the[br]ellipsoid, it looks horrible.

0:41:29.780,0:41:32.740
And the hint of[br]this problem-- well,

0:41:32.740,0:41:38.460
if you were to be given[br]such a thing on an exam,

0:41:38.460,0:41:41.990
the hint would be[br]that a projection--

0:41:41.990,0:41:42.990
look at the picture.

0:41:42.990,0:41:50.815
The projection of the curve of[br]intersection on the ground--

0:41:50.815,0:41:53.610
ground means the[br]plane on the equator.

0:41:53.610,0:41:54.660
How shall I say that?

0:41:54.660,0:42:01.010
The x, y plane is this.

0:42:01.010,0:42:02.730
It looks horrible.

0:42:02.730,0:42:10.400


0:42:10.400,0:42:12.110
And it looks like an egg.

0:42:12.110,0:42:13.568
It's not supposed[br]to be an egg, OK?

0:42:13.568,0:42:14.670
It's a circle.

0:42:14.670,0:42:16.780
I'm sorry if it[br]looks like an egg.

0:42:16.780,0:42:20.540


0:42:20.540,0:42:24.892
OK, and that would be the[br]only hint you would get.

0:42:24.892,0:42:29.430
You would be asked to[br]figure out this circle

0:42:29.430,0:42:31.745
in polar coordinates.

0:42:31.745,0:42:37.395
And I'm not sure if all of[br]you would know how to do that.

0:42:37.395,0:42:41.070
And this is what worried me.

0:42:41.070,0:42:44.570
So before we do everything,[br]before everything,

0:42:44.570,0:42:49.930
can we express this[br]in polar coordinates?

0:42:49.930,0:42:54.280
How are you going to set[br]up something in r theta

0:42:54.280,0:42:59.159
for the same domain[br]inside this disc?

0:42:59.159,0:43:00.034
STUDENT: [INAUDIBLE].

0:43:00.034,0:43:27.807


0:43:27.807,0:43:29.390
PROFESSOR: So if we[br]were, for example,

0:43:29.390,0:43:32.820
to say x is r cosine[br]theta, can we do that?

0:43:32.820,0:43:35.600
And i to be r sine[br]theta, what would

0:43:35.600,0:43:37.655
we get instead of this equation?

0:43:37.655,0:43:41.390
Because it looks horrible.

0:43:41.390,0:43:45.190
We would get-- this equation,[br]let's brush it up a little bit

0:43:45.190,0:43:45.810
first.

0:43:45.810,0:43:48.430
It's x squared plus y squared.

0:43:48.430,0:43:49.450
And that's nice.

0:43:49.450,0:43:57.320
But then it's minus twice--[br]it's just x plus 1/4

0:43:57.320,0:44:00.840
equals 1/4, the heck with it.

0:44:00.840,0:44:02.440
My son says, don't say "heck."

0:44:02.440,0:44:03.270
That's a bad word.

0:44:03.270,0:44:04.700
I didn't know that.

0:44:04.700,0:44:07.530
But he says that he's being[br]told in school it's a bad word.

0:44:07.530,0:44:09.870
So he must know what[br]he's talking about.

0:44:09.870,0:44:13.390
So this is r squared.

0:44:13.390,0:44:15.265
And x is r cosine theta.

0:44:15.265,0:44:20.970
Aha, so there we almost[br]did it in the sense

0:44:20.970,0:44:24.170
that r squared equals[br]r cosine theta is

0:44:24.170,0:44:27.460
the polar equation,[br]equation of the circle

0:44:27.460,0:44:28.620
in polar coordinates.

0:44:28.620,0:44:31.690
But we hate r.

0:44:31.690,0:44:33.280
Let's simplify by an r.

0:44:33.280,0:44:36.950
Because r is positive--[br]cannot be 0, right?

0:44:36.950,0:44:38.190
It would be a point.

0:44:38.190,0:44:44.200
So divide by r and get[br]r equals cosine theta.

0:44:44.200,0:44:45.770
So what is r equals[br]cosine theta?

0:44:45.770,0:44:49.590
r equals cosine theta[br]is your worst nightmare.

0:44:49.590,0:44:52.020
So I'm going to make a face.

0:44:52.020,0:44:55.263
That was your worst[br]nightmare in Calculus II.

0:44:55.263,0:44:58.522
And I was just talking to a[br]few colleagues in Calculus II

0:44:58.522,0:45:01.280
telling me that the[br]students don't know that,

0:45:01.280,0:45:06.270
and they have a big[br]hard time with that.

0:45:06.270,0:45:12.270
So the equation of this circle[br]is r equals cosine theta.

0:45:12.270,0:45:16.400
So if I were to[br]express this domain,

0:45:16.400,0:45:19.466
which in Cartesian[br]coordinates would be written--

0:45:19.466,0:45:22.378
I don't know if you want to--[br]as double integral, We'd?

0:45:22.378,0:45:24.710
Have to do the[br]vertical strip thingy.

0:45:24.710,0:45:28.620
But if I want to do it[br]in polar coordinates,

0:45:28.620,0:45:33.562
I'm going to say,[br]I start-- well,

0:45:33.562,0:45:36.420
you have to tell[br]me what you think.

0:45:36.420,0:45:42.330


0:45:42.330,0:45:44.930
We have an r that[br]starts with the origin.

0:45:44.930,0:45:50.500
And that's dr.[br]How far does r go?

0:45:50.500,0:45:55.942
For the domain inside, r goes[br]between 0 and cosine theta.

0:45:55.942,0:45:57.650
STUDENT: Why were you[br]able to divide by r

0:45:57.650,0:45:59.060
if it could have equaled 0?

0:45:59.060,0:46:00.143
PROFESSOR: We already did.

0:46:00.143,0:46:02.960
STUDENT: Yes, but[br]then you just said

0:46:02.960,0:46:05.200
you could only do that[br]because it never equaled 0.

0:46:05.200,0:46:08.840
PROFESSOR: Right, and for[br]0 we pull out one point

0:46:08.840,0:46:12.550
where we take the[br]angle that we want.

0:46:12.550,0:46:15.760
We will still get[br]the same thing.

0:46:15.760,0:46:16.635
STUDENT: [INAUDIBLE].

0:46:16.635,0:46:19.882


0:46:19.882,0:46:22.140
PROFESSOR: No, r will be any--

0:46:22.140,0:46:24.739


0:46:24.739,0:46:25.530
STUDENT: Oh, I see.

0:46:25.530,0:46:27.730
PROFESSOR: Yeah,[br]so little r, what

0:46:27.730,0:46:30.396
is the r of any[br]little point inside?

0:46:30.396,0:46:33.702
The r of any little[br]point inside is

0:46:33.702,0:46:37.880
between 0 and N cosine theta.

0:46:37.880,0:46:42.430
Cosine theta would be the r[br]corresponding to the boundary.

0:46:42.430,0:46:45.300
Say it again-- so every[br]point on the boundary

0:46:45.300,0:46:49.010
will have that r[br]equals cosine theta.

0:46:49.010,0:46:55.920
The points inside the domain--[br]and this is on the circle,

0:46:55.920,0:46:59.170
on C. This is the circle.

0:46:59.170,0:47:03.125
Let's call it C ground.

0:47:03.125,0:47:08.060
That is the C.

0:47:08.060,0:47:13.710
So the r, the points[br]inside have one property,

0:47:13.710,0:47:16.320
that their r is between[br]0 and cosine theta.

0:47:16.320,0:47:18.940
If I take r theta[br]with this property,

0:47:18.940,0:47:21.480
I should be able to[br]get all the domain.

0:47:21.480,0:47:27.360
But theta, you have to be a[br]little bit careful about theta.

0:47:27.360,0:47:30.299
STUDENT: It goes from pi[br]over 2 to negative pi over 2.

0:47:30.299,0:47:31.340
PROFESSOR: Actually, yes.

0:47:31.340,0:47:38.140
So you have theta will be[br]between minus pi over 2

0:47:38.140,0:47:39.230
and pi over 2.

0:47:39.230,0:47:42.100
And you have to think[br]a little bit about how

0:47:42.100,0:47:43.740
you set up the double integral.

0:47:43.740,0:47:44.840
But you're not there yet.

0:47:44.840,0:47:48.170
So when we'll be there[br]at the double integral

0:47:48.170,0:47:50.070
we will have to think about it.

0:47:50.070,0:47:54.020


0:47:54.020,0:47:56.012
What else did I want?

0:47:56.012,0:48:06.000


0:48:06.000,0:48:12.230
All right, so did I give[br]you the right form of F?

0:48:12.230,0:48:13.688
Yes.

0:48:13.688,0:48:18.065
I'd like you to compute curl[br]F and N all by yourselves.

0:48:18.065,0:48:18.880
So compute.

0:48:18.880,0:48:24.270


0:48:24.270,0:48:26.740
This is going to be F1.

0:48:26.740,0:48:28.788
This is going to be F2.

0:48:28.788,0:48:31.178
This is going to be F3.

0:48:31.178,0:48:34.660
And I'd like you to[br]realize that this

0:48:34.660,0:48:42.091
is nothing but integral over[br]C F dR. So who is this animal?

0:48:42.091,0:48:43.444
This is the work, guys.

0:48:43.444,0:48:46.150


0:48:46.150,0:48:49.930
All right, so I should[br]be able to set up

0:48:49.930,0:48:53.530
some integral, double[br]integral, over a surface

0:48:53.530,0:49:02.320
where I have curl F times N dS.

0:49:02.320,0:49:06.135
So what I want you to[br]do is simply-- maybe

0:49:06.135,0:49:08.060
I'm a little bit too lazy.

0:49:08.060,0:49:10.490
Take the curl of F and[br]tell me what it is.

0:49:10.490,0:49:14.720
Take the unit normal vector[br]field and tell me what it is.

0:49:14.720,0:49:22.630


0:49:22.630,0:49:24.462
And then we will[br]figure out the rest.

0:49:24.462,0:49:27.840


0:49:27.840,0:49:29.855
So you say, wait a[br]minute, Magdalena, now

0:49:29.855,0:49:33.860
you want me to look at this[br]Stokes' theorem over what

0:49:33.860,0:49:34.500
surface?

0:49:34.500,0:49:37.260
Because C is the red boundary.

0:49:37.260,0:49:44.080
So you want me to look at[br]this surface, right, the cap?

0:49:44.080,0:49:46.360
So the surface could be the cap.

0:49:46.360,0:49:49.310
But what did I tell you before?

0:49:49.310,0:49:56.700
I told you that Stokes' theorem[br]works for any kind of domain

0:49:56.700,0:50:02.490
that is bounded by the[br]curve C. So is this the way

0:50:02.490,0:50:05.890
you're going to do it-- take[br]the cap, put the normals,

0:50:05.890,0:50:09.500
find the normals, and do all[br]the horrible computation?

0:50:09.500,0:50:12.450
Or you will simplify[br]your life and understand

0:50:12.450,0:50:17.700
that this is exactly the same[br]as the integral evaluated

0:50:17.700,0:50:22.050
over any surface bounded by C.

0:50:22.050,0:50:25.210
Well, this horrible thing[br]is going to kill us.

0:50:25.210,0:50:27.466
So what's the simplest[br]way to do this?

0:50:27.466,0:50:32.360


0:50:32.360,0:50:35.150
It would be to do it[br]over another surface.

0:50:35.150,0:50:37.060
It doesn't matter[br]what surface you have.

0:50:37.060,0:50:41.180
This is the C. You can take any[br]surface that's bounded by C.

0:50:41.180,0:50:43.050
You can take this balloon.

0:50:43.050,0:50:44.540
You can take this one.

0:50:44.540,0:50:47.182
You can take the[br]disc bounded by C.

0:50:47.182,0:50:49.612
You can take any surface[br]that's bounded by C.

0:50:49.612,0:50:53.986
So in particular,[br]what if you take

0:50:53.986,0:51:00.030
the surface inside[br]this red disc,

0:51:00.030,0:51:03.730
the planar surface[br]inside that red disc?

0:51:03.730,0:51:07.660
OK, do you see it?

0:51:07.660,0:51:10.460
OK, that's going to[br]be part of a plane.

0:51:10.460,0:51:12.250
What is that plane?

0:51:12.250,0:51:15.180
x plus z equals 1.

0:51:15.180,0:51:18.064
So you guys have to[br]tell me who N will be

0:51:18.064,0:51:19.992
and who the curl will be.

0:51:19.992,0:51:25.300


0:51:25.300,0:51:30.230
And let me show you again[br]with my hands what you have.

0:51:30.230,0:51:34.430
You have a surface that's[br]curvilinear and round

0:51:34.430,0:51:35.410
and has boundary C.

0:51:35.410,0:51:39.540
The boundary is C. You[br]have another surface that's

0:51:39.540,0:51:45.320
an ellipse that has[br]C as a boundary.

0:51:45.320,0:51:47.140
And this is sitting in a plane.

0:51:47.140,0:51:50.800
And I want-- it's very hard[br]to model with my hands.

0:51:50.800,0:51:51.460
But this is it.

0:51:51.460,0:51:52.272
You see it?

0:51:52.272,0:51:53.084
You see it?

0:51:53.084,0:51:56.780
OK, when you project[br]this on the ground,

0:51:56.780,0:52:00.690
this is going to become that[br]circle that I just erased,

0:52:00.690,0:52:02.650
so this and that.

0:52:02.650,0:52:04.740
We have a surface integral.

0:52:04.740,0:52:08.650
Remember, you have dS here up,[br]and you have dA here down--

0:52:08.650,0:52:10.490
dS here up, dA here down.

0:52:10.490,0:52:13.370
So that shouldn't be[br]hard to do at all.

0:52:13.370,0:52:17.740
Now what is N?

0:52:17.740,0:52:22.080
N, for such an individual,[br]will be really nice and sassy.

0:52:22.080,0:52:27.010
x plus z equals 1.

0:52:27.010,0:52:33.820
So what is the normal[br]to the plane x plus z?

0:52:33.820,0:52:34.320
[INAUDIBLE]

0:52:34.320,0:52:38.820


0:52:38.820,0:52:46.940
So who is this normal for[br]D curl F times [INAUDIBLE]

0:52:46.940,0:52:54.570
but N d-- I don't know,[br]another S, S tilde.

0:52:54.570,0:53:01.295
So for this kind of surface,[br]I have another dS, dS tilde.

0:53:01.295,0:53:04.654
So who's going to[br]tell me who N is?

0:53:04.654,0:53:07.380


0:53:07.380,0:53:13.730
Well, it should be[br]x plus z equals 1.

0:53:13.730,0:53:14.660
What do we keep?

0:53:14.660,0:53:16.136
What do we throw away?

0:53:16.136,0:53:18.420
The plane is x plus z equals 1.

0:53:18.420,0:53:19.296
What's the normal?

0:53:19.296,0:53:22.870


0:53:22.870,0:53:27.820
So the plane is x plus[br]0y plus 1z equals 1.

0:53:27.820,0:53:29.808
What's the normal to the plane?

0:53:29.808,0:53:31.724
STUDENT: Is it i plus k[br]over square root of 2?

0:53:31.724,0:53:33.720
PROFESSOR: i plus k,[br]very good, but why

0:53:33.720,0:53:37.670
does Alexander say the[br]over square root of 2?

0:53:37.670,0:53:40.090
Because it says,[br]remember guys, that that

0:53:40.090,0:53:42.740
has to be a unit normal.

0:53:42.740,0:53:47.900
We cannot take i plus k based[br]on being perpendicular to the x

0:53:47.900,0:53:48.750
plus z.

0:53:48.750,0:53:51.200
Because you need[br]to normalize it.

0:53:51.200,0:53:52.020
So he did.

0:53:52.020,0:53:57.632
So he got i plus k[br]over square root of 2.

0:53:57.632,0:54:00.464
How much is curl F?

0:54:00.464,0:54:02.352
You have to do[br]this by yourselves.

0:54:02.352,0:54:04.030
I'll just give it to you.

0:54:04.030,0:54:06.180
I'll give you three[br]minutes, and then I'll

0:54:06.180,0:54:10.392
check your work based on[br]the answers that we have.

0:54:10.392,0:54:16.200


0:54:16.200,0:54:19.070
And in the end, I'll have to do[br]the dot product and keep going.

0:54:19.070,0:55:32.650


0:55:32.650,0:55:33.840
Is it hard?

0:55:33.840,0:55:36.822
I should do it[br]along with you guys.

0:55:36.822,0:55:41.792
I have i jk d/dx, d/dy, d/dz.

0:55:41.792,0:55:45.790


0:55:45.790,0:55:50.710
Who were the guys? y[br]squared over 2 was F1.

0:55:50.710,0:55:53.364
z was F2.

0:55:53.364,0:55:54.663
x was F3.

0:55:54.663,0:55:58.882


0:55:58.882,0:56:00.866
And let's see what you got.

0:56:00.866,0:56:05.330


0:56:05.330,0:56:09.320
I'm checking to see if[br]you get the same thing.

0:56:09.320,0:56:11.350
Minus psi is the first guy.

0:56:11.350,0:56:13.996
[INAUDIBLE] the next one?

0:56:13.996,0:56:14.950
STUDENT: Minus j.

0:56:14.950,0:56:16.704
PROFESSOR: Minus j.

0:56:16.704,0:56:19.020
STUDENT: Minus yk.

0:56:19.020,0:56:20.540
PROFESSOR: yk.

0:56:20.540,0:56:23.700
And I think that's[br]what it is, yes.

0:56:23.700,0:56:28.120
So when you do the integral,[br]what are you going to get?

0:56:28.120,0:56:29.958
I'm going to erase this here.

0:56:29.958,0:56:35.326


0:56:35.326,0:56:41.450
You have your N.[br]And your N is nice.

0:56:41.450,0:56:43.430
What was it again, Alexander?

0:56:43.430,0:56:47.170
i plus k over square[br]root of 2, right?

0:56:47.170,0:56:49.950
So let's write[br]down the integral W

0:56:49.950,0:56:55.540
will be-- double[br]integral over the domain.

0:56:55.540,0:57:02.830
Now, in our case, the domain[br]is this domain, this one here.

0:57:02.830,0:57:07.120
Let's call it-- do you want to[br]call it D or D star or D tilde?

0:57:07.120,0:57:08.060
I don't know what.

0:57:08.060,0:57:10.655
Because we use to call[br]the domain on the ground

0:57:10.655,0:57:15.329
D. Let's put here D star.

0:57:15.329,0:57:21.100
So over D star, and the cap[br]doesn't exist in your life

0:57:21.100,0:57:21.710
anymore.

0:57:21.710,0:57:23.540
You said, bye-bye bubble.

0:57:23.540,0:57:27.210
I can do the whole[br]computation on D star.

0:57:27.210,0:57:29.000
I get the same answer.

0:57:29.000,0:57:32.400
So you help me right?

0:57:32.400,0:57:37.700
I get minus 1 times[br]1 over root 2.

0:57:37.700,0:57:39.630
Am I right?

0:57:39.630,0:57:48.832
A 0 for the middle term, and[br]a minus y times 1 over root 2,

0:57:48.832,0:57:53.540
good-- this is the[br]whole thing over here.

0:57:53.540,0:57:57.450
My worry is about dS star.

0:57:57.450,0:58:00.170
What was dS star?

0:58:00.170,0:58:06.880
dS star is the area[br]limit for the plane-- are

0:58:06.880,0:58:10.760
limit for how can I call this?

0:58:10.760,0:58:15.361
For disc, for D star, not for D.

0:58:15.361,0:58:16.970
It's a little bit complicated.

0:58:16.970,0:58:18.966
D is a projection.

0:58:18.966,0:58:22.700
So who reminds me how we did it?

0:58:22.700,0:58:28.760
dS star was what times dA?

0:58:28.760,0:58:30.710
This is the surface area.

0:58:30.710,0:58:35.930
And if you have a surface that's[br]nice-- your surface is nice.

0:58:35.930,0:58:39.150
STUDENT: It's area, so r?

0:58:39.150,0:58:41.616
PROFESSOR: What was this[br]equation of this surface

0:58:41.616,0:58:42.550
up here?

0:58:42.550,0:58:48.304
This is the ellipse that goes[br]projected on the surface.

0:58:48.304,0:58:50.240
STUDENT: Cosine of theta.

0:58:50.240,0:58:52.180
PROFESSOR: The equation[br]of the plane, see?

0:58:52.180,0:58:53.580
The equation of the plane.

0:58:53.580,0:58:54.630
So I erased it.

0:58:54.630,0:58:57.410
So was it x plus z equals 1?

0:58:57.410,0:58:58.230
STUDENT: Yes.

0:58:58.230,0:59:01.660
PROFESSOR: So z[br]must be 1 minus x.

0:59:01.660,0:59:09.750
So this is going to be the[br]square root of 1 plus-- minus 1

0:59:09.750,0:59:10.970
is the first partial.

0:59:10.970,0:59:12.722
Are you guys with me?

0:59:12.722,0:59:19.440
Partial with respect to x of[br]this guy is minus 1 squared

0:59:19.440,0:59:22.785
plus the partial of[br]this with respect to y

0:59:22.785,0:59:25.710
is missing 0 squared.

0:59:25.710,0:59:28.240
And then comes[br]dA, and who is dA?

0:59:28.240,0:59:32.800
dA is dxdy in the floor plane.

0:59:32.800,0:59:40.130
This is the [INAUDIBLE] that[br]projects onto the floor.

0:59:40.130,0:59:44.340
Good, ds star is going to[br]be then square root 2dA.

0:59:44.340,0:59:46.250
Again, the old[br]trick that I taught

0:59:46.250,0:59:49.600
you guys is that[br]this will always

0:59:49.600,0:59:53.810
have to simplify with[br][INAUDIBLE] on the bottom

0:59:53.810,0:59:56.420
of the N. Say what?

0:59:56.420,0:59:57.740
Magdalena, say it again.

0:59:57.740,1:00:02.770
Square root of 2DA, this[br]is that magic square root

1:00:02.770,1:00:06.460
of 1 plus [INAUDIBLE].

1:00:06.460,1:00:10.430
This guy, no matter what[br]exercise you are doing,

1:00:10.430,1:00:20.210
will always simplify with[br]the bottom of N [INAUDIBLE],

1:00:20.210,1:00:22.326
so you can do this[br]simplification

1:00:22.326,1:00:23.730
from the beginning.

1:00:23.730,1:00:26.100
And so in the end, what[br]are you going to have?

1:00:26.100,1:00:35.800
You're going to have[br]W is minus y minus 1

1:00:35.800,1:00:40.175
over the domain D in the[br]plane that this will claim.

1:00:40.175,1:00:43.430


1:00:43.430,1:00:45.620
The square root of[br][INAUDIBLE], and then you'll

1:00:45.620,1:00:50.800
have dA, which is dxdy

1:00:50.800,1:00:56.020
OK, at this point suppose[br]that you are taking the 5.

1:00:56.020,1:00:58.510
And this is why I[br]got to this point

1:00:58.510,1:01:04.650
because I wanted[br]to emphasize this.

1:01:04.650,1:01:09.170
Whether you stop here[br]or you do one more step,

1:01:09.170,1:01:11.360
I would be happy.

1:01:11.360,1:01:12.780
Let's see what I mean.

1:01:12.780,1:01:22.140
So you would have minus who[br]is y r cosine theta minus 1.

1:01:22.140,1:01:24.565
dA will become instead[br]of dxdy, you have--

1:01:24.565,1:01:25.440
STUDENT: [INAUDIBLE].

1:01:25.440,1:01:28.430
PROFESSOR: r, very[br]good. r dr is theta.

1:01:28.430,1:01:31.200


1:01:31.200,1:01:33.766
So you're thinking--

1:01:33.766,1:01:35.170
STUDENT: [INAUDIBLE].

1:01:35.170,1:01:40.125
PROFESSOR: --well,[br]so you're thinking--

1:01:40.125,1:01:47.610
I'm looking here what we have--[br]r was from 0 to cosine theta,

1:01:47.610,1:01:51.050
and theta is from[br]minus [INAUDIBLE].

1:01:51.050,1:01:54.200
Please stop here, all right?

1:01:54.200,1:01:59.330
So in the exam, we will not[br]expect-- on some integrals who

1:01:59.330,1:02:02.460
are not expected to[br]go on and do them,

1:02:02.460,1:02:04.531
which they set up the[br]integral and leave it.

1:02:04.531,1:02:05.030
Yes, sir?

1:02:05.030,1:02:08.260
STUDENT: Why did you throw[br]r cosine theta for y?

1:02:08.260,1:02:10.600
PROFESSOR: OK, because[br]let me remind you,

1:02:10.600,1:02:15.290
when you project the image[br]of this ellipse on the plane,

1:02:15.290,1:02:20.240
we got this fellow, which[br]is drawn in the book

1:02:20.240,1:02:22.780
as being this.

1:02:22.780,1:02:27.135
So we said, I want to[br]see how I set this up

1:02:27.135,1:02:28.800
in [INAUDIBLE] coordinates.

1:02:28.800,1:02:31.700
The equation of the[br]plane of the circle

1:02:31.700,1:02:34.280
was r equals cosine theta,[br]and this was calculus too.

1:02:34.280,1:02:37.780
That's why we[br]actually [INAUDIBLE].

1:02:37.780,1:02:40.120
So if somebody[br]would ask you guys,

1:02:40.120,1:02:45.880
compute me instead of an[br]area over the domain, what

1:02:45.880,1:02:48.812
if you compute for me the[br]linear area of the domain?

1:02:48.812,1:02:50.108
How would you do that?

1:02:50.108,1:02:57.530
Well, double integral of 1[br]or whatever-that-is integral

1:02:57.530,1:03:04.630
of r drd theta,[br]instead of 1, you

1:03:04.630,1:03:07.980
can have some other ugly[br]integral looking at you.

1:03:07.980,1:03:10.880
I put the stop here.

1:03:10.880,1:03:15.810
Theta is between minus[br]pie over 2n pi over 2

1:03:15.810,1:03:23.170
because I'm moving from here[br]to here, from here to here, OK?

1:03:23.170,1:03:27.655
Nr is between 0 and the margin.

1:03:27.655,1:03:29.380
Who is on the margin?

1:03:29.380,1:03:29.940
I started 0.

1:03:29.940,1:03:31.370
I ended cosine theta.

1:03:31.370,1:03:33.720
I started 0, ended cosine theta.

1:03:33.720,1:03:38.570
Cosine theta happens[br]online for the boundary,

1:03:38.570,1:03:39.777
so that's what you do.

1:03:39.777,1:03:40.860
Do we want you to do that?

1:03:40.860,1:03:43.130
No, we want you to leave it.

1:03:43.130,1:03:43.630
Yes?

1:03:43.630,1:03:49.500
STUDENT: He was asking why you[br]had a negative y minus 1 r sine

1:03:49.500,1:03:52.055
theta, not r cosine theta.

1:03:52.055,1:03:53.950
PROFESSOR: You are so right.

1:03:53.950,1:04:00.680
I forgot that x was r cosine[br]theta, and y was r sine theta.

1:04:00.680,1:04:02.180
You are correct.

1:04:02.180,1:04:05.350
And you have the group[br]good observation.

1:04:05.350,1:04:09.900
So r was [INAUDIBLE][br]cosine theta.

1:04:09.900,1:04:13.930
And x was r cosine theta.

1:04:13.930,1:04:16.920
y was r sine theta.

1:04:16.920,1:04:19.440
Very good.

1:04:19.440,1:04:23.410
OK, so if you get something[br]like that, we will now

1:04:23.410,1:04:28.935
want you to go on, we[br]will want you to stop.

1:04:28.935,1:04:34.410
Let me show you one[br]where we wanted to go on,

1:04:34.410,1:04:37.510
and we indicate it[br]like this, example 3.

1:04:37.510,1:04:44.280


1:04:44.280,1:04:48.550
So here, we just dont'[br]want you to show some work,

1:04:48.550,1:04:52.702
we wanted to actually[br]get the exact answer.

1:04:52.702,1:04:56.993
And I'll draw the picture,[br]and don't be afraid of it.

1:04:56.993,1:04:58.534
It's going to look[br]a little bit ugly.

1:04:58.534,1:05:01.936


1:05:01.936,1:05:07.960
You have the surface[br]Z equals 1 minus x

1:05:07.960,1:05:11.960
squared minus 2y squared.

1:05:11.960,1:05:19.630
And you have to evaluate[br]over double the integral

1:05:19.630,1:05:23.511
of the surface S.[br]This is the surface.

1:05:23.511,1:05:24.510
Let me draw the surface.

1:05:24.510,1:05:30.308
We will have to understand[br]what kind of surface that is.

1:05:30.308,1:05:34.619


1:05:34.619,1:05:42.250
Double integral of curl F[br][INAUDIBLE] dS evaluated

1:05:42.250,1:05:55.600
where F equals xI plus y squared[br]J plus-- this looks like a Z e

1:05:55.600,1:05:57.592
to the xy.

1:05:57.592,1:05:58.415
It's very tiny.

1:05:58.415,1:06:00.430
I bet you won't see it.

1:06:00.430,1:06:07.640
[INAUDIBLE] xy k and S. Is[br]that part of the surface?

1:06:07.640,1:06:13.590


1:06:13.590,1:06:16.504
Let me change the marker so[br]the video can see better.

1:06:16.504,1:06:19.910


1:06:19.910,1:06:22.188
Z-- this is a bad marker.

1:06:22.188,1:06:26.530


1:06:26.530,1:06:33.800
Z equals-- what was it, guys?

1:06:33.800,1:06:40.984
1 minus x squared minus 2y[br]squared with Z positive or 0.

1:06:40.984,1:06:44.470


1:06:44.470,1:06:48.210
And the [? thing ?] is[br]I think we may give you

1:06:48.210,1:06:50.130
this hint on the exam.

1:06:50.130,1:06:56.060
Think of the Stokes theorem[br]and the typical-- think

1:06:56.060,1:07:06.456
of the Stokes theorem[br]and the typical tools.

1:07:06.456,1:07:09.935
You have learned them.

1:07:09.935,1:07:11.426
OK, what does it mean?

1:07:11.426,1:07:13.800
We have like an[br]eggshell, which is coming

1:07:13.800,1:07:15.870
from the parabola [INAUDIBLE].

1:07:15.870,1:07:20.270
This parabola [INAUDIBLE][br]is S minus x squared

1:07:20.270,1:07:23.960
minus 2y squared,[br]and we call that S,

1:07:23.960,1:07:27.320
but you see, we have[br]two surfaces that are

1:07:27.320,1:07:29.830
in this picture bounded by c.

1:07:29.830,1:07:33.580
The other one is the domain[br]D, and it's a simple problem

1:07:33.580,1:07:37.590
because your domain D is[br]sitting on the xy plane.

1:07:37.590,1:07:43.326
So it's a blessing that you[br]already know what D will be.

1:07:43.326,1:07:48.260
D will be those pairs[br]xy with what property?

1:07:48.260,1:07:52.710
Can you guys tell[br]me what D will be?

1:07:52.710,1:07:54.900
Z should be 0, right?

1:07:54.900,1:07:58.010
If you impose it[br]to be 0, then this

1:07:58.010,1:08:02.050
has to satisfy x squared[br]plus 2y squared less than

1:08:02.050,1:08:03.480
or equal to 1.

1:08:03.480,1:08:05.510
Who is the C?

1:08:05.510,1:08:08.090
C are the points[br]on the boundary,

1:08:08.090,1:08:13.250
which means exactly x squared[br]plus 2y squared is equal to 1.

1:08:13.250,1:08:15.250
What in the world is this curve?

1:08:15.250,1:08:16.580
STUDENT: [INAUDIBLE].

1:08:16.580,1:08:18.050
PROFESSOR: It's an ellipse.

1:08:18.050,1:08:19.260
Is it an ugly ellipse?

1:08:19.260,1:08:21.500
Uh, not really.

1:08:21.500,1:08:22.348
It's a nice ellipse.

1:08:22.348,1:08:26.020


1:08:26.020,1:08:28.850
OK, what do they give us?

1:08:28.850,1:08:32.444
They give us xy[br]squared and Z times

1:08:32.444,1:08:37.591
e to the xy, so this[br]is F1, F2, and F3.

1:08:37.591,1:08:42.569


1:08:42.569,1:08:46.590
So the surface itself[br]is just the part

1:08:46.590,1:08:51.694
that corresponds to Z[br]positive, not all the surface

1:08:51.694,1:08:55.069
because the whole surface[br]will be infinitely large.

1:08:55.069,1:08:59.649
It's a paraboloid that keeps[br]going down to minus infinite,

1:08:59.649,1:09:01.942
so you only take this part.

1:09:01.942,1:09:07.899
It's a finite patch that I stop.

1:09:07.899,1:09:12.310
So this is a problem[br]that's amazingly simple

1:09:12.310,1:09:13.990
once you solve it one time.

1:09:13.990,1:09:20.970
You don't even have to show your[br]work much in the actual exam,

1:09:20.970,1:09:22.710
and I'll show you why.

1:09:22.710,1:09:26.830
So Stokes theorem tells[br]you what in this case?

1:09:26.830,1:09:29.305
Let's review what[br]Stokes theorem says.

1:09:29.305,1:09:34.250
Stokes theorem says, OK,[br]you have the work performed

1:09:34.250,1:09:37.550
by the four steps that's[br]given to you as a vector value

1:09:37.550,1:09:42.029
function along the path[br]C, which is given to you

1:09:42.029,1:09:44.380
as this wonderful ellipse.

1:09:44.380,1:09:48.505
Let me put C like I did it[br]before, C. This is not L,

1:09:48.505,1:09:53.210
it's C, which what is that?

1:09:53.210,1:09:57.540
It's the same as[br]double integral over S,

1:09:57.540,1:10:05.440
the round paraboloid [INAUDIBLE][br]like church roof, S curl F

1:10:05.440,1:10:09.261
times N dS.

1:10:09.261,1:10:11.310
But what does it[br]say, this happens

1:10:11.310,1:10:15.066
for any-- for every, for[br]any, do you know the sign?

1:10:15.066,1:10:15.940
STUDENT: [INAUDIBLE].

1:10:15.940,1:10:22.950
PROFESSOR: Surface[br]is bounded by C.

1:10:22.950,1:10:28.134
And here is that winking[br]emoticon from-- how

1:10:28.134,1:10:30.370
is that in Facebook?

1:10:30.370,1:10:31.700
Something like that?

1:10:31.700,1:10:35.430
A wink would be a good[br]hint on the final.

1:10:35.430,1:10:38.180
What are you going to do[br]when you see that wink?

1:10:38.180,1:10:40.520
If it's not on the[br]final, I will wink at you

1:10:40.520,1:10:43.480
until you understand[br]what I'm trying to say.

1:10:43.480,1:10:48.560
It means that you can change[br]the surface to any other surface

1:10:48.560,1:10:51.920
that has the boundary[br]C. What's the simplest

1:10:51.920,1:10:53.216
surface you may think of?

1:10:53.216,1:10:54.090
STUDENT: [INAUDIBLE].

1:10:54.090,1:10:57.180
PROFESSOR: The D.[br]So I'm going to say,

1:10:57.180,1:11:02.960
double integral over D. Curl[br]left, God knows what that is.

1:11:02.960,1:11:06.120
We still have to[br]do some work here.

1:11:06.120,1:11:08.880
I'm making a sad[br]face because I really

1:11:08.880,1:11:12.021
wanted no work whatsoever.

1:11:12.021,1:11:17.444
N becomes-- we've done this[br]argument three times today.

1:11:17.444,1:11:18.110
STUDENT: It's k.

1:11:18.110,1:11:19.140
PROFESSOR: It's a k.

1:11:19.140,1:11:21.380
That is your blessing.

1:11:21.380,1:11:24.310
That's what you have[br]to indicate on the exam

1:11:24.310,1:11:27.780
that N is k when I look[br]at the plane or domain.

1:11:27.780,1:11:28.800
STUDENT: And dS is DA.

1:11:28.800,1:11:31.880
PROFESSOR: And dS is dA.

1:11:31.880,1:11:34.539
It's much simpler than[br]before because you

1:11:34.539,1:11:35.717
don't have to project.

1:11:35.717,1:11:37.130
You are already projecting.

1:11:37.130,1:11:38.543
You are all to the floor.

1:11:38.543,1:11:40.430
You are on the ground.

1:11:40.430,1:11:42.430
What else do you have to do?

1:11:42.430,1:11:47.440
Not much, you just have to be[br]patient and compute with me

1:11:47.440,1:11:49.290
something I don't like to.

1:11:49.290,1:11:51.960
Last time I asked you[br]to do it by yourself,

1:11:51.960,1:11:55.890
but now I shouldn't be lazy.

1:11:55.890,1:11:58.480
I have to help you.

1:11:58.480,1:11:59.780
You have to help me.

1:11:59.780,1:12:08.954
i j k z dx z dx z dz of what?

1:12:08.954,1:12:13.930
x y squared and[br]this horrible guy.

1:12:13.930,1:12:18.845


1:12:18.845,1:12:20.825
What do we get?

1:12:20.825,1:12:24.290


1:12:24.290,1:12:28.290
Well, it's not so[br]obvious anymore.

1:12:28.290,1:12:29.276
STUDENT: [INAUDIBLE].

1:12:29.276,1:12:34.720
PROFESSOR: It's Z prime this[br]guy with respect to y Zx,

1:12:34.720,1:12:35.660
very good.

1:12:35.660,1:12:43.500
The x into the xy times i, and[br]I don't care about the rest.

1:12:43.500,1:12:45.000
Why don't I care about the rest?

1:12:45.000,1:12:49.100
Because when I prime y squared[br]with respect to Z goes away.

1:12:49.100,1:12:51.940
So I'm done with the first term.

1:12:51.940,1:12:56.640
I'm going very slow as you[br]can see, but I don't care.

1:12:56.640,1:12:57.945
So I'm going to erase more.

1:12:57.945,1:13:00.520


1:13:00.520,1:13:07.400
Next guy, minus and then[br]we'll make an observation.

1:13:07.400,1:13:11.090
The same thing here,[br]I go [INAUDIBLE].

1:13:11.090,1:13:17.715
So I have x to the[br]Z Zy e to the xy.

1:13:17.715,1:13:22.070
Are you guys with me, or[br]am I talking nonsense?

1:13:22.070,1:13:25.970
So what am I saying?

1:13:25.970,1:13:31.560
I'm saying that I expand with[br]respect to the j element here.

1:13:31.560,1:13:33.560
I have a minus because[br]of that, and then I

1:13:33.560,1:13:36.400
have the derivative of[br]this animal with respect

1:13:36.400,1:13:42.806
to x, which is Zy into[br]the x y j, correct?

1:13:42.806,1:13:43.620
STUDENT: Yes.

1:13:43.620,1:13:48.220
PROFESSOR: Finally, last but[br]not least, and actually that's

1:13:48.220,1:13:50.950
the most important guy, and[br]I'll tell you in a second why.

1:13:50.950,1:13:52.190
What is the last guy?

1:13:52.190,1:13:53.065
STUDENT: [INAUDIBLE].

1:13:53.065,1:13:54.060
PROFESSOR: 0.

1:13:54.060,1:13:59.870
So one of you will[br]hopefully realize what

1:13:59.870,1:14:01.880
I'm going to ask you right now.

1:14:01.880,1:14:08.820
No matter what I got here,[br]this was-- what is that called?

1:14:08.820,1:14:14.530
Work that is not necessary,[br]it's some stupid word.

1:14:14.530,1:14:16.830
So why is that not necessary?

1:14:16.830,1:14:24.055
Why could I have said star[br]i plus start j-- God knows

1:14:24.055,1:14:27.404
what that is-- plus 0k.

1:14:27.404,1:14:32.870
Because in the end, I have to[br]multiply that product with k,

1:14:32.870,1:14:36.736
so no matter what we do[br]here, and we sweat a lot.

1:14:36.736,1:14:39.001
And so no matter[br]what we put here

1:14:39.001,1:14:41.940
it would not have made a[br]difference because I have

1:14:41.940,1:14:46.640
to take this whole curl and[br]multiply as a dot with k,

1:14:46.640,1:14:52.630
and what matters is[br]only what's left over.

1:14:52.630,1:14:57.493
So my observation is this[br]whole thing is how much?

1:14:57.493,1:14:58.200
STUDENT: 0.

1:14:58.200,1:14:59.260
PROFESSOR: 0, thank God.

1:14:59.260,1:15:01.350
So the answer is 0.

1:15:01.350,1:15:03.100
And we've given[br]this problem where

1:15:03.100,1:15:09.176
the answer is 0 about four[br]times on four different finals.

1:15:09.176,1:15:12.210
The thing is that many[br]students won't study,

1:15:12.210,1:15:14.730
and they didn't know the trick.

1:15:14.730,1:15:18.630
When you have a surface like[br]that, that bounds the curve

1:15:18.630,1:15:21.440
C. Instead of doing[br]Stokes over the surface,

1:15:21.440,1:15:25.660
you do Stokes over the domain[br]and plane, and you'll get zero.

1:15:25.660,1:15:28.800
So poor kids, they[br]went ahead and tried

1:15:28.800,1:15:31.740
to compute this from[br]scratch for the surface,

1:15:31.740,1:15:33.750
and they got nowhere.

1:15:33.750,1:15:36.955
And then I started the fights[br]with, of course, [INAUDIBLE],

1:15:36.955,1:15:38.990
but they don't want to[br]give them any credit.

1:15:38.990,1:15:41.600
And I wanted to give[br]them at least some credit

1:15:41.600,1:15:43.790
for knowing the[br]theorem, the statement,

1:15:43.790,1:15:51.040
and trying to do something for[br]the nasty surface, the roof

1:15:51.040,1:15:53.396
that is a paraboloid.

1:15:53.396,1:15:55.680
They've done something,[br]so in the end,

1:15:55.680,1:15:57.710
I said I want to[br]do whatever I want,

1:15:57.710,1:15:59.700
and I gave partial credit.

1:15:59.700,1:16:03.130
But normally, I was told[br]not to give partial credit

1:16:03.130,1:16:08.310
for this kind of a thing because[br]the whole key of the problem

1:16:08.310,1:16:11.990
is to be smart,[br]understand the idea,

1:16:11.990,1:16:16.890
and get 0 without doing any[br]work, and that was nice.

1:16:16.890,1:16:17.666
Yes, sir?

1:16:17.666,1:16:19.665
STUDENT: Does that mean[br]that all we would really

1:16:19.665,1:16:22.490
needed to do compute[br]the curl is the k part?

1:16:22.490,1:16:24.790
Because if k would[br]have been something,

1:16:24.790,1:16:26.440
then there would have[br]been a dot on it.

1:16:26.440,1:16:28.270
PROFESSOR: Exactly,[br]but only if-- guys,

1:16:28.270,1:16:35.340
no matter what, if we give you,[br]if your surface has a planer

1:16:35.340,1:16:37.216
boundary-- say it again?

1:16:37.216,1:16:38.797
If your surface,[br]no matter what it

1:16:38.797,1:16:44.890
is-- it could look like[br]geography-- if your surface has

1:16:44.890,1:16:49.590
a boundary in the plane xy[br]like it is in geography,

1:16:49.590,1:16:52.520
imagine you have a[br]hill or something,

1:16:52.520,1:16:54.270
and that's the sea level.

1:16:54.270,1:16:58.454
And around the hill you have[br]the rim of the [INAUDIBLE].

1:16:58.454,1:17:00.340
OK, that's your planar curve.

1:17:00.340,1:17:03.520
Then you can reduce[br]to the plane,

1:17:03.520,1:17:07.130
and all the arguments[br]will be like that.

1:17:07.130,1:17:15.170
So the thing is you get 0[br]when the curl has 0 here,

1:17:15.170,1:17:17.170
and there is [INAUDIBLE].

1:17:17.170,1:17:18.080
Say it again?

1:17:18.080,1:17:22.720
When the F is given to you[br]so that the last component

1:17:22.720,1:17:26.970
of the curl is zero, you[br]will get 0 for the work.

1:17:26.970,1:17:32.300
Otherwise, you can get something[br]else, but not bad at all.

1:17:32.300,1:17:35.790
You can get something[br]that-- let's do

1:17:35.790,1:17:40.083
another example like that where[br]you have a simplification.

1:17:40.083,1:17:43.210
I'm going to go ahead[br]and erase the whole--

1:17:43.210,1:17:49.725
STUDENT: So, let's say if I[br]knew the [INAUDIBLE] equal to 0,

1:17:49.725,1:17:50.687
so I--

1:17:50.687,1:17:55.016
PROFESSOR: Eh, you cannot know[br]unless you look at the F first.

1:17:55.016,1:17:55.980
You see--

1:17:55.980,1:17:58.100
STUDENT: Let's say that[br]I put the F on stop,

1:17:58.100,1:18:00.940
and I put the equation,[br]which is F d r,

1:18:00.940,1:18:05.042
and I put the curl[br]F [INAUDIBLE], so

1:18:05.042,1:18:06.775
and then I said--[br]I looked at it.

1:18:06.775,1:18:08.755
I said, oh, it's a 0.

1:18:08.755,1:18:10.735
PROFESSOR: If you[br]see that's a big 0,

1:18:10.735,1:18:12.710
you can go at them[br]to 0 at the end.

1:18:12.710,1:18:13.210
STUDENT: OK.

1:18:13.210,1:18:15.685
PROFESSOR: Because the[br]dot product between k,

1:18:15.685,1:18:17.170
that's what matters.

1:18:17.170,1:18:21.130
The dot product between k and[br]the last component of the curl.

1:18:21.130,1:18:23.605
And in the end,[br]integral of 0 is 0.

1:18:23.605,1:18:26.080
And that is the lesson.

1:18:26.080,1:18:28.555
STUDENT: We should[br]also have N equal to k

1:18:28.555,1:18:30.570
if we don't have that.

1:18:30.570,1:18:35.470
PROFESSOR: Yeah, so I'm saying[br]if-- um, that's a problem.

1:18:35.470,1:18:39.170
This is not going to happen, but[br]assume that somebody gives you

1:18:39.170,1:18:45.040
a hill that looks like that,[br]and this is not a planar curve.

1:18:45.040,1:18:47.990
This would be a really[br]nasty curve in space.

1:18:47.990,1:18:50.000
You cannot do that anymore.

1:18:50.000,1:18:54.120
You have to apply [INAUDIBLE][br]for the general surface.

1:18:54.120,1:19:01.150
But if your boundary sees a[br]planar boundary [INAUDIBLE],

1:19:01.150,1:19:05.608
then you can do that,[br]and simplify your life.

1:19:05.608,1:19:07.440
So let me give you[br]another example.

1:19:07.440,1:19:11.126


1:19:11.126,1:19:15.690
This time it's not going to be--[br]OK, you will see the surprise.

1:19:15.690,1:19:49.806


1:19:49.806,1:19:53.520
And you have a sphere, and[br]you have a spherical cap,

1:19:53.520,1:20:00.490
the sphere of radius R, and[br]this is going to be, let's say,

1:20:00.490,1:20:01.810
R to be 5.

1:20:01.810,1:20:06.690
And this is z equals 3.

1:20:06.690,1:20:08.642
You have the surface.

1:20:08.642,1:20:15.000
Somebody gives[br]you the surface S.

1:20:15.000,1:20:22.570
That is the spherical cap[br]of the sphere x squared

1:20:22.570,1:20:30.522
plus y squared plus z squared[br]equals 25 above the plane z

1:20:30.522,1:20:31.494
equals 3.

1:20:31.494,1:20:37.820


1:20:37.820,1:20:52.750
Compute double[br]integral of F times--

1:20:52.750,1:20:55.558
how did we phrase this[br]if we phrase it as a--

1:20:55.558,1:20:58.360
STUDENT: Curl FN?

1:20:58.360,1:21:00.340
PROFESSOR: No, he said, curl FN.

1:21:00.340,1:21:10.060
I'm sorry, if we rephrase[br]it as work curl FN

1:21:10.060,1:21:15.160
over S, whereas this[br]is the spherical cap.

1:21:15.160,1:21:20.140
This is S.

1:21:20.140,1:21:23.090
So you're going to[br]have this on the final.

1:21:23.090,1:21:24.600
First thing is, stay calm.

1:21:24.600,1:21:26.182
Don't freak out.

1:21:26.182,1:21:29.830
This is a typical--[br]you have to say, OK.

1:21:29.830,1:21:30.830
She prepared me well.

1:21:30.830,1:21:32.650
I did review, [INAUDIBLE].

1:21:32.650,1:21:35.228
For God's sake, I'm[br]going to do fine.

1:21:35.228,1:21:37.520
Just keep in mind that[br]no matter what we do,

1:21:37.520,1:21:40.260
it's not going to involve[br]a heavy computation like we

1:21:40.260,1:21:42.470
saw in that horrible[br]first example

1:21:42.470,1:21:44.670
I gave you-- second[br]example I gave you.

1:21:44.670,1:21:48.530
So the whole idea is to make[br]your life easier rather than

1:21:48.530,1:21:49.030
harder.

1:21:49.030,1:21:50.850
So what's the[br]first thing you do?

1:21:50.850,1:21:56.160
You take curl F, and you want[br]to see what that will be.

1:21:56.160,1:22:05.130
i j k is going to[br]be d dx, d dy, d dz.

1:22:05.130,1:22:07.170
And you say, all[br]right, then I'll

1:22:07.170,1:22:16.444
have x squared yz xy[br]squared z and xy z squared.

1:22:16.444,1:22:19.811
And then you say, well,[br]this look ugly, right?

1:22:19.811,1:22:21.740
That's what you're going to say.

1:22:21.740,1:22:31.070
So what times i minus what times[br]j plus what times k remains up

1:22:31.070,1:22:35.120
to you to clue the computation,[br]and you say, wait a minute.

1:22:35.120,1:22:38.980
The first minor is it math?

1:22:38.980,1:22:42.640
No, the first minor-- minor is[br]the name of such a determinant

1:22:42.640,1:22:44.900
is just a silly path.

1:22:44.900,1:22:47.470
So you do x yz squared[br]with respect to y,

1:22:47.470,1:23:01.860
it's xz squared minus prime[br]with respect to z dz xy squared.

1:23:01.860,1:23:05.630
Next guy, what do we have?

1:23:05.630,1:23:06.890
Who tells me?

1:23:06.890,1:23:08.640
He's sort of significant[br]but not really--

1:23:08.640,1:23:09.473
STUDENT: yz squared?

1:23:09.473,1:23:11.126
PROFESSOR:yz squared, good.

1:23:11.126,1:23:13.102
It's symmetric in a way.

1:23:13.102,1:23:16.560


1:23:16.560,1:23:19.350
x squared y, right guys?

1:23:19.350,1:23:20.274
Are you with me?

1:23:20.274,1:23:20.940
STUDENT: Mh-hmm.

1:23:20.940,1:23:22.606
PROFESSOR: And for[br]the k, you will have?

1:23:22.606,1:23:24.004
STUDENT: y squared z.

1:23:24.004,1:23:25.952
PROFESSOR: y squared z.

1:23:25.952,1:23:30.050
STUDENT: And x squared z.

1:23:30.050,1:23:36.470
PROFESSOR: z squared z because[br]if you look at this guy--

1:23:36.470,1:23:38.410
so we [INAUDIBLE] again.

1:23:38.410,1:23:43.340
And you say, well, I have[br]derivative with respect

1:23:43.340,1:23:45.490
to x is y squared z.

1:23:45.490,1:23:49.110
The derivative with[br]respect to y is

1:23:49.110,1:23:56.314
x squared y squared z and[br]then minus x squared z.

1:23:56.314,1:24:00.554
Then I have-- [INAUDIBLE].

1:24:00.554,1:24:05.090


1:24:05.090,1:24:05.790
I did, right?

1:24:05.790,1:24:08.020
So this is squared.

1:24:08.020,1:24:12.000
What matters is that I[br]check what I'm going to do,

1:24:12.000,1:24:17.510
so now I say, my[br]c is a boundaries.

1:24:17.510,1:24:21.650
That's a circle, so the[br]meaning of this integral given

1:24:21.650,1:24:24.550
by Stokes is actually[br]a path integral

1:24:24.550,1:24:27.600
along the c at the[br]level z equals 3.

1:24:27.600,1:24:30.810
I'm at the third floor looking[br]at the world from up there.

1:24:30.810,1:24:34.770
I have the circle on the[br]third floor z equals 3.

1:24:34.770,1:24:40.230
And then I say, that's going[br]to be F dot dR God knows what.

1:24:40.230,1:24:42.210
That was originally the work.

1:24:42.210,1:24:44.620
And Stokes theorem[br]says, no matter

1:24:44.620,1:24:46.770
what surfaces[br]you're going to take

1:24:46.770,1:24:50.250
to have a regular surface[br]without controversy,

1:24:50.250,1:24:54.690
without holes that bounded[br]family by the circles c,

1:24:54.690,1:24:57.100
you're going to be in business.

1:24:57.100,1:25:03.040
So I say, the heck with[br]the S. I want the D,

1:25:03.040,1:25:07.820
and I want that D to be colorful[br]because life is great enough.

1:25:07.820,1:25:10.670
Let's make it D.

1:25:10.670,1:25:13.620
That D has what meaning?

1:25:13.620,1:25:14.850
z equals 3.

1:25:14.850,1:25:18.370
I'm at the level[br]three, but also x

1:25:18.370,1:25:23.012
squared plus y squared must[br]be less than or equal to sum.

1:25:23.012,1:25:25.442
Could anybody tell[br]me what that is?

1:25:25.442,1:25:26.900
STUDENT: 16.

1:25:26.900,1:25:28.550
PROFESSOR: So how do I know?

1:25:28.550,1:25:31.960
I will just plug in a 3 here.

1:25:31.960,1:25:34.240
3 squared is 9.

1:25:34.240,1:25:40.660
25 minus 9, so I get x squared[br]plus y squared equals 16.

1:25:40.660,1:25:45.646
So from here to here,[br]how much do I have?

1:25:45.646,1:25:46.145
STUDENT: 4.

1:25:46.145,1:25:47.960
PROFESSOR: 4, right?

1:25:47.960,1:25:52.950
So that little radius of that[br]yellow domain, [INAUDIBLE].

1:25:52.950,1:25:56.850


1:25:56.850,1:26:00.095
OK, so let's write[br]down the thing.

1:26:00.095,1:26:08.810
Let's go with D. This domain[br]is going to be called D.

1:26:08.810,1:26:12.230
And then I have this curl F.

1:26:12.230,1:26:15.420
And who is N?

1:26:15.420,1:26:16.250
N is k.

1:26:16.250,1:26:16.750
Why?

1:26:16.750,1:26:20.660
Because I'm in a[br]plane that's upstairs,

1:26:20.660,1:26:24.040
and I have dA because[br]whether the plane is

1:26:24.040,1:26:27.210
upstairs or downstairs[br]on the first floor,

1:26:27.210,1:26:29.162
dA will still be dxdy.

1:26:29.162,1:26:34.042


1:26:34.042,1:26:40.890
OK, so now let's compute[br]what we have backwards.

1:26:40.890,1:26:44.080
So this times k will[br]give me double integral

1:26:44.080,1:26:52.400
over D of y squared times.

1:26:52.400,1:26:55.780
Who is z?

1:26:55.780,1:26:58.311
I'm in a domain, d,[br]where z is fixed.

1:26:58.311,1:26:58.810
STUDENT: 3.

1:26:58.810,1:27:01.360
PROFESSOR: z is 3.

1:27:01.360,1:27:05.520
Minus x squared times 3.

1:27:05.520,1:27:07.050
And--

1:27:07.050,1:27:07.837
STUDENT: dx2.

1:27:07.837,1:27:09.670
PROFESSOR: I just came[br]up with this problem.

1:27:09.670,1:27:11.390
If I were to write[br]it for the final,

1:27:11.390,1:27:13.340
I would write it even simpler.

1:27:13.340,1:27:17.280
But let's see, 3 and[br]3, and then nothing.

1:27:17.280,1:27:20.400
And then da, dx dy, right?

1:27:20.400,1:27:23.300
Over the d, which[br]is x squared plus y

1:27:23.300,1:27:26.964
squared this is[br][INAUDIBLE] over 16.

1:27:26.964,1:27:28.940
How do I solve such a integral?

1:27:28.940,1:27:30.916
I'm going to make it nicer.

1:27:30.916,1:27:32.970
OK.

1:27:32.970,1:27:34.910
How would I solve[br]such an integral?

1:27:34.910,1:27:39.638
Is it a painful thing?

1:27:39.638,1:27:41.509
STUDENT: [INAUDIBLE].

1:27:41.509,1:27:43.050
PROFESSOR: Well,[br]they're coordinates.

1:27:43.050,1:27:45.160
And somebody's going to help me.

1:27:45.160,1:27:47.090
And as soon as we are[br]done, we are done.

1:27:47.090,1:27:49.980
3 gets out.

1:27:49.980,1:27:53.170
And instead of x squared[br]plus y squared is then 16,

1:27:53.170,1:28:00.270
I have r between 0 and 4,[br]theta between 0 and 2pi.

1:28:00.270,1:28:03.120
I have to take[br]advantage of everything

1:28:03.120,1:28:06.710
I've learned all the semester.

1:28:06.710,1:28:07.880
Knowledge is power.

1:28:07.880,1:28:08.780
What's missing?

1:28:08.780,1:28:10.300
r.

1:28:10.300,1:28:12.005
A 3 gets out.

1:28:12.005,1:28:14.720
And here I have to be just[br]smart and pay attention

1:28:14.720,1:28:15.573
to what you told me.

1:28:15.573,1:28:18.970
Because you told me, Magdalena,[br]why is our sine theta not

1:28:18.970,1:28:20.240
our cosine theta?

1:28:20.240,1:28:21.170
[INAUDIBLE]

1:28:21.170,1:28:27.190
This is r squared, sine[br]squared theta minus r

1:28:27.190,1:28:30.940
squared cosine squared theta.

1:28:30.940,1:28:34.820
So what have I taught[br]you about integrals

1:28:34.820,1:28:37.750
that can be expressed as[br]products of a function of theta

1:28:37.750,1:28:39.830
and function of r?

1:28:39.830,1:28:41.480
That they have a[br]blessing from God.

1:28:41.480,1:28:49.640
So you have 3 integral[br]from 0 to 2pi minus 1.

1:28:49.640,1:28:52.160
I have my plan when[br]it comes to this guy.

1:28:52.160,1:28:55.480
Because it goes on my nerves.

1:28:55.480,1:28:56.100
All right.

1:28:56.100,1:28:57.080
Do you see this?

1:28:57.080,1:28:57.830
[INAUDIBLE] theta.

1:28:57.830,1:28:58.614
STUDENT: That's the [INAUDIBLE].

1:28:58.614,1:28:59.447
[INTERPOSING VOICES]

1:28:59.447,1:29:01.735
PROFESSOR: Do you know[br]what I'm coming up with?

1:29:01.735,1:29:02.780
[INTERPOSING VOICES]

1:29:02.780,1:29:05.260
PROFESSOR: Cosine[br]of a double angle.

1:29:05.260,1:29:06.280
Very good.

1:29:06.280,1:29:07.950
I'm proud of you guys.

1:29:07.950,1:29:10.520
If I were to test--[br]oh, there was a test.

1:29:10.520,1:29:14.020
But [INAUDIBLE] next[br]for the whole nation.

1:29:14.020,1:29:18.520
Only about 10% of the[br]students remembered that

1:29:18.520,1:29:22.180
by the end of the[br]calculus series.

1:29:22.180,1:29:25.800
But I think that's not--[br]that doesn't show weakness

1:29:25.800,1:29:27.200
of the [INAUDIBLE] programs.

1:29:27.200,1:29:32.580
It shows a weakness in[br]the trigonometry classes

1:29:32.580,1:29:36.560
that are either missing from[br]high school or whatever.

1:29:36.560,1:29:38.785
So you know that[br]you want in power.

1:29:38.785,1:29:45.364
Now, times what[br]integral from 0 to 4?

1:29:45.364,1:29:46.252
STUDENT: r squared.

1:29:46.252,1:29:47.140
Or r cubed.

1:29:47.140,1:29:51.309
PROFESSOR: r cubed, which again[br]is wonderful that we have.

1:29:51.309,1:29:55.221
And we should be able to[br]compute the whole thing easily.

1:29:55.221,1:29:57.769
Now if I'm smart,

1:29:57.769,1:29:58.644
STUDENT: [INAUDIBLE].

1:29:58.644,1:29:59.727
PROFESSOR: How can we see?

1:29:59.727,1:30:03.534
STUDENT: Because the cosine to[br]the integral is sine to theta.

1:30:03.534,1:30:04.512
And [INAUDIBLE].

1:30:04.512,1:30:05.490
PROFESSOR: Right.

1:30:05.490,1:30:09.600
So the sine to theta, whether[br]I put it here or here,

1:30:09.600,1:30:10.630
is still going to be 0.

1:30:10.630,1:30:12.580
The whole thing will be 0.

1:30:12.580,1:30:13.810
So I play the game.

1:30:13.810,1:30:18.120
Maybe I should've given[br]such a problem when we

1:30:18.120,1:30:20.570
wrote this edition of the book.

1:30:20.570,1:30:24.145
I think it's nicer than[br]the computational one

1:30:24.145,1:30:25.680
you saw before.

1:30:25.680,1:30:30.720
But I told you this trick so[br]you remember it for the final.

1:30:30.720,1:30:33.160
And you are to promise[br]that you'll remember it.

1:30:33.160,1:30:40.090
And that was the whole[br]essence of understanding

1:30:40.090,1:30:44.610
that the Stokes' theorem can[br]become Green's theorem very

1:30:44.610,1:30:47.160
easily when you work[br]with a surface that's

1:30:47.160,1:30:51.120
a domain in plane, a[br]planar domain. [INAUDIBLE].

1:30:51.120,1:30:53.441
Are you done with this?

1:30:53.441,1:30:53.940
OK.

1:30:53.940,1:31:02.060


1:31:02.060,1:31:05.340
So you say, OK, so what else?

1:31:05.340,1:31:07.580
This was something[br]that's sort of fun.

1:31:07.580,1:31:09.010
I understand it.

1:31:09.010,1:31:12.180
Is there anything left[br]in this whole chapter?

1:31:12.180,1:31:16.160
Fortunately or unfortunately,[br]there is only one section left.

1:31:16.160,1:31:18.559
And I'm going to[br]go over it today.

1:31:18.559,1:31:21.100
STUDENT: Can I ask you a quick[br]question about [INAUDIBLE] 6--

1:31:21.100,1:31:21.350
PROFESSOR: Yes, sire.

1:31:21.350,1:31:22.659
STUDENT: --before you move on?

1:31:22.659,1:31:23.450
PROFESSOR: Move on?

1:31:23.450,1:31:24.525
STUDENT: I was an idiot.

1:31:24.525,1:31:25.650
PROFESSOR: No, you are not.

1:31:25.650,1:31:27.441
STUDENT: And when I[br]was writing these down,

1:31:27.441,1:31:29.070
I missed the variable.

1:31:29.070,1:31:32.645
So I have the[br]integral of fdr over c

1:31:32.645,1:31:36.627
equals double integral[br]over f, curl f dot n.

1:31:36.627,1:31:37.210
PROFESSOR: ds.

1:31:37.210,1:31:39.966
STUDENT: I didn't[br]write down what c was.

1:31:39.966,1:31:41.465
I didn't write down[br]what this c was.

1:31:41.465,1:31:45.530
PROFESSOR: The c was[br]the whatever boundary

1:31:45.530,1:31:49.090
you had there of the surface s.

1:31:49.090,1:31:51.115
And that was in[br]the beginning when

1:31:51.115,1:31:55.650
we defined the sphere, when[br]we gave the general statement

1:31:55.650,1:31:59.480
for the function.

1:31:59.480,1:32:02.245
So I'm going to try[br]and draw a potato.

1:32:02.245,1:32:05.140
We don't do a very[br]good job in the book

1:32:05.140,1:32:07.520
drawing the solid body.

1:32:07.520,1:32:11.626
But I'll try and draw[br]a very nice solid body.

1:32:11.626,1:32:12.126
Let's see.

1:32:12.126,1:32:17.940


1:32:17.940,1:32:21.570
You have a solid body.

1:32:21.570,1:32:26.490
Imagine it as a potato,[br]topologically a sphere.

1:32:26.490,1:32:28.240
It's a balloon that you blow.

1:32:28.240,1:32:29.290
It's a closed surface.

1:32:29.290,1:32:31.560
It closes in itself.

1:32:31.560,1:32:36.050
And we call that r in the book.

1:32:36.050,1:32:43.360
It's a solid region enclosed[br]by the closed surfaces.

1:32:43.360,1:32:46.650


1:32:46.650,1:32:48.862
Sometimes we call[br]such a surface compact

1:32:48.862,1:32:51.640
for some topological reasons.

1:32:51.640,1:32:53.850
Let's put s.

1:32:53.850,1:32:55.896
s is the boundary of r.

1:32:55.896,1:33:00.660


1:33:00.660,1:33:05.053
We as you know our old friend[br]to be a vector value function.

1:33:05.053,1:33:09.750


1:33:09.750,1:33:14.230
And again, if you[br]like a force field,

1:33:14.230,1:33:15.770
think of it as a force field.

1:33:15.770,1:33:17.520
Now, I'm not going to[br]tell you what it is.

1:33:17.520,1:33:21.480
It's [INAUDIBLE] function[br]differential [INAUDIBLE]

1:33:21.480,1:33:24.930
the partial here is continuous.

1:33:24.930,1:33:28.660
The magic thing is that this[br]surface must be orientable.

1:33:28.660,1:33:34.185
And if we are going to immerse[br]it, it's a regular surface.

1:33:34.185,1:33:35.646
Then of course, n exists.

1:33:35.646,1:33:40.770
And your [INAUDIBLE], guys,[br]doesn't have to be outwards.

1:33:40.770,1:33:44.552
It could be inwards [INAUDIBLE].

1:33:44.552,1:33:51.890
Let's make the convention that n[br]will be outwards by convention.

1:33:51.890,1:33:54.990
So we have to have an agreement[br]like they do in politics,

1:33:54.990,1:33:57.840
between Fidel Castro and Obama.

1:33:57.840,1:34:00.910
By convention, whether[br]we like it or not,

1:34:00.910,1:34:07.172
let's assume the normal[br]will be pointing out.

1:34:07.172,1:34:09.580
Then something magic happens.

1:34:09.580,1:34:13.040
And that magic thing, I'm not[br]going to tell you what it is.

1:34:13.040,1:34:16.400
But you should tell[br]me if you remember

1:34:16.400,1:34:20.760
what the double integral was[br]in this case, intolerance

1:34:20.760,1:34:22.660
of physics.

1:34:22.660,1:34:23.740
Shut up, Magdalena.

1:34:23.740,1:34:26.710
Don't tell them everything.

1:34:26.710,1:34:29.230
Let people remember[br]what this was.

1:34:29.230,1:34:31.530
So what is the second term?

1:34:31.530,1:34:34.670
This is the so-called[br]famous divergence theorem.

1:34:34.670,1:34:37.700


1:34:37.700,1:34:39.850
So this is the divergence.

1:34:39.850,1:34:42.680
If you don't remember[br]that, we will review it.

1:34:42.680,1:34:44.550
dV is the volume integral.

1:34:44.550,1:34:48.350
I have a [INAUDIBLE] integral[br]over the solid potato,

1:34:48.350,1:34:49.023
of course.

1:34:49.023,1:34:53.024
What is this animal [INAUDIBLE]?

1:34:53.024,1:34:53.790
OK.

1:34:53.790,1:34:59.590
Take some milk and strain[br]it and make cheese.

1:34:59.590,1:35:02.950
And you have that kind[br]of piece of cloth.

1:35:02.950,1:35:04.900
And you hang it.

1:35:04.900,1:35:09.330
And the water goes through[br]that piece of cloth.

1:35:09.330,1:35:12.570
[INAUDIBLE] have this[br]kind of suggestive image

1:35:12.570,1:35:14.430
should make you[br]think of something we

1:35:14.430,1:35:16.030
talked about before.

1:35:16.030,1:35:20.920
Whether that was fluid[br]dynamics or electromagnetism,

1:35:20.920,1:35:24.890
[INAUDIBLE], this[br]has the same name.

1:35:24.890,1:35:32.220
f is some sort of field,[br]vector [INAUDIBLE] field.

1:35:32.220,1:35:34.816
N is the outer[br]normal in this case.

1:35:34.816,1:35:40.170
What is the meaning of that, for[br]a dollar, which I don't have?

1:35:40.170,1:35:41.460
It's a four-letter word.

1:35:41.460,1:35:42.640
It's an F word.

1:35:42.640,1:35:43.526
STUDENT: Flux.

1:35:43.526,1:35:44.400
PROFESSOR: Very good.

1:35:44.400,1:35:45.390
I'm proud of you.

1:35:45.390,1:35:47.780
Who said it first?

1:35:47.780,1:35:49.339
Aaron said it first?

1:35:49.339,1:35:50.130
I owe you a dollar.

1:35:50.130,1:35:51.427
You can stop by my office.

1:35:51.427,1:35:52.385
I'll give you a dollar.

1:35:52.385,1:35:53.744
STUDENT: He said it[br]five minutes ago.

1:35:53.744,1:35:55.166
PROFESSOR: So the flux-- He did?

1:35:55.166,1:35:56.114
STUDENT: Yeah, he did.

1:35:56.114,1:35:57.536
Silently.

1:35:57.536,1:36:00.380
PROFESSOR: Aaron[br]is a mindreader.

1:36:00.380,1:36:01.328
OK.

1:36:01.328,1:36:03.059
So the flux in the[br]left-hand side.

1:36:03.059,1:36:04.600
This thing you don't[br]know what it is.

1:36:04.600,1:36:06.508
But it's some sort of potato.

1:36:06.508,1:36:08.416
What is the divergence[br]of something?

1:36:08.416,1:36:11.290


1:36:11.290,1:36:20.125
So if somebody gives you the[br]vector field F1, [INAUDIBLE],

1:36:20.125,1:36:26.830
where these are functions[br]of xyz, [INAUDIBLE].

1:36:26.830,1:36:29.810
What is the divergence[br]of F by definition?

1:36:29.810,1:36:34.810
Remember section 13.1?

1:36:34.810,1:36:36.924
Keep it in mind for the final.

1:36:36.924,1:36:40.660


1:36:40.660,1:36:41.950
So what do we do?

1:36:41.950,1:36:44.840
Differentiate the[br]first component respect

1:36:44.840,1:36:50.100
to x plus differentiate[br]the second component

1:36:50.100,1:36:52.815
respect to y plus differentiate[br]the third component

1:36:52.815,1:36:54.780
respect to c, sum them up.

1:36:54.780,1:36:56.165
And that's your divergence.

1:36:56.165,1:36:58.940


1:36:58.940,1:37:01.540
OK?

1:37:01.540,1:37:05.428
How do engineers[br]write divergence?

1:37:05.428,1:37:08.460
Not like a mathematician[br]or like a geometer.

1:37:08.460,1:37:09.950
I'm doing differential geometry.

1:37:09.950,1:37:11.280
How do they write?

1:37:11.280,1:37:12.530
STUDENT: Kinds of [INAUDIBLE].

1:37:12.530,1:37:16.850
PROFESSOR: [INAUDIBLE] dot if.

1:37:16.850,1:37:19.409
This is how engineers[br]write divergence.

1:37:19.409,1:37:22.283
And when they write curl,[br]how do they write it?

1:37:22.283,1:37:24.678
They write [INAUDIBLE][br]cross product.

1:37:24.678,1:37:25.915
Because it has a meaning.

1:37:25.915,1:37:29.982
If you think about[br]operator, you have ddx

1:37:29.982,1:37:34.730
applied to F1, ddy applied[br]to F2, ddz applied to F3.

1:37:34.730,1:37:39.165
So it's like having the dot[br]product between ddx, ddy,

1:37:39.165,1:37:43.640
ddz operators, which would[br]be the [INAUDIBLE] operator

1:37:43.640,1:37:46.510
acting on F1, F2, F3.

1:37:46.510,1:37:49.850
So you go first first,[br]plus second second,

1:37:49.850,1:37:52.340
plus third third, right?

1:37:52.340,1:37:58.426
It's exactly the same idea that[br]you inherited from dot product.

1:37:58.426,1:38:02.300
Now let's see the last two[br]problems of this semester.

1:38:02.300,1:38:04.470
except for step the review.

1:38:04.470,1:38:07.090
But the review's another story.

1:38:07.090,1:38:12.940
So I'm going to pick one[br]of your favorite problems.

1:38:12.940,1:38:21.780


1:38:21.780,1:38:22.280
OK.

1:38:22.280,1:38:26.478
Example one, remember[br]your favorite tetrahedron.

1:38:26.478,1:38:28.370
I'm going to erase it.

1:38:28.370,1:38:31.681


1:38:31.681,1:38:35.730
Instead of the potato, you can[br]have something like a pyramid.

1:38:35.730,1:38:37.575
And you have example one.

1:38:37.575,1:38:40.275


1:38:40.275,1:38:44.170
Let's say, [INAUDIBLE][br]we have that.

1:38:44.170,1:38:46.070
Somebody gives you the F.

1:38:46.070,1:38:48.730
I'm going to make[br]it nice and sassy.

1:38:48.730,1:38:52.995
Because the final is coming[br]and I want simple examples.

1:38:52.995,1:38:56.180
And don't expect anything[br][INAUDIBLE] really nice

1:38:56.180,1:38:59.030
examples also on the final.

1:38:59.030,1:39:12.740
Apply divergence[br]theorem in order

1:39:12.740,1:39:22.460
to compute double integral[br]of F dot n ds over s,

1:39:22.460,1:39:37.992
where s is the surface of the[br]tetrahedron in the picture.

1:39:37.992,1:39:40.880
And that's your[br]favorite tetrahedron.

1:39:40.880,1:39:43.950
We've done that like[br]a million times.

1:39:43.950,1:39:51.130
Somebody gave you a--[br]shall I put 1 or a?

1:39:51.130,1:39:55.920
1, because [INAUDIBLE] is[br][INAUDIBLE] is [INAUDIBLE].

1:39:55.920,1:39:59.315
So you have the plane[br]x plus y plus z.

1:39:59.315,1:40:02.780
Plus 1 you intersect[br]with the axis'.

1:40:02.780,1:40:05.500
The coordinates, you take[br]the place of coordinates

1:40:05.500,1:40:08.000
and you form a tetrahedron.

1:40:08.000,1:40:10.500
Next tetrahedron is a[br]little bit beautiful

1:40:10.500,1:40:14.645
that it has 90 degree[br]angles at the vertex.

1:40:14.645,1:40:18.300
And it has a name, OABC.

1:40:18.300,1:40:20.940
OABC is the tetrahedron.

1:40:20.940,1:40:24.800
And the surface of[br]the tetrahedron is s.

1:40:24.800,1:40:28.640
How are you going[br]to do this problem?

1:40:28.640,1:40:32.068
You're going to say, oh[br]my god, I don't know.

1:40:32.068,1:40:32.748
It's not hard.

1:40:32.748,1:40:34.498
STUDENT: It looks like[br]you're going to use

1:40:34.498,1:40:35.956
the formula you just gave us.

1:40:35.956,1:40:37.414
PROFESSOR: The[br]divergence theorem.

1:40:37.414,1:40:39.580
STUDENT: And the divergence[br]for that is really easy.

1:40:39.580,1:40:40.816
It's just a constant.

1:40:40.816,1:40:41.788
PROFESSOR: Right.

1:40:41.788,1:40:45.820
And we have to give a name to[br]the tetrahedron, [INAUDIBLE]

1:40:45.820,1:40:47.480
T, with the solid tetrahedron.

1:40:47.480,1:40:53.010


1:40:53.010,1:40:59.350
And its area, its[br]surface is this.

1:40:59.350,1:41:02.640
Instead of a potato, you[br]have the solid tetrahedron.

1:41:02.640,1:41:03.920
So what do you write?

1:41:03.920,1:41:10.095
Exactly what [INAUDIBLE] told[br]you, triple integral over T.

1:41:10.095,1:41:11.009
Of what?

1:41:11.009,1:41:16.270
The divergence of F, because[br]that's the divergence theorem,

1:41:16.270,1:41:16.770
dv.

1:41:16.770,1:41:19.950


1:41:19.950,1:41:20.950
Well, it should be easy.

1:41:20.950,1:41:24.816
Because just as you[br]said, divergence of F

1:41:24.816,1:41:25.808
would be a constant.

1:41:25.808,1:41:27.300
How come?

1:41:27.300,1:41:30.730
Differentiate this[br]with respect to x, 2.

1:41:30.730,1:41:32.690
This with respect to y, 3.

1:41:32.690,1:41:36.512
This with respect to z, 5.

1:41:36.512,1:41:40.816
Last time I checked this was[br]10 when I was [INAUDIBLE].

1:41:40.816,1:41:43.800


1:41:43.800,1:41:48.485
So 10 says I'm going for a walk.

1:41:48.485,1:41:54.038
And then triple integral of[br]the volume of 1dv over T,

1:41:54.038,1:41:55.034
what is this?

1:41:55.034,1:41:59.018


1:41:59.018,1:42:02.010
[INTERPOSING VOICES]

1:42:02.010,1:42:06.630
PROFESSOR: Well, because I[br]taught you how to cheat, yes.

1:42:06.630,1:42:14.435
But what if I were to ask[br]you to express this as--

1:42:14.435,1:42:15.310
STUDENT: [INAUDIBLE]?

1:42:15.310,1:42:15.976
PROFESSOR: Yeah.

1:42:15.976,1:42:17.040
Integrate one at a time.

1:42:17.040,1:42:22.380
So you have 1dz, dy, dx--[br]I'm doing review with you--

1:42:22.380,1:42:27.644
from 0 to 1 minus x[br]minus y from 0 to--

1:42:27.644,1:42:29.100
STUDENT: [INTERPOSING VOICES]

1:42:29.100,1:42:33.430
PROFESSOR: --1 minus[br]x from zero to 1.

1:42:33.430,1:42:36.472
And how did I teach[br]you how to cheat?

1:42:36.472,1:42:39.700
I taught you that in[br]this case you shouldn't

1:42:39.700,1:42:42.180
bother to compute that.

1:42:42.180,1:42:44.790
Remember that you[br]were in school and we

1:42:44.790,1:42:47.310
learned the volume[br]of a tetrahedron

1:42:47.310,1:42:54.340
was the area of the base[br]times the height divided by 3,

1:42:54.340,1:43:01.700
which was one half[br]times 1 divided by 3.

1:43:01.700,1:43:03.380
So you guys right.

1:43:03.380,1:43:06.350
The answer is 10 times 1 over 6.

1:43:06.350,1:43:07.520
Do I leave it like that?

1:43:07.520,1:43:08.019
No.

1:43:08.019,1:43:09.710
Because it's not nice.

1:43:09.710,1:43:11.130
So the answer is 5/3.

1:43:11.130,1:43:13.740


1:43:13.740,1:43:19.280
Expect something like[br]that on the final,

1:43:19.280,1:43:21.100
something very similar.

1:43:21.100,1:43:24.970
So you'll have to apply[br]the divergence theorem

1:43:24.970,1:43:26.130
and do a good job.

1:43:26.130,1:43:29.102
And of course, you[br]have to be careful.

1:43:29.102,1:43:30.880
But it shouldn't be hard.

1:43:30.880,1:43:34.940
It's something that[br]should be easy to do.

1:43:34.940,1:43:38.700
Now, the last problem of the[br]semester that I want to do

1:43:38.700,1:43:41.410
is an application of[br]the divergence theorem

1:43:41.410,1:43:43.030
is over a cube.

1:43:43.030,1:43:49.160
So I'm going to erase[br]it, the whole thing.

1:43:49.160,1:43:53.040
And I'm going to[br]draw a cube, which is

1:43:53.040,1:43:56.160
an open-topped box upside down.

1:43:56.160,1:44:01.130
Say it again, an open-topped[br]box upside down, which

1:44:01.130,1:44:04.460
means somebody gives[br]you a cubic box

1:44:04.460,1:44:06.530
and tells you to[br]turn it upside down.

1:44:06.530,1:44:11.540


1:44:11.540,1:44:13.900
And you have from[br]here to here, 1.

1:44:13.900,1:44:16.970
All the dimensions[br]of the cube are 1.

1:44:16.970,1:44:23.490
The top is missing, so[br]there's faces missing.

1:44:23.490,1:44:25.350
The bottom face is missing.

1:44:25.350,1:44:27.930
Bottom face is missing.

1:44:27.930,1:44:31.084
Let's call it-- you know,[br]what shall we call it?

1:44:31.084,1:44:38.180


1:44:38.180,1:44:39.980
F1.

1:44:39.980,1:44:44.140
Because it was the top,[br]but now it's the bottom.

1:44:44.140,1:44:45.080
OK?

1:44:45.080,1:44:55.797
And the rest are F2, F3, F4,[br]F5, and F6, which is the top.

1:44:55.797,1:44:57.681
And I'm going to erase.

1:44:57.681,1:45:00.520


1:45:00.520,1:45:07.440
And the last thing before this[br]section is to do the following.

1:45:07.440,1:45:08.980
What do I want?

1:45:08.980,1:45:17.634
Evaluate the flux double[br]integral over s F dot n ds.

1:45:17.634,1:45:20.940
You have to evaluate that.

1:45:20.940,1:45:25.490
For the case when F-- I usually[br]don't take the exact data

1:45:25.490,1:45:26.100
from the book.

1:45:26.100,1:45:27.500
But in this case, I want to.

1:45:27.500,1:45:29.600
Because I know you'll read it.

1:45:29.600,1:45:33.550
And I don't want you to have any[br]difficulty with this problem.

1:45:33.550,1:45:35.470
I hate the data myself.

1:45:35.470,1:45:36.700
I didn't like it very much.

1:45:36.700,1:45:41.530


1:45:41.530,1:45:44.640
It's unit cube, OK.

1:45:44.640,1:45:47.710
So x must be between 0 and 1.

1:45:47.710,1:45:51.080
y must be between 0[br]and 1 including them.

1:45:51.080,1:45:57.810
But z-- attention guys-- must[br]be between 0 [INAUDIBLE],

1:45:57.810,1:45:59.530
without 0.

1:45:59.530,1:46:02.580
Because you remove the[br]face on the ground.

1:46:02.580,1:46:08.700
z is greater than 0 and[br]less than or equal to 1.

1:46:08.700,1:46:12.590
And do we want anything else?

1:46:12.590,1:46:13.150
No.

1:46:13.150,1:46:15.280
That is all.

1:46:15.280,1:46:20.750
So let's compute[br]the whole thing.

1:46:20.750,1:46:29.429
Now, assume the box[br]would be complete.

1:46:29.429,1:46:35.700


1:46:35.700,1:46:38.540
If the box were[br]complete, then I would

1:46:38.540,1:46:43.640
have the following, double[br]integral over all the

1:46:43.640,1:46:50.120
faces F2 union with[br]F3 union with F4 union

1:46:50.120,1:46:58.490
with F5 union with--[br]oh my God-- F6 of F

1:46:58.490,1:47:05.236
dot 10 ds plus double integral[br]over what's missing guys, F1?

1:47:05.236,1:47:09.610


1:47:09.610,1:47:14.956
Of n dot n ds-- F,[br]Magdalena, that's the flux.

1:47:14.956,1:47:17.900
F dot n ds.

1:47:17.900,1:47:20.820
If it were complete,[br]that would mean

1:47:20.820,1:47:25.060
I have the double integral[br]over all the six faces.

1:47:25.060,1:47:29.820
In that case, this sum would[br]be-- I can apply finally

1:47:29.820,1:47:31.740
the divergence theorem.

1:47:31.740,1:47:35.206
That would be triple[br]integral of-- God

1:47:35.206,1:47:42.570
knows what that is-- divergence[br]of F dv over the cube.

1:47:42.570,1:47:46.130
What do you want us[br]to call the cube?

1:47:46.130,1:47:47.490
STUDENT: C.

1:47:47.490,1:47:51.250
PROFESSOR: C is usually what[br]we denote for the curve.

1:47:51.250,1:47:53.108
STUDENT: How about q?

1:47:53.108,1:47:54.000
PROFESSOR: Beautiful.

1:47:54.000,1:47:54.740
Sounds like.

1:47:54.740,1:47:56.030
Oh, I like that. q.

1:47:56.030,1:48:00.830
q is the cube inside[br]the whole thing.

1:48:00.830,1:48:03.415
Unfortunately, this[br]is not very nicely

1:48:03.415,1:48:09.900
picked just to make[br]your life miserable.

1:48:09.900,1:48:14.420
So you have dv x over y.

1:48:14.420,1:48:16.790
There is no j, at least that.

1:48:16.790,1:48:20.495
ddz minus this way.

1:48:20.495,1:48:22.387
As soon as we are[br]done with this,

1:48:22.387,1:48:24.790
since I gave you no break,[br]I'm going to let you go.

1:48:24.790,1:48:26.050
So what do we have?

1:48:26.050,1:48:29.280
y minus 2z.

1:48:29.280,1:48:30.470
Does it look good?

1:48:30.470,1:48:31.530
No.

1:48:31.530,1:48:32.750
Does it look bad?

1:48:32.750,1:48:36.920
No, not really bad either.

1:48:36.920,1:48:39.360
If I were to solve[br]the problem, I

1:48:39.360,1:48:46.250
would have to say triple,[br]triple, triple y minus 2z.

1:48:46.250,1:48:53.140
And now-- oh my[br]God-- dz, dy, dx.

1:48:53.140,1:48:59.125
I sort of hate when a little bit[br]of computation 0 to 1, 0 to 1,

1:48:59.125,1:49:00.180
0 to 1.

1:49:00.180,1:49:02.924
But this is for[br][INAUDIBLE] theorem.

1:49:02.924,1:49:05.600
Is there anybody[br]missing from the picture

1:49:05.600,1:49:09.516
so I can reduce it[br]to a double integral?

1:49:09.516,1:49:10.230
STUDENT: x.

1:49:10.230,1:49:11.229
PROFESSOR: x is missing.

1:49:11.229,1:49:14.620
So I say there is no x inside.

1:49:14.620,1:49:18.801
I go what is integral[br]from 0 to 1 of 1dx?

1:49:18.801,1:49:19.300
STUDENT: 1

1:49:19.300,1:49:20.060
PROFESSOR: 1.

1:49:20.060,1:49:24.457
So I will rewrite it[br]as integral from 0 to1,

1:49:24.457,1:49:30.260
integral from 0 to1,[br]y minus 2z, dz, dy.

1:49:30.260,1:49:32.022
Is this hard?

1:49:32.022,1:49:33.187
Eh, no.

1:49:33.187,1:49:34.520
But it's a little bit obnoxious.

1:49:34.520,1:49:37.638


1:49:37.638,1:49:41.510
When I integrate with[br]respect to z, what do I get?

1:49:41.510,1:49:44.420


1:49:44.420,1:49:49.640
Yz minus z squared.

1:49:49.640,1:49:58.774
No, not that-- between z equals[br]0 1 down, z equals 1 up to z

1:49:58.774,1:50:03.080
equals 0 down dy.

1:50:03.080,1:50:06.550
So z goes from 0[br]to 1 [INAUDIBLE].

1:50:06.550,1:50:09.530
When z is 0 down,[br]I have nothing.

1:50:09.530,1:50:11.990
STUDENT: Yeah, 1[br]minus-- y minus 1.

1:50:11.990,1:50:15.780
PROFESSOR: 1. y[br]minus 1, not so bad.

1:50:15.780,1:50:18.210
Not so bad, dy.

1:50:18.210,1:50:24.970
So I get y squared[br]over 2 minus y.

1:50:24.970,1:50:30.504
Between 0 and 1, what do I get?

1:50:30.504,1:50:31.420
STUDENT: Negative 1/2.

1:50:31.420,1:50:32.419
PROFESSOR: Negative 1/2.

1:50:32.419,1:50:36.680


1:50:36.680,1:50:37.260
All right.

1:50:37.260,1:50:38.550
Let's see what we've got here.

1:50:38.550,1:50:39.050
Yeah.

1:50:39.050,1:50:40.590
They got [INAUDIBLE].

1:50:40.590,1:50:43.290
And now I'm asking you[br]what's going to happen.

1:50:43.290,1:50:46.590
Our contour is the[br]open-topped box upside down.

1:50:46.590,1:50:48.900
This is what we need.

1:50:48.900,1:50:54.020
This is what we--

1:50:54.020,1:50:57.490
STUDENT: Couldn't you[br]just the double integral?

1:50:57.490,1:51:01.360
PROFESSOR: We just have[br]to compute this fellow.

1:51:01.360,1:51:03.860
We need to compute that fellow.

1:51:03.860,1:51:06.935
So how do we do that?

1:51:06.935,1:51:09.905
How do we do that?

1:51:09.905,1:51:12.890
STUDENT: What is the[br]problem asking for again?

1:51:12.890,1:51:16.700
PROFESSOR: So the problem[br]is asking over this flux,

1:51:16.700,1:51:21.910
but only over the box'[br]walls and the top.

1:51:21.910,1:51:25.085
The top, one, two, three,[br]four, without the bottom,

1:51:25.085,1:51:26.830
which is missing.

1:51:26.830,1:51:29.150
In order to apply[br]divergence theorem,

1:51:29.150,1:51:36.190
I have to put the bottom back[br]and have a closed surface that

1:51:36.190,1:51:38.330
is enclosing the whole cube.

1:51:38.330,1:51:41.140
So this is what I want.

1:51:41.140,1:51:42.740
This is what I know.

1:51:42.740,1:51:43.680
How much is it guys?

1:51:43.680,1:51:44.710
Minus 1/2.

1:51:44.710,1:51:47.660
And this is, again, what I need.

1:51:47.660,1:51:48.160
Right?

1:51:48.160,1:51:53.417
That's the last thing[br]I'm going to do today.

1:51:53.417,1:51:54.250
[INTERPOSING VOICES]

1:51:54.250,1:51:57.930


1:51:57.930,1:52:00.293
STUDENT: F times k da.

1:52:00.293,1:52:02.610
PROFESSOR: Let's compute it.

1:52:02.610,1:52:06.000
k is a blessing, as[br]you said, [INAUDIBLE].

1:52:06.000,1:52:07.490
It's actually minus k.

1:52:07.490,1:52:09.050
Why is it minus k?

1:52:09.050,1:52:10.820
Because it's upside down.

1:52:10.820,1:52:12.130
And it's an altered normal.

1:52:12.130,1:52:13.274
STUDENT: Oh, it is the[br][INAUDIBLE] normal.

1:52:13.274,1:52:13.716
OK.

1:52:13.716,1:52:14.158
Yeah.

1:52:14.158,1:52:14.600
That's right.

1:52:14.600,1:52:15.042
PROFESSOR: [INAUDIBLE].

1:52:15.042,1:52:15.541
So minus k.

1:52:15.541,1:52:17.640
But it doesn't [INAUDIBLE].

1:52:17.640,1:52:18.610
The sign matters.

1:52:18.610,1:52:20.560
So I have to be careful.

1:52:20.560,1:52:25.620
F is-- z is 0, thank God.

1:52:25.620,1:52:26.600
So that does away.

1:52:26.600,1:52:36.000
So I have x y i dot[br]product with minus k.

1:52:36.000,1:52:37.520
What's the beauty of this?

1:52:37.520,1:52:38.170
0.

1:52:38.170,1:52:39.580
STUDENT: 0.

1:52:39.580,1:52:40.610
PROFESSOR: Yay.

1:52:40.610,1:52:42.060
0.

1:52:42.060,1:52:46.030
So the answer to this[br]problem is minus 1/2.

1:52:46.030,1:52:50.880
So the answer is minus 1.

1:52:50.880,1:52:52.685
And we are done with[br]the last section

1:52:52.685,1:52:57.480
of the book, which is 13.7.

1:52:57.480,1:52:58.630
It was a long way.

1:52:58.630,1:53:01.430
We came a long way[br]to what I'm going

1:53:01.430,1:53:05.521
to do next time and[br]the times to come.

1:53:05.521,1:53:08.370
First of all, ask me from now[br]on you want a break or not.

1:53:08.370,1:53:10.750
Because I didn't give[br]you a break today.

1:53:10.750,1:53:12.880
We are not in a hurry.

1:53:12.880,1:53:15.180
I will pick up exams.

1:53:15.180,1:53:18.150
And I will go over[br]them together with you.

1:53:18.150,1:53:21.620
And by the time we[br]finish this review,

1:53:21.620,1:53:26.210
we will have solved two or[br]three finals completely.

1:53:26.210,1:53:29.810
We will be [INAUDIBLE].

1:53:29.810,1:53:38.810
And so the final is on the 11th,[br]May 11 at 10:30 in the morning.

1:53:38.810,1:53:39.435
I think.

1:53:39.435,1:53:40.310
STUDENT: It's the 11.

1:53:40.310,1:53:41.226
The 12 is [INAUDIBLE].

1:53:41.226,1:53:43.610
The 12th is the other class.

1:53:43.610,1:53:44.210
STUDENT: Yeah.

1:53:44.210,1:53:44.810
I'm positive.

1:53:44.810,1:53:46.310
PROFESSOR: We are[br]switching the two classes.

1:53:46.310,1:53:47.210
STUDENT: [INAUDIBLE].

1:53:47.210,1:53:51.410
PROFESSOR: And it's May 11[br]at 10:30 in the morning.

1:53:51.410,1:53:55.610
On May 12, there are other[br]math courses that have a final.

1:53:55.610,1:53:59.510
But fortunately for[br]them, they start at 4:30.

1:53:59.510,1:54:02.210
I'm really blessed that I[br]don't have that [INAUDIBLE].

1:54:02.210,1:54:05.510
They start at 4:30,[br]and they end at 7:00.

1:54:05.510,1:54:07.610
Can you imagine how[br]frisky you feel when you

1:54:07.610,1:54:09.110
take that final in the night?

1:54:09.110,1:54:11.810


1:54:11.810,1:54:13.010
Good luck with the homework.

1:54:13.010,1:54:16.660
Ask me questions about the[br]homework if you have them.

1:54:16.660,1:54:19.777