0:00:00.000,0:00:02.020
PROFESSOR: I'll[br]go over the exam.

0:00:02.020,0:00:04.170
It's good review for[br]the final, and it's

0:00:04.170,0:00:09.530
a good feedback for you in[br]case you have questions.

0:00:09.530,0:00:15.074
I do not change grades,[br]I do not curve your exam.

0:00:15.074,0:00:19.493
I do not make adjustments[br]after I give you the grade.

0:00:19.493,0:00:22.193
Therefore, it's very[br]important for me

0:00:22.193,0:00:25.385
to explain why you[br]got what you got.

0:00:25.385,0:00:30.300
Not everybody did[br]well on this exam.

0:00:30.300,0:00:35.060
Most people did[br]pretty good, and I'm

0:00:35.060,0:00:41.800
quite happy with what I see[br]as a average for the class.

0:00:41.800,0:00:44.120
However, there are[br]many open questions

0:00:44.120,0:00:47.920
from many people, things[br]they didn't quite understand,

0:00:47.920,0:00:51.180
and I would like[br]to discuss those.

0:00:51.180,0:00:54.420


0:00:54.420,0:01:09.165
First of all, the midterm[br]exam was 11 questions.

0:01:09.165,0:01:16.480
10 were mandatory, so[br]the maximum possible

0:01:16.480,0:01:20.800
percentage-wise was 110%.

0:01:20.800,0:01:25.400
So for somebody who[br]did perfectly fine,

0:01:25.400,0:01:26.430
they would have 110%.

0:01:26.430,0:01:29.130


0:01:29.130,0:01:33.810
There is one person[br]only who got the high.

0:01:33.810,0:01:36.750
I didn't disclose[br]his name, but I would

0:01:36.750,0:01:39.576
like to say congratulations.

0:01:39.576,0:01:46.827
And I'm going to go ahead and[br]solve each problem with you,

0:01:46.827,0:01:48.276
for you.

0:01:48.276,0:01:53.600
So you have the[br]function f of x, y,

0:01:53.600,0:01:55.954
to be x squared minus y squared.

0:01:55.954,0:02:01.800
And the differential was f[br]sub x dx plus f sub y dy,

0:02:01.800,0:02:06.240
whihc is 2x dx, minus 2y dy.

0:02:06.240,0:02:09.070


0:02:09.070,0:02:11.190
That was something very easy.

0:02:11.190,0:02:18.324
It was not supposed to give you[br]any headache, and most of you

0:02:18.324,0:02:21.726
did a fine job on this one.

0:02:21.726,0:02:24.440
What created some[br]problems to most students

0:02:24.440,0:02:26.650
was the second problem, though.

0:02:26.650,0:02:29.990
And I sorry to hear[br]that, sorry to see that.

0:02:29.990,0:02:33.455
Find the directional[br]derivative of a function,

0:02:33.455,0:02:35.930
the same function as before.

0:02:35.930,0:02:39.890


0:02:39.890,0:02:45.520
So I have taken advantage[br]of the previous problem,

0:02:45.520,0:02:50.492
in order to make[br]your do time shorter.

0:02:50.492,0:02:56.444
At the point p of coordinates[br]x equals 0, y equals 1,

0:02:56.444,0:02:59.420
you will have a direction[br]given by the vector.

0:02:59.420,0:03:07.090
What does it mean[br]direction given?

0:03:07.090,0:03:14.702
Analyze that direction given[br]by the vector means what?

0:03:14.702,0:03:20.840
Not the vector i minus j is y,[br]because it's not a unit vector.

0:03:20.840,0:03:24.690
What is the corresponding[br]direction given by it?

0:03:24.690,0:03:27.285
A corresponding[br]direction given by it

0:03:27.285,0:03:32.600
is 1 over square root of 2i[br]minus 1 over square root of 2j.

0:03:32.600,0:03:36.900
So it's a collinear vector--[br]that is, unit varies.

0:03:36.900,0:03:38.400
Say it again, [INAUDIBLE]?

0:03:38.400,0:03:43.800
The direction u represents[br]a collinear vector.

0:03:43.800,0:03:47.209
So pointing in the[br]same direction as v,

0:03:47.209,0:03:49.144
but it has to be unitary.

0:03:49.144,0:03:49.644
Why?

0:03:49.644,0:03:54.514
Because the definition of[br]the directional derivative

0:03:54.514,0:03:59.850
is a function along the[br]direction u at the point p,

0:03:59.850,0:04:06.256
was given by the formula[br]partial derivative at p

0:04:06.256,0:04:10.350
and at 1, plus partial[br]derivative at p times 2.

0:04:10.350,0:04:12.420
Did I expect to[br]write all this down?

0:04:12.420,0:04:15.350
Yes, I did, as I[br]showed you last time.

0:04:15.350,0:04:19.470
So you have 2x evaluated[br]at-- what is x?

0:04:19.470,0:04:20.240
0.

0:04:20.240,0:04:24.488
1 times u1.

0:04:24.488,0:04:31.440
This is u1 minus 2y,[br]evaluated at 0, 1 times

0:04:31.440,0:04:36.232
minus 1 over root[br]2, which is u2.

0:04:36.232,0:04:45.540


0:04:45.540,0:04:48.120
Well that means the[br]first term goes away,

0:04:48.120,0:04:50.420
because this is going to be 0.

0:04:50.420,0:04:54.250
And after the second[br]term, you have a plus.

0:04:54.250,0:04:57.570
y is 1, thank god, that's easy.

0:04:57.570,0:05:01.670
2 over square root of[br]2, the answer is root 2.

0:05:01.670,0:05:05.050
So any other answer would[br]normally receiving a 0.

0:05:05.050,0:05:10.610
The answer was b, square root 2.

0:05:10.610,0:05:17.790
Now, on number three, the[br]function given is different.

0:05:17.790,0:05:24.875
f of x, y equals e to the xy.

0:05:24.875,0:05:29.330


0:05:29.330,0:05:34.600
And they say, the[br]gradient of this function

0:05:34.600,0:05:37.010
is at an arbitrary point.

0:05:37.010,0:05:38.461
Say is again?

0:05:38.461,0:05:42.150
The gradient of this function[br]is at an arbitrary point.

0:05:42.150,0:05:44.625
That was only part[br]of the problem.

0:05:44.625,0:05:48.585
A little bit of credit for[br]just writing the gradient.

0:05:48.585,0:05:52.050
This is actually[br]easy, a piece of cake.

0:05:52.050,0:05:53.040
y is that.

0:05:53.040,0:05:57.495
You have f sub x[br]i plus f sub y j.

0:05:57.495,0:06:11.130
It equals y to the xy[br]i plus x to the xyj.

0:06:11.130,0:06:12.546
That's very good.

0:06:12.546,0:06:18.690


0:06:18.690,0:06:20.668
Alright, OK?

0:06:20.668,0:06:39.660
Then, which direction-- it's[br]just the gradient right?

0:06:39.660,0:06:41.410
The direction corresponds[br]to the gradient.

0:06:41.410,0:06:43.710
They don't ask you for the u.

0:06:43.710,0:06:45.580
Actually, you don't need the u.

0:06:45.580,0:06:49.039
You just need the tangent[br]plane in this case.

0:06:49.039,0:06:53.260
And if you know the equation[br]of the tangent plane,

0:06:53.260,0:06:56.980
as I told you to remember that,[br]that would be very helpful.

0:06:56.980,0:06:58.960
Write your answer in[br]the space provided.

0:06:58.960,0:07:01.326
So what did I expect you to do?

0:07:01.326,0:07:09.220
First this, and then write[br]the equation z minus z0

0:07:09.220,0:07:17.110
equals f sub x times x minus[br]x0, plus f sub y, y minus y0.

0:07:17.110,0:07:22.274
Here at the point p,[br]evaluate it at the point p.

0:07:22.274,0:07:24.470
But attention, what[br]is the point p?

0:07:24.470,0:07:29.210
Well, p is the origin,[br]because we say at the origin.

0:07:29.210,0:07:32.250
Oh, so that makes things easier.

0:07:32.250,0:07:33.429
I'm not done.

0:07:33.429,0:07:37.101
Half of the problem[br]is still coming.

0:07:37.101,0:07:40.830
If you did until this point, I[br]can only give you 5 out of 10

0:07:40.830,0:07:41.823
or something like that.

0:07:41.823,0:07:44.538


0:07:44.538,0:07:48.620
Many people made[br]a mistake at z0.

0:07:48.620,0:07:53.060
Attention guys, you plus[br]in that 0, you don't get 0.

0:07:53.060,0:07:55.810
For god's sake, it's 1, right?

0:07:55.810,0:07:58.720
So z0 is 1.

0:07:58.720,0:08:01.890
Now you're getting[br]the sense that you

0:08:01.890,0:08:07.395
have z minus 1 equals[br]f sub x, computed as 0,

0:08:07.395,0:08:10.650
0 will be 0, lucky you.

0:08:10.650,0:08:18.380
f sub y computed at 0, 0, you[br]were expected to say that.

0:08:18.380,0:08:23.540
So the answer for this[br]problem was z equals 1.

0:08:23.540,0:08:28.450
Still, if you messed up, I[br]gave some partial credit,

0:08:28.450,0:08:32.539
because I didn't want to punish[br]you too much, too harshly.

0:08:32.539,0:08:36.380


0:08:36.380,0:08:42.870
On number four, find[br]the direction u-- now

0:08:42.870,0:08:45.868
you're using number[br]three, so I should not

0:08:45.868,0:08:50.050
erase number three completely.

0:08:50.050,0:08:52.260
On number four, you[br]use number three, so

0:08:52.260,0:08:54.628
the same type of function.

0:08:54.628,0:08:58.300
But it says find the[br]direction in which

0:08:58.300,0:09:05.820
this function increases most[br]rapidly, at the point 1, 1.

0:09:05.820,0:09:06.740
OK.

0:09:06.740,0:09:16.320
So what do you do? you compute[br]the gradient at the point 1,1,

0:09:16.320,0:09:20.006
and you say, this[br]is a piece of cake.

0:09:20.006,0:09:23.408
It's going to be ei plus ej, ee.

0:09:23.408,0:09:33.130


0:09:33.130,0:09:35.680
Wonderful.

0:09:35.680,0:09:40.155
So what you do is a u is the[br]gradient f over the length

0:09:40.155,0:09:43.340
of the gradient of f at p.

0:09:43.340,0:09:48.730
Which is ee divided[br]by the length of it.

0:09:48.730,0:09:52.120
But you say, I don't have[br]to compute the length of it.

0:09:52.120,0:09:56.670
I know what is pulling[br]your two e's is what?

0:09:56.670,0:10:00.600
And no matter what you have here[br]you get the same unique result.

0:10:00.600,0:10:02.840
Remember we talked[br]about that uniqueness?

0:10:02.840,0:10:04.780
This is what I[br]tried to emphasize,

0:10:04.780,0:10:10.074
that you can have 77,[br]ee, 99, 55, 100 and 100.

0:10:10.074,0:10:13.768
If you divide by the norm,[br]you still get the same answer.

0:10:13.768,0:10:17.620
Not 11, but 1 over[br]root 2, 1 over root 2.

0:10:17.620,0:10:19.490
So no matter what[br]you had there--

0:10:19.490,0:10:22.470
it could have had a million,[br]or something instead of e,

0:10:22.470,0:10:24.419
you still have the same u.

0:10:24.419,0:10:29.580


0:10:29.580,0:10:31.300
Yes, put it back.

0:10:31.300,0:10:32.940
Give yourself[br]points, modify that.

0:10:32.940,0:10:35.840


0:10:35.840,0:10:37.626
OK, so let me tell you.

0:10:37.626,0:10:40.800
Normally I should penalize,[br]because I say write the answer

0:10:40.800,0:10:41.950
in the space provided.

0:10:41.950,0:10:45.160
And thank god you had[br]enough space, right?

0:10:45.160,0:10:48.326
Look, this person wrote--[br]I shouldn't show you who

0:10:48.326,0:10:50.160
he is, he's not in here anyway.

0:10:50.160,0:10:52.660
He has space and he[br]provided last year

0:10:52.660,0:10:54.710
with no square root[br]of 2, because only two

0:10:54.710,0:10:57.220
rows are enough to write that.

0:10:57.220,0:10:59.710
It's OK, I understand[br]you forgot to copy.

0:10:59.710,0:11:02.040
My son did the same thing.

0:11:02.040,0:11:03.990
He got a scantron at the UIL.

0:11:03.990,0:11:07.520
Come to visit my son, I wanted[br]to kill him, but it's OK.

0:11:07.520,0:11:10.322
He got all the answers[br]right, and then

0:11:10.322,0:11:14.530
the teacher-- that reminds[br]me of a movie with Mr. Bean.

0:11:14.530,0:11:17.260
So the teacher comes[br]to him and says,

0:11:17.260,0:11:19.810
wow your scantron is blank.

0:11:19.810,0:11:21.750
So what was I supposed to do?

0:11:21.750,0:11:24.489
Adjust for all the answers[br]you got in the box,

0:11:24.489,0:11:25.805
put them in the scantron.

0:11:25.805,0:11:26.305
Oh, really?

0:11:26.305,0:11:27.564
So he goes quickly.

0:11:27.564,0:11:30.735
And then he got only[br]75% of them transferred.

0:11:30.735,0:11:32.950
The rest of them[br]were not transferred.

0:11:32.950,0:11:34.295
I don't know what they did.

0:11:34.295,0:11:36.690
I have no idea.

0:11:36.690,0:11:40.144
But the professor would[br]have given full credit,

0:11:40.144,0:11:43.476
even for the answers[br]that he had in the box.

0:11:43.476,0:11:45.860
From what I understood,[br]the rule for scantrons,

0:11:45.860,0:11:49.110
exams like you I only say,[br]if you don't have them

0:11:49.110,0:11:52.430
on the scantron,[br]they don't count.

0:11:52.430,0:11:56.180
This is very harsh,[br]because we don't do that.

0:11:56.180,0:11:58.710
For example, the[br]final-- if you--

0:11:58.710,0:12:02.940
that's why I'm trying[br]to read everything.

0:12:02.940,0:12:07.975
Suppose you box you answer and[br]it's 1 over square root of 2.

0:12:07.975,0:12:09.454
If that's the right answer.

0:12:09.454,0:12:10.933
Then if have the[br]multiple choice,

0:12:10.933,0:12:14.384
and they forgot to circle[br]1 over square root of 2.

0:12:14.384,0:12:17.342
I still give you[br]100% on that problem.

0:12:17.342,0:12:20.300
Some professor do not.

0:12:20.300,0:12:24.244
So this is at the latitude[br]at whoever makes the rules,

0:12:24.244,0:12:29.167
or whoever writes the exam.

0:12:29.167,0:12:29.667
OK.

0:12:29.667,0:12:32.180


0:12:32.180,0:12:39.400
So again for the final, even for[br]the multiple choice problems,

0:12:39.400,0:12:41.580
I still need the solutions.

0:12:41.580,0:12:46.002
I'm going to ask you[br]to use a bluebook.

0:12:46.002,0:12:50.880
Some professors do not[br]ask you to use a bluebook.

0:12:50.880,0:12:54.770
They say, as long as you can[br]write on the sheet, circle

0:12:54.770,0:12:56.690
the answer, I'm fine.

0:12:56.690,0:12:57.440
I'm not fine.

0:12:57.440,0:13:00.010
I want to keep what's[br]in the bluebook.

0:13:00.010,0:13:01.850
So buy-- how much is it?

0:13:01.850,0:13:03.320
Like a dollar?

0:13:03.320,0:13:06.260
Buy the books ahead of time,[br]make sure you have them.

0:13:06.260,0:13:13.120
Now, number five was a piece of[br]cake once you did number four.

0:13:13.120,0:13:15.570
You have a question?

0:13:15.570,0:13:18.330
STUDENT: What size bluebook[br]do you need for the final?

0:13:18.330,0:13:20.020
PROFESSOR: The big one.

0:13:20.020,0:13:22.960
Bigger than that, right?

0:13:22.960,0:13:27.913
The direction u for five.

0:13:27.913,0:13:33.314
With the problem four was i[br]plus j over root 2, right?

0:13:33.314,0:13:36.751
This is what you remember[br]that you did in problem four.

0:13:36.751,0:13:40.690
If you didn't do problem four,[br]you cannot do problem five.

0:13:40.690,0:13:44.272
Problem five says, this[br]is parallel to one line.

0:13:44.272,0:13:49.706
This is parallel to--[br]what is i plus j?

0:13:49.706,0:13:51.372
Of course, you don't[br]have to draw that.

0:13:51.372,0:13:52.830
I'm not expecting[br]you to draw that.

0:13:52.830,0:13:54.670
y equals x is the[br]first bisection.

0:13:54.670,0:13:59.460


0:13:59.460,0:14:01.830
All you had to do was[br]circle C, and that

0:14:01.830,0:14:04.740
was-- once you circled[br]C, you get full credit.

0:14:04.740,0:14:09.220
If you don't do that, you[br]don't get credit for anything.

0:14:09.220,0:14:11.690
Now six.

0:14:11.690,0:14:16.043
What is the maximum rate[br]of increase of the function

0:14:16.043,0:14:22.031
z the same of your friend,[br]your fellow z equals

0:14:22.031,0:14:27.520
e to the xy at p0,[br]coordinates 1, 1?

0:14:27.520,0:14:33.520
Then the value of the[br]maximum rate of change is?

0:14:33.520,0:14:34.060
A noun.

0:14:34.060,0:14:36.862


0:14:36.862,0:14:38.600
What's the simplest[br]way to do it?

0:14:38.600,0:14:40.600
There are two ways to do it.

0:14:40.600,0:14:42.955
One is the long way,[br]one is the short way.

0:14:42.955,0:14:45.800
What's the short way, guys?

0:14:45.800,0:14:51.070
Just compute the[br]length of the gradient.

0:14:51.070,0:14:54.620
The length of the[br]gradient at the point P.

0:14:54.620,0:15:03.210
So you have whatever[br]that was, ee in length.

0:15:03.210,0:15:06.380
So the answer was e root 2.

0:15:06.380,0:15:07.900
Am I right?

0:15:07.900,0:15:09.290
What was the long way?

0:15:09.290,0:15:11.177
I saw somebody do it.

0:15:11.177,0:15:14.110
This is a lot more[br]work, but of course,

0:15:14.110,0:15:18.110
would be to compute the[br]directional derivative

0:15:18.110,0:15:20.510
at the point p[br]for this function.

0:15:20.510,0:15:24.030
In the direction of u,[br]where u is the gradient

0:15:24.030,0:15:27.980
divided by the length.

0:15:27.980,0:15:28.780
at the point p.

0:15:28.780,0:15:30.745
And you get, of course,[br]the same answer.

0:15:30.745,0:15:31.245
Why?

0:15:31.245,0:15:35.700
Because we proved that actually[br]the maximum rate of change

0:15:35.700,0:15:38.670
represented directional[br]derivative exactly

0:15:38.670,0:15:41.640
in the direction[br]given by the gradient.

0:15:41.640,0:15:43.620
This is something we proved.

0:15:43.620,0:15:48.075
One of the few things[br]we proved in this class.

0:15:48.075,0:15:50.055
Alright.

0:15:50.055,0:15:54.015
So the answer was e root 2.

0:15:54.015,0:15:55.500
Let's move on to number seven.

0:15:55.500,0:16:01.935
Number seven-- and remind[br]me of your five points.

0:16:01.935,0:16:07.380
Can you email me, so[br]I have an Excel sheet,

0:16:07.380,0:16:09.220
and I'll put it in.

0:16:09.220,0:16:15.510
Consider the function f of x,[br]y e to the negative x squared,

0:16:15.510,0:16:17.750
y squared.

0:16:17.750,0:16:20.990
What can you tell me about[br]this type of function?

0:16:20.990,0:16:22.740
It's the headache function.

0:16:22.740,0:16:25.620
If I would ask you to do[br]an anti-derivative of each

0:16:25.620,0:16:28.620
of the negative squares,[br]you would say Magdalene,

0:16:28.620,0:16:31.580
didn't you say that[br]this is impossible?

0:16:31.580,0:16:38.980
While the anti-derivative[br]exists, it cannot be expressed.

0:16:38.980,0:16:42.205
It cannot be expressed as[br]an elementary function.

0:16:42.205,0:16:44.145
And that's a big headache.

0:16:44.145,0:16:47.055
This problem is beautiful,[br]why is it beautiful?

0:16:47.055,0:16:50.935
Because in the end,[br]it becomes magic.

0:16:50.935,0:16:53.845
So it's a positive function.

0:16:53.845,0:16:56.990
It's like a bell on top[br]of the church something.

0:16:56.990,0:17:00.930
And then, you have to[br]compute double integral

0:17:00.930,0:17:06.368
over the unit disk of[br]centers of 0 and radius 1.

0:17:06.368,0:17:09.770
Of e to the negative x[br]squared minus y squared dx/dy.

0:17:09.770,0:17:12.280


0:17:12.280,0:17:15.618
Well then you say, well[br]I've done this kind of thing

0:17:15.618,0:17:19.829
before, but not with[br]Cartesian coordinates.

0:17:19.829,0:17:24.880
We did it with the Jacobian[br]r, that changes everything

0:17:24.880,0:17:28.790
into polar coordinates.

0:17:28.790,0:17:34.130
So this guy becomes e[br]to the minus r squared.

0:17:34.130,0:17:39.140
Each of the numbers are[br]squared dr, d theta.

0:17:39.140,0:17:42.000
D on the unit[br][INAUDIBLE] disk means

0:17:42.000,0:17:44.730
the radius goes from 0 to 1.

0:17:44.730,0:17:48.540
This is a blessing for us,[br]because it's easy data.

0:17:48.540,0:17:51.030
Then we have 0 to 2 pi.

0:17:51.030,0:17:53.790
You could have put[br]it in any order.

0:17:53.790,0:17:58.870
For u, it's easier to close your[br]eyes when it comes to theta.

0:17:58.870,0:18:01.530
Say, theta is independent.

0:18:01.530,0:18:05.350
He is like a partition[br]that has to do nothing

0:18:05.350,0:18:07.440
with what's inside here.

0:18:07.440,0:18:10.710
So let's pull him[br]out of this picture.

0:18:10.710,0:18:14.132
And he wants to live by himself.

0:18:14.132,0:18:18.614
An integral from 0 to 2 pi of[br]d theta was of course 2 pi.

0:18:18.614,0:18:22.100
He's happy to go[br]out, having fun.

0:18:22.100,0:18:26.590
This guy inside has to[br]be thoroughly computed.

0:18:26.590,0:18:30.100
In the sense that you[br]perform the substitution.

0:18:30.100,0:18:39.590
I was actually amused that half[br]of you did u equals r squared,

0:18:39.590,0:18:42.840
and half of you did u[br]equals minus r squared.

0:18:42.840,0:18:44.600
It really doesn't[br]matter which one.

0:18:44.600,0:18:46.940
But the problem is[br]that some of you

0:18:46.940,0:18:51.670
made a mess when you put the[br]limit points back in place,

0:18:51.670,0:18:53.640
and you made mistakes.

0:18:53.640,0:18:56.490
Somebody even got[br]negative answers,

0:18:56.490,0:18:59.860
I was about to[br]fall off the chair.

0:18:59.860,0:19:04.130
Of course, I was in a good[br]mood because it was a holiday,

0:19:04.130,0:19:05.270
I graded them.

0:19:05.270,0:19:09.040
Fortunately, I graded[br]them over the break.

0:19:09.040,0:19:12.330
So after I came[br]back from Georgia.

0:19:12.330,0:19:18.680
I have minus r dr. rdr[br]was minus a half du.

0:19:18.680,0:19:21.360


0:19:21.360,0:19:24.656
This fellow is just[br]into the u, and he's

0:19:24.656,0:19:31.104
a blessing because[br]the [INAUDIBLE]

0:19:31.104,0:19:36.560
So into the u, however,[br]take it between 1 and what?

0:19:36.560,0:19:38.048
Not 0 and 1.

0:19:38.048,0:19:41.520
But when you have 0[br]here, you have 0 here.

0:19:41.520,0:19:44.496
When you have 1,[br]you have minus 1.

0:19:44.496,0:19:48.470
So pay attention to[br]that, otherwise, you

0:19:48.470,0:19:52.440
get something that[br]makes no sense.

0:19:52.440,0:19:55.010
Times minus a half.

0:19:55.010,0:19:59.475
That, you will have[br]to be careful about.

0:19:59.475,0:19:59.975
Why?

0:19:59.975,0:20:04.097
Because there will be a minus[br]from here and here, in the end,

0:20:04.097,0:20:05.310
the answer will be positive.

0:20:05.310,0:20:08.270
And that's reminding me[br]of that city plumber joke

0:20:08.270,0:20:11.926
when he doesn't pay attention[br]to the limits of integration.

0:20:11.926,0:20:16.910
And you can get a minus[br]volume, or a minus area.

0:20:16.910,0:20:24.010
So e to the minus 1 minus 1.

0:20:24.010,0:20:25.800
But that leaves a[br]negative number,

0:20:25.800,0:20:34.270
but when you multiply it by a[br]minus, you have 1 minus 1/e.

0:20:34.270,0:20:35.060
1 minus 1/e.

0:20:35.060,0:20:37.850


0:20:37.850,0:20:38.540
Good, thank god.

0:20:38.540,0:20:41.130
This is a nice guy, less than 1.

0:20:41.130,0:20:45.085
And this is key to your[br]answer, because 2 goes away

0:20:45.085,0:20:48.373
and pi stays in place[br]and this is less than pi.

0:20:48.373,0:20:52.480
So the answer to this question[br]was an answer less than pi.

0:20:52.480,0:20:55.070
And if you didn't get[br]it, I'm very sorry,

0:20:55.070,0:21:01.130
if you didn't get less than[br]pi, you didn't get any points.

0:21:01.130,0:21:04.520
But, there are enough chances[br]for you to get another point.

0:21:04.520,0:21:14.476
I was brokenhearted for 10[br]people or more out of 25

0:21:14.476,0:21:24.094
did not remember what I[br]taught in class about the area

0:21:24.094,0:21:28.486
of a collateral triangle.

0:21:28.486,0:21:31.902
And it broke my heart,[br]and I was about to cry,

0:21:31.902,0:21:35.318
but I said, c'mon, they'll[br]do it better in the final.

0:21:35.318,0:21:39.222
Honestly, I was[br]so brokenhearted.

0:21:39.222,0:21:41.700
So this is 1, 0, 0.

0:21:41.700,0:21:42.620
This was 0, 1, 0.

0:21:42.620,0:21:44.945
This was 0, 0, 1.

0:21:44.945,0:21:49.332
On number eight.

0:21:49.332,0:21:50.316
Thank you.

0:21:50.316,0:22:03.108


0:22:03.108,0:22:04.092
Beautiful.

0:22:04.092,0:22:10.380
It's an equilateral[br]triangle, and the l side

0:22:10.380,0:22:14.594
of that equilateral triangle[br]is the square root of 2.

0:22:14.594,0:22:17.106
I even taught you how to cheat.

0:22:17.106,0:22:17.980
That's why I was mad.

0:22:17.980,0:22:22.120
I taught you how to cheat,[br]and you didn't take advantage.

0:22:22.120,0:22:29.550
So the area was l squared,[br]square root of 2, 4.

0:22:29.550,0:22:33.840
Which we did this together[br]in fifth or sixth grade

0:22:33.840,0:22:40.616
by multiplying that height and[br]the width and divided by 2.

0:22:40.616,0:22:43.520
And then we came up with this[br]formula with the Pythagorean

0:22:43.520,0:22:45.456
theorem in the classroom.

0:22:45.456,0:22:49.131
If eligible to, you can[br]very quickly get an answer.

0:22:49.131,0:22:54.382
So that's going to be 2 root[br]2 over 4, just root 3 over 2.

0:22:54.382,0:22:56.840
And when I saw that people got[br]something else except root 3

0:22:56.840,0:22:59.400
over 2, that broke my heart.

0:22:59.400,0:23:01.240
Really.

0:23:01.240,0:23:06.200
You have plenty of[br]time to catch up

0:23:06.200,0:23:07.665
with that being on your final.

0:23:07.665,0:23:10.320


0:23:10.320,0:23:14.230
Did I expect you to really[br]do the surface integral?

0:23:14.230,0:23:20.220
Some people again, need to write[br]integral over the shaded domain

0:23:20.220,0:23:24.213
d a square root[br]of f sub x squared

0:23:24.213,0:23:28.440
plus f sub y squared plus 1.

0:23:28.440,0:23:31.100
That was the right track,[br]because this is root 3.

0:23:31.100,0:23:37.611
And then the area, you get the[br]area of the 1 times 1 over 2,

0:23:37.611,0:23:38.110
right?

0:23:38.110,0:23:40.438
1/3 is the area.

0:23:40.438,0:23:45.398
Root 3 gets out of this, so[br]you have-- when you integrate,

0:23:45.398,0:23:49.366
you have the area of the[br]shaded base that I have.

0:23:49.366,0:23:51.350
And you get the same answer.

0:23:51.350,0:23:54.326
No matter how you do[br]it, with calculators

0:23:54.326,0:23:58.790
or without calculators, you[br]still could have passed.

0:23:58.790,0:24:02.540
Am I if you didn't[br]get the answer?

0:24:02.540,0:24:03.080
No.

0:24:03.080,0:24:04.844
Absolutely.

0:24:04.844,0:24:08.990
But it hurts me as if, I[br]don't know, a relative of mine

0:24:08.990,0:24:13.730
messed up some task.

0:24:13.730,0:24:16.930
That's why it's better that[br]you don't know your students,

0:24:16.930,0:24:21.010
because when you[br]know your students,

0:24:21.010,0:24:23.000
you know that they[br]could have done better,

0:24:23.000,0:24:24.130
because you know them.

0:24:24.130,0:24:27.050
So we can say, OK,[br]it really hurts

0:24:27.050,0:24:30.010
when you know that they messed[br]up, not because they are not

0:24:30.010,0:24:34.580
smart or educated, but because[br]they just either didn't

0:24:34.580,0:24:37.622
pay attention or they[br]were stressed out.

0:24:37.622,0:24:41.840
However, my substitute,[br]the guy who came here,

0:24:41.840,0:24:43.220
was my Ph.D. student.

0:24:43.220,0:24:47.564
He got a doctoral degree[br]mathematics with me last year.

0:24:47.564,0:24:50.115
And he told me you were[br]not stressed out at all.

0:24:50.115,0:24:51.625
And I said, thank god.

0:24:51.625,0:24:55.100
I'm glad that they were calm.

0:24:55.100,0:24:58.235
And he said, I didn't[br]look at the exam,

0:24:58.235,0:25:01.205
but it seemed like they did very[br]well and they were comfortable.

0:25:01.205,0:25:02.690
And I was so happy.

0:25:02.690,0:25:04.670
I was in Athens, Georgia.

0:25:04.670,0:25:06.155
And reading this[br]email I said, yay!

0:25:06.155,0:25:07.640
Everybody's going to get an A!

0:25:07.640,0:25:10.610
So I come home and[br]I start grading it.

0:25:10.610,0:25:16.055
I was sad to see that my[br]prediction was not correct.

0:25:16.055,0:25:20.026
But anyway, [INAUDIBLE][br]with an average of B.

0:25:20.026,0:25:21.930
For an honors class, it's OK.

0:25:21.930,0:25:24.786
I just expected a lot better.

0:25:24.786,0:25:29.560
And I know it's going to be[br]a lot better in the final.

0:25:29.560,0:25:32.430
Number nine.

0:25:32.430,0:25:34.744
This was done by[br]almost everybody,

0:25:34.744,0:25:36.910
except for a few people who[br]messed up on the limits.

0:25:36.910,0:25:38.365
I don't know why.

0:25:38.365,0:25:44.185
When they compute-- when they[br]drew, they drew x squared,

0:25:44.185,0:25:46.125
and they drew square root of xn.

0:25:46.125,0:25:49.520
Of course, you were supposed--[br]the answer was 0 to 1,

0:25:49.520,0:25:50.980
integral of.

0:25:50.980,0:25:56.140
Now, if you do first[br]x, you have x from y

0:25:56.140,0:25:57.564
squared to square root of y.

0:25:57.564,0:25:58.460
You guys with me?

0:25:58.460,0:26:02.170
Because this is[br]smaller than that.

0:26:02.170,0:26:02.930
OK?

0:26:02.930,0:26:08.190
So you have 1 and dx dy equals[br]to integral from 0 to 1,

0:26:08.190,0:26:11.690
integral x squared to[br]square root of x1 dy dx.

0:26:11.690,0:26:17.190


0:26:17.190,0:26:24.270
Now, what a few people did--[br]and I just forgave them.

0:26:24.270,0:26:29.216
They just-- one[br]put this like that.

0:26:29.216,0:26:31.790
And here, he put root 2.

0:26:31.790,0:26:33.380
Root y and y squared.

0:26:33.380,0:26:34.600
Don't do that.

0:26:34.600,0:26:37.495
It's like chasing that[br]a positive number equals

0:26:37.495,0:26:41.880
a negative number, which[br]is all complete nonsense.

0:26:41.880,0:26:45.770
So the correct answer was[br]we put y squared down,

0:26:45.770,0:26:48.805
and square root of y[br]because this guy is

0:26:48.805,0:26:54.000
bigger than this guy for[br]something between 0 and 1.

0:26:54.000,0:26:54.960
Because I told you.

0:26:54.960,0:27:04.130
Square root of 0.04 is bigger[br]than the square of that.

0:27:04.130,0:27:06.040
OK.

0:27:06.040,0:27:15.170
Now am I happy with that?

0:27:15.170,0:27:16.450
I'm quite happy.

0:27:16.450,0:27:21.578
In general, people understood[br]the vertical strip method

0:27:21.578,0:27:23.940
compared to the[br]horizontal strip method.

0:27:23.940,0:27:25.350
And why am I happy?

0:27:25.350,0:27:30.110
Because I was asked by three[br]people from other classes

0:27:30.110,0:27:33.610
to help them, over[br]there, on the corridor.

0:27:33.610,0:27:35.225
And I asked them,[br]who is your teacher?

0:27:35.225,0:27:36.190
This and that.

0:27:36.190,0:27:40.820
But we did not understand[br]reversing the order

0:27:40.820,0:27:42.158
of integration in class.

0:27:42.158,0:27:43.652
And I said, how come?

0:27:43.652,0:27:45.644
Well, they didn't[br]explain it very well.

0:27:45.644,0:27:47.636
So I started[br]explaining it to them.

0:27:47.636,0:27:50.126
And then I realized that[br]it's a conflict of interest.

0:27:50.126,0:27:52.616
I'm not allowed to do that.

0:27:52.616,0:27:55.604
And then I go, oh my god, I[br]cannot do the homework for you.

0:27:55.604,0:27:56.600
I'm not allowed.

0:27:56.600,0:27:59.610
But I was already talking.

0:27:59.610,0:28:04.350
So I said, guys, can you do it?

0:28:04.350,0:28:05.120
I don't know.

0:28:05.120,0:28:07.060
I said, do you draw?

0:28:07.060,0:28:07.950
Why would we draw?

0:28:07.950,0:28:10.050
They didn't teach[br]us how to draw.

0:28:10.050,0:28:13.235
I said, but how do you[br]know about vertical strips

0:28:13.235,0:28:14.460
and horizontal strips?

0:28:14.460,0:28:15.440
No.

0:28:15.440,0:28:17.890
And how do you do this?

0:28:17.890,0:28:18.870
We don't know.

0:28:18.870,0:28:21.830
We felt like we have[br]to figure it out.

0:28:21.830,0:28:25.446
Without drawing, without[br]understanding how the vertical

0:28:25.446,0:28:27.816
strips are drawn[br]between two functions,

0:28:27.816,0:28:31.070
and how you switch[br]the horizontal strips,

0:28:31.070,0:28:33.340
you cannot do this[br]problem, period.

0:28:33.340,0:28:36.318
So if you don't have--[br]maybe some people have

0:28:36.318,0:28:39.180
enough imagination--[br]but that's very rare--

0:28:39.180,0:28:40.937
That they can close[br]their eyes and they

0:28:40.937,0:28:44.960
can see a picture[br]with their eyes closed

0:28:44.960,0:28:46.050
and they can solve that.

0:28:46.050,0:28:48.175
But that's not the way to learn.

0:28:48.175,0:28:51.790
The way to learn is a very[br]visual learning thing.

0:28:51.790,0:28:54.459
So that's why we[br]draw all the time.

0:28:54.459,0:28:57.324


0:28:57.324,0:28:59.449
STUDENT: Professor, you[br]can cheat these with Cal 2.

0:28:59.449,0:29:00.447
PROFESSOR: Yes.

0:29:00.447,0:29:02.443
You can do that with Cal 2.

0:29:02.443,0:29:03.441
What's the problem?

0:29:03.441,0:29:05.936
You have integral from 0 to 1.

0:29:05.936,0:29:09.380
Square root of y[br]minus y squared.

0:29:09.380,0:29:17.440
Well, they learn to[br]do the other one.

0:29:17.440,0:29:21.750
The one with square root x[br]minus x squared, 0,1 and so on.

0:29:21.750,0:29:26.246
But they were told explicitly[br]to write-- the professor even

0:29:26.246,0:29:29.772
left these empty and put[br]spaces, fill in the spaces.

0:29:29.772,0:29:32.022
And they say, how the heck[br]do we fill in those spaces?

0:29:32.022,0:29:35.004
Plus the whiteboard problems[br]have the empty spaces.

0:29:35.004,0:29:36.992
And they couldn't[br]believe that at all.

0:29:36.992,0:29:40.970
And one of them went to the[br]tutoring center and was lucky.

0:29:40.970,0:29:43.130
Because he got--[br]this is like when

0:29:43.130,0:29:45.827
you go to a medical[br]doctor, sometimes you

0:29:45.827,0:29:48.607
are lucky and get a good[br]doctor who takes care of you,

0:29:48.607,0:29:49.982
figures out what[br]your problem is.

0:29:49.982,0:29:54.140
And sometimes, they give[br]you the wrong medicine.

0:29:54.140,0:29:58.010
So one of them got the right[br]tutor who knew how to explain

0:29:58.010,0:29:59.860
and sort of knew something.

0:29:59.860,0:30:04.176
But the other one got a tutor[br]who never took Calculus 3

0:30:04.176,0:30:09.040
and said, I don't know what the[br]heck these multiple snakes are.

0:30:09.040,0:30:11.652
So I'm not going to[br]be able to help you.

0:30:11.652,0:30:14.991
So he was very disappointed.

0:30:14.991,0:30:15.490
OK.

0:30:15.490,0:30:20.110
Compute the area of the domain[br]D from the previous problem.

0:30:20.110,0:30:23.160
This was something that[br]nobody's telling you,

0:30:23.160,0:30:26.270
hey, you have to do it[br]with the double snakes.

0:30:26.270,0:30:28.715
You can do it with just[br]with a simple snake

0:30:28.715,0:30:31.110
and you're still fine.

0:30:31.110,0:30:37.020
So in Calc 1-- this[br]Calc 1, whatever it is.

0:30:37.020,0:30:41.842
In Calc 1, you learn that[br]you have to integrate this

0:30:41.842,0:30:49.000
and you'll get 2/3 x[br]to the 3/2 minus 1/3 x

0:30:49.000,0:30:56.007
cubed at x equals 1 minus[br]whatever you have with 0.

0:30:56.007,0:30:59.430
But at 0, you have 0, so[br]you say, forget about it.

0:30:59.430,0:31:05.290
And you have 2/3 minus 1/3[br]equals 1/3, then you're done.

0:31:05.290,0:31:05.790
OK?

0:31:05.790,0:31:08.130
Did I expect you[br]to show me work?

0:31:08.130,0:31:09.380
No.

0:31:09.380,0:31:11.890
For everybody who[br]wrote 1.3-- and there

0:31:11.890,0:31:13.990
were many people who[br]did this mentally,

0:31:13.990,0:31:17.800
and they came up with 1/3.

0:31:17.800,0:31:20.900
They got 10 pionts[br]on the problem.

0:31:20.900,0:31:25.990
Finally, number 11.

0:31:25.990,0:31:29.130
Without computing the[br]volume inside the sphere,

0:31:29.130,0:31:35.094
x squared plus y squared[br]plus z squared equals 2.

0:31:35.094,0:31:37.850


0:31:37.850,0:31:42.860
Set up a triple integral[br]corresponding to it

0:31:42.860,0:31:44.748
in the space provided below.

0:31:44.748,0:31:48.241


0:31:48.241,0:31:52.689
Some people, a few[br]people, messed up.

0:31:52.689,0:31:53.730
They forgot the Jacobian.

0:31:53.730,0:31:57.223
So they put the 1 instead of[br]r squared [? side-side. ?]

0:31:57.223,0:32:01.340
When you work in[br]three components,

0:32:01.340,0:32:04.290
they do fine setting[br]up the limits.

0:32:04.290,0:32:05.990
[INAUDIBLE] 1 here.

0:32:05.990,0:32:07.380
Don't look at it in the final.

0:32:07.380,0:32:09.370
You can ruin your life this way.

0:32:09.370,0:32:11.950
So we have r squared sine phi.

0:32:11.950,0:32:16.150
Phi was the latitude[br]from the North Pole.

0:32:16.150,0:32:18.510
it doesn't matter in[br]which order you do it.

0:32:18.510,0:32:21.920
But I would do to[br]er b phi b theta.

0:32:21.920,0:32:25.380
You tell me what the end[br]points are, and we are done.

0:32:25.380,0:32:26.370
STUDENT: From 0 to 5.

0:32:26.370,0:32:27.360
PROFESSOR: 0 to--

0:32:27.360,0:32:29.340
STUDENT: No, on the first one.

0:32:29.340,0:32:31.315
PROFESSOR: 0 to--

0:32:31.315,0:32:31.815
STUDENT: Dr?

0:32:31.815,0:32:33.300
It's the square root of 2.

0:32:33.300,0:32:34.290
PROFESSOR: Mm-hmm.

0:32:34.290,0:32:35.775
STUDENT: And b theta--

0:32:35.775,0:32:37.260
PROFESSOR: 0.

0:32:37.260,0:32:38.745
2pi.

0:32:38.745,0:32:41.540
And theta, all around.

0:32:41.540,0:32:42.508
STUDENT: 2pi.

0:32:42.508,0:32:45.880
PROFESSOR: Longitude[br]360 meridian degrees.

0:32:45.880,0:32:46.380
OK.

0:32:46.380,0:32:47.832
0 to 2pi.

0:32:47.832,0:32:49.284
So good.

0:32:49.284,0:32:50.330
So we are done.

0:32:50.330,0:32:52.000
Did I expect you[br]to write it down?

0:32:52.000,0:32:52.560
No.

0:32:52.560,0:32:57.130
I had three people who[br]were nice and wrote down 4.

0:32:57.130,0:32:58.850
I mean, they actually[br]did the work.

0:32:58.850,0:33:00.810
Maybe they had[br]nothing better to do.

0:33:00.810,0:33:02.400
I have no idea why.

0:33:02.400,0:33:04.820
4pi i cubed over 3, right?

0:33:04.820,0:33:09.808
And then they proved the formula[br]in general using the Jacobian.

0:33:09.808,0:33:14.638
Using the formula, they got[br]the correct formula for r

0:33:14.638,0:33:16.087
equals square root of 2.

0:33:16.087,0:33:18.030
And I was very happy.

0:33:18.030,0:33:19.731
But did I ask you to do that?

0:33:19.731,0:33:20.230
No.

0:33:20.230,0:33:22.070
Did I give you extra credit.

0:33:22.070,0:33:22.990
No.

0:33:22.990,0:33:26.810
So all the extra credit[br]was just one problem to

0:33:26.810,0:33:30.121
asked to do exactly what[br]you were told to do.

0:33:30.121,0:33:32.960


0:33:32.960,0:33:36.490
I don't know about how[br]you feel about this exam,

0:33:36.490,0:33:39.150
but it wasn't a hard exam.

0:33:39.150,0:33:41.138
It was not an easy exam.

0:33:41.138,0:33:45.106
It was an exam that[br]was supposed to test

0:33:45.106,0:33:49.570
what you learned until now[br]all through the course.

0:33:49.570,0:33:53.538
And that was the whole idea.

0:33:53.538,0:33:55.890
I think you've[br]learned very much,

0:33:55.890,0:33:58.920
and I think you did fine,[br]the majority of you.

0:33:58.920,0:34:02.920
And that should ease[br]the pressure on you

0:34:02.920,0:34:05.420
when it comes to[br]preparing for the final.

0:34:05.420,0:34:10.080
I was thinking last night, I'm[br]going to send you, probably

0:34:10.080,0:34:13.121
by email or in-person[br]in class, two

0:34:13.121,0:34:17.049
or three samples of the[br]final from old finals

0:34:17.049,0:34:22.449
that inspire us when[br]we write the final.

0:34:22.449,0:34:25.887
A few of us will provide[br]problems and comments

0:34:25.887,0:34:29.324
and suggestions when we write[br]out the departmental final.

0:34:29.324,0:34:33.252
But the final will be[br]departmental for all sections.

0:34:33.252,0:34:37.670
I don't expect more than[br]15 problems on the final.

0:34:37.670,0:34:44.264
I have yet to think and[br]decide if I want to [? lift ?]

0:34:44.264,0:34:46.045
probably the same policy.

0:34:46.045,0:34:47.889
I mean, the final is[br]the same for everybody.

0:34:47.889,0:34:51.538
But the policy about how[br]to give partial credit

0:34:51.538,0:34:54.090
or not give partial[br]credit. [INAUDIBLE].

0:34:54.090,0:34:57.609
And I already decided that[br]I'm going to read everything,

0:34:57.609,0:35:00.794
so in case that you mess up[br]at the end with your miracle

0:35:00.794,0:35:02.509
answer, you still[br]get partial credit

0:35:02.509,0:35:05.939
for your integrals[br][INAUDIBLE] shown.

0:35:05.939,0:35:10.360
Also, one of those 15 problems.[br]might be for extra credit.

0:35:10.360,0:35:12.550
I have to think a[br]little bit better

0:35:12.550,0:35:17.580
how-- what is the maximum[br]weight I want to put.

0:35:17.580,0:35:22.600
What I would say, since I never[br][INAUDIBLE] open a homework,

0:35:22.600,0:35:27.030
and I never curve[br]exams, I would think

0:35:27.030,0:35:33.714
I could make 110% as[br]the possible maximum.

0:35:33.714,0:35:37.102
In this case, you[br]have some cushion

0:35:37.102,0:35:42.360
to make a mistake or two and[br]still get a perfect score.

0:35:42.360,0:35:43.670
OK.

0:35:43.670,0:35:46.841
I'm going to move[br]on to a new chapter.

0:35:46.841,0:35:49.296
I have actually[br]moved on already,

0:35:49.296,0:35:51.751
but nobody believed me.

0:35:51.751,0:35:56.170
Last time, I started Chapter 13.

0:35:56.170,0:36:04.200
Chapter 13 is a mixture of[br]mathematics and physics.

0:36:04.200,0:36:07.083
You will be surprised[br]how many things

0:36:07.083,0:36:10.751
are coming from solid[br]mechanics, fluid mechanics.

0:36:10.751,0:36:12.218
Yes, Regan.

0:36:12.218,0:36:14.670
STUDENT: [INAUDIBLE]

0:36:14.670,0:36:16.330
PROFESSOR: For a job?

0:36:16.330,0:36:18.236
You want me to come with you?

0:36:18.236,0:36:19.670
[LAUGHTER]

0:36:19.670,0:36:22.060
STUDENT: Because I tried[br]to talk to you [INAUDIBLE].

0:36:22.060,0:36:24.928
PROFESSOR: Yes, yes.

0:36:24.928,0:36:26.840
Yes.

0:36:26.840,0:36:28.290
Yeah.

0:36:28.290,0:36:29.865
And you have to sign up.

0:36:29.865,0:36:32.131
Start a [? sheet, ?] attend[br][? the sheet, ?] and sign

0:36:32.131,0:36:34.777
your name and good luck[br]with the interview.

0:36:34.777,0:36:37.182
You should have told me before!

0:36:37.182,0:36:39.106
I could have said[br]a prayer for you.

0:36:39.106,0:36:41.030
This things are very stressful!

0:36:41.030,0:36:43.429
I remember my own interviews.

0:36:43.429,0:36:44.220
There were several.

0:36:44.220,0:36:47.671
I didn't know anything about it,[br]and my hands were all sweaty.

0:36:47.671,0:36:49.962
And you know you should never[br]shake hands with somebody

0:36:49.962,0:36:51.954
when your hands are sweaty.

0:36:51.954,0:36:54.942
You have to do like this first.

0:36:54.942,0:36:57.432
Be confident and[br]don't be nervous.

0:36:57.432,0:36:59.424
Don't sweat or anything.

0:36:59.424,0:37:01.414
Because they can see that.

0:37:01.414,0:37:01.914
All right.

0:37:01.914,0:37:05.610
You just be yourself.

0:37:05.610,0:37:06.830
Do you have earrings?

0:37:06.830,0:37:10.074
Because after my[br]several job interviews--

0:37:10.074,0:37:13.500
those are good earrings-- I[br]was told that I should never

0:37:13.500,0:37:17.630
wear dangling earrings at the[br]interviews, which I did not,

0:37:17.630,0:37:18.930
because I didn't have any.

0:37:18.930,0:37:20.990
But I love dangling earrings.

0:37:20.990,0:37:25.972
And I was asking some academics[br]why that was [? our ?] problem.

0:37:25.972,0:37:27.780
And they say they[br]are distracting.

0:37:27.780,0:37:30.666
Because mathematicians[br]are like cats.

0:37:30.666,0:37:32.292
[LAUGHTER]

0:37:32.292,0:37:33.792
PROFESSOR: --pendulum,[br]and then they

0:37:33.792,0:37:36.438
get hypnotized by the dangling.

0:37:36.438,0:37:37.320
So I don't know.

0:37:37.320,0:37:40.914
I think most of the[br]interviewers have some problems

0:37:40.914,0:37:45.306
and they find some things[br]distracting or annoying.

0:37:45.306,0:37:46.770
Otherwise, I think you are fine.

0:37:46.770,0:37:49.686
You're dressed fine[br]for an interview.

0:37:49.686,0:37:50.186
OK.

0:37:50.186,0:37:52.990
So now serious job.

0:37:52.990,0:37:57.886
We have to remember some of[br]the things we don't remember.

0:37:57.886,0:38:03.742
Which are the gradient for[br]a function of let's say

0:38:03.742,0:38:05.206
three variables.

0:38:05.206,0:38:08.134
Let's grow up a little bit.

0:38:08.134,0:38:13.290
And that was what[br]the vector field

0:38:13.290,0:38:19.630
F sub xi plus F sub[br][? I j ?] plus F sub z k.

0:38:19.630,0:38:20.170
Right?

0:38:20.170,0:38:24.354
At an arbitrary point[br]xyz in your domain.

0:38:24.354,0:38:29.880
So where xyz is in some[br]domain, you are in a potato.

0:38:29.880,0:38:34.940
And the meaning of the gradient,[br]the geometric meaning of this,

0:38:34.940,0:38:36.910
doesn't look like a[br]theta [INAUDIBLE].

0:38:36.910,0:38:41.170
It's some sort of solid[br]that it corresponds

0:38:41.170,0:38:42.674
to a closed surface.

0:38:42.674,0:38:46.530
And this closed surface[br]that closes up on its own

0:38:46.530,0:38:49.660
is having a hard[br]time [INAUDIBLE].

0:38:49.660,0:38:51.900
It has a normal.

0:38:51.900,0:38:56.650
And this normal is given by[br]the gradient of this function,

0:38:56.650,0:38:59.530
we can increase[br][? it ?] like that.

0:38:59.530,0:39:00.970
You remember that.

0:39:00.970,0:39:02.970
And that was a long time ago.

0:39:02.970,0:39:06.810
But you should[br]still master that.

0:39:06.810,0:39:11.980
Last time, I gave you[br]the z equals f of xy,

0:39:11.980,0:39:14.490
z equals little f[br]of xy, as a graph

0:39:14.490,0:39:18.240
of the function of two variables[br]over a domain in plane.

0:39:18.240,0:39:19.980
We computed the[br]gradient of that.

0:39:19.980,0:39:22.860
But that's what we did all[br]through the [? meter ?].

0:39:22.860,0:39:24.620
So that's no fun.

0:39:24.620,0:39:27.135
We know that too well.

0:39:27.135,0:39:33.677
On this problem, I[br]gave you some new piece

0:39:33.677,0:39:34.830
of information last time.

0:39:34.830,0:39:38.192
So I said, if you have[br]a vector field that

0:39:38.192,0:39:44.480
looks F 1i plus F[br]2j plus F 3k, where

0:39:44.480,0:39:50.444
Fi is C1, that means[br]that the differentiable

0:39:50.444,0:39:52.924
and the derivatives[br]are continuous,

0:39:52.924,0:39:56.892
what was the divergence of it?

0:39:56.892,0:39:59.372
Well, that was before[br]the Easter break.

0:39:59.372,0:40:01.356
And I know we had a long break.

0:40:01.356,0:40:06.316
I cannot recover from this break[br]so easily, because it was long.

0:40:06.316,0:40:08.300
And I also traveled last week.

0:40:08.300,0:40:13.700
But before I traveled, I[br]remember that I gave you this.

0:40:13.700,0:40:15.850
And you memorized it.

0:40:15.850,0:40:17.670
Most of you memorised it.

0:40:17.670,0:40:19.859
How was it?

0:40:19.859,0:40:23.310
The first component[br]differentiated with respect

0:40:23.310,0:40:28.733
to the first variable[br]plus the second component

0:40:28.733,0:40:33.170
differentiated with respect[br]to the second variable.

0:40:33.170,0:40:37.607
Plus the third component[br]differentiated with respect

0:40:37.607,0:40:41.058
to the third variable.

0:40:41.058,0:40:46.270
So I'm asking you, as[br]an exercise, like I

0:40:46.270,0:40:49.390
did last time, the same thing.

0:40:49.390,0:40:54.852
Exercise one for this section.

0:40:54.852,0:40:57.317
Compute divergence[br]of the gradient

0:40:57.317,0:41:04.519
of F, where F is a[br]C1 function of xyz.

0:41:04.519,0:41:07.004
That means F is[br][? like this ?] differentiable

0:41:07.004,0:41:08.992
and with continuous derivatives.

0:41:08.992,0:41:10.483
What does it mean?

0:41:10.483,0:41:15.453
It means that you have to[br]compute divergence of F sub xi

0:41:15.453,0:41:20.423
plus F sub yj plus F sub zk.

0:41:20.423,0:41:24.010
And you're thinking,[br]I can do that!

0:41:24.010,0:41:29.752
By definition, I take the[br]first component-- who was that?

0:41:29.752,0:41:30.748
Hmm?

0:41:30.748,0:41:32.242
STUDENT: Brian.

0:41:32.242,0:41:33.238
PROFESSOR: Oh, right.

0:41:33.238,0:41:35.230
I thought that somebody[br]wanted to come in

0:41:35.230,0:41:37.720
and then he heard me[br]and changed his mind.

0:41:37.720,0:41:39.214
[LAUGHTER]

0:41:39.214,0:41:41.020
PROFESSOR: F sub x[br]parentheses [INAUDIBLE]

0:41:41.020,0:41:45.010
x plus F sub-- like when[br]you go on a blind date

0:41:45.010,0:41:47.290
and you see, change your mind.

0:41:47.290,0:41:47.910
OK.

0:41:47.910,0:41:53.780
F sub y y plus F sub z z.

0:41:53.780,0:41:57.570
Do you remember that[br]I gave away 95 cents

0:41:57.570,0:41:59.990
for this type of question?

0:41:59.990,0:42:03.230
So what was this operator?

0:42:03.230,0:42:05.450
We can write it better.

0:42:05.450,0:42:09.342
We can write it using the[br]second partial derivatives

0:42:09.342,0:42:12.357
with respect to z, y, and z.

0:42:12.357,0:42:15.766
And we gave a name to this one.

0:42:15.766,0:42:17.410
We called this names--

0:42:17.410,0:42:18.201
STUDENT: Laplacian.

0:42:18.201,0:42:19.175
PROFESSOR: Laplacian.

0:42:19.175,0:42:21.123
Laplace operator.

0:42:21.123,0:42:22.584
Laplace.

0:42:22.584,0:42:25.506
Laplace.

0:42:25.506,0:42:26.480
Laplacian.

0:42:26.480,0:42:29.060
That's how you spell it.

0:42:29.060,0:42:30.826
Laplac-ian.

0:42:30.826,0:42:33.260
OK?

0:42:33.260,0:42:44.060
Of F. And then what do you have?

0:42:44.060,0:42:48.910


0:42:48.910,0:42:51.862
You have to introduce[br]a new notation in.

0:42:51.862,0:42:54.020
When you see this[br]triangle that looks

0:42:54.020,0:42:56.375
like an equilateral[br]triangle, this

0:42:56.375,0:42:58.650
means Laplacian of something.

0:42:58.650,0:43:01.085
So if you have a function[br]of two variables-- so

0:43:01.085,0:43:03.377
let's say z equals F of xy.

0:43:03.377,0:43:07.796
What is the Laplacian[br]of this little f?

0:43:07.796,0:43:11.724
Little f x x plus little f y y.

0:43:11.724,0:43:14.670
So we could be second[br]partial with respect

0:43:14.670,0:43:17.460
to x plus the second[br]partial with respect to y.

0:43:17.460,0:43:20.530
What if I have something else?

0:43:20.530,0:43:26.202
Like let me give you a[br]more general function.

0:43:26.202,0:43:28.886
Let's say I have a[br]differentiable function

0:43:28.886,0:43:31.570
of N variables with[br]continuous derivatives.

0:43:31.570,0:43:33.522
And it looks like crazy.

0:43:33.522,0:43:35.474
It looks like that.

0:43:35.474,0:43:39.480
x1, x2, x n minus what?

0:43:39.480,0:43:46.743
Well, the Laplace operator in[br]this case will be F sub x1 x1

0:43:46.743,0:43:48.430
plus [? A of ?] sub x2 x2.

0:43:48.430,0:43:53.260
Which means the partial of[br]F, the second derivative

0:43:53.260,0:43:55.240
with respect to x2.

0:43:55.240,0:44:03.160
And plus the last derivative[br]with respect-- two [INAUDIBLE]

0:44:03.160,0:44:04.645
with respect to[br]the same variable.

0:44:04.645,0:44:07.120
The last variable is xm minus 1.

0:44:07.120,0:44:09.100
This could be one million and 1.

0:44:09.100,0:44:10.620
I don't know.

0:44:10.620,0:44:14.270
You can have this as many[br]variables as you want.

0:44:14.270,0:44:17.480
Now, actually in[br]engineering, there

0:44:17.480,0:44:20.440
are functions that[br]have many parameters.

0:44:20.440,0:44:22.090
You have three[br]special opponents.

0:44:22.090,0:44:22.900
Then you have time.

0:44:22.900,0:44:25.080
Then you have temperature,[br]then you have pressure,

0:44:25.080,0:44:27.020
then you have god knows what.

0:44:27.020,0:44:29.780
The surface tension[br]of the membrane.

0:44:29.780,0:44:32.070
Many things.

0:44:32.070,0:44:34.350
You really have a[br]million parameters.

0:44:34.350,0:44:35.700
Actually, it's impossible.

0:44:35.700,0:44:38.560
It's even hard to work[br]with 10 parameters.

0:44:38.560,0:44:41.600
Imagine always[br]working with equations

0:44:41.600,0:44:47.960
that have lots of variables[br]and having do deal with that.

0:44:47.960,0:44:52.690
In fluid flows,[br]hydrodynamical problems,

0:44:52.690,0:44:56.180
most the time in[br]3D turbulent flows,

0:44:56.180,0:44:58.930
for example, then you have[br]xyz spatial coordinates

0:44:58.930,0:45:02.530
and time T. So even[br]with four variables,

0:45:02.530,0:45:07.110
once you get those operators,[br]you could have something like F

0:45:07.110,0:45:14.254
sub x x x t plus g sub[br]x x t plus and so on.

0:45:14.254,0:45:17.982
All sorts of ugly components.

0:45:17.982,0:45:21.780
Sometimes you'll have[br]equations of fluid flows

0:45:21.780,0:45:24.320
in dynamic software.

0:45:24.320,0:45:26.170
Fluid flows with[br]turbulence are really

0:45:26.170,0:45:30.680
an area of[br]mathematics in itself,

0:45:30.680,0:45:36.566
of really complicated equations[br]with most of the operators.

0:45:36.566,0:45:38.554
I was looking at[br]them in Georgia,

0:45:38.554,0:45:40.045
where I went to this conference.

0:45:40.045,0:45:43.027
Most of those[br]equations were order 4.

0:45:43.027,0:45:47.003
Of course, most of them you[br]cannot even think about solving

0:45:47.003,0:45:50.000
by hand, or with[br]any known methods.

0:45:50.000,0:45:54.230
You can solve them numerically[br]with computational software.

0:45:54.230,0:45:58.100
That is the only [INAUDIBLE][br]that modern mathematics

0:45:58.100,0:46:01.392
has in some areas right now.

0:46:01.392,0:46:04.270
The right software, in[br]order to find solutions

0:46:04.270,0:46:06.866
to a fluid flow with turbulence.

0:46:06.866,0:46:09.306
That is the solution to[br]this type of equation.

0:46:09.306,0:46:13.700
Like [INAUDIBLE], for example.

0:46:13.700,0:46:18.712
Now we are going to see--[br]well, you are going to see.

0:46:18.712,0:46:20.876
I'm too old and I saw[br]that 20 years ago.

0:46:20.876,0:46:23.654
When you're going[br]3350 [INAUDIBLE]

0:46:23.654,0:46:25.097
differential equations.

0:46:25.097,0:46:29.907
And then, if you do PD[br]3350 one in engineering,

0:46:29.907,0:46:32.312
You're going to see[br]lots of equations

0:46:32.312,0:46:34.430
that are hard to solve.

0:46:34.430,0:46:37.050
But in many of them, you're[br]going to see partials,

0:46:37.050,0:46:38.120
like that.

0:46:38.120,0:46:40.145
And you're going to[br]say, oh, thank god

0:46:40.145,0:46:42.180
that I like partials[br]in Calc Three

0:46:42.180,0:46:43.530
so they became my friends.

0:46:43.530,0:46:47.350
And you'll never have[br]headaches-- [? you know what ?]

0:46:47.350,0:46:50.740
would be easy, if you understood[br]that notion of differential

0:46:50.740,0:46:55.380
well, the notion of partial[br]derivatives very well.

0:46:55.380,0:46:59.376
So I'm going to erase this one.

0:46:59.376,0:47:13.640


0:47:13.640,0:47:14.460
OK.

0:47:14.460,0:47:17.880
And then I'll say, I[br]don't how many of you--

0:47:17.880,0:47:21.688
I'll try to make this[br]formula more visible.

0:47:21.688,0:47:25.010
Some of you maybe, who[br]are engineering majors

0:47:25.010,0:47:27.991
know about curl.

0:47:27.991,0:47:30.510
Have you heard about curl?

0:47:30.510,0:47:32.831
Curl of a vector value function.

0:47:32.831,0:47:33.331
No.

0:47:33.331,0:47:34.822
You haven't.

0:47:34.822,0:47:38.301
Suppose that you have a[br]vector value function.

0:47:38.301,0:47:44.762


0:47:44.762,0:47:49.732
That is F of coordinates[br]x, y, z, the coordinates.

0:47:49.732,0:47:53.470
The C1 of over[br]seven domain omega.

0:47:53.470,0:47:57.530
Omega is the domain that your[br]special coordinates live in.

0:47:57.530,0:47:59.878
Xyz living some potato.

0:47:59.878,0:48:02.070
That's it.

0:48:02.070,0:48:06.830
Whose solid body enclosed[br]by a closed surface.

0:48:06.830,0:48:11.150
In that potato, F is a[br]differentiable function

0:48:11.150,0:48:16.200
with respect to xyz, and the[br]derivatives are continuous.

0:48:16.200,0:48:19.478
Now, in most cases, if[br]you work with Laplacian,

0:48:19.478,0:48:21.609
this is not enough C1.

0:48:21.609,0:48:24.104
If you work with Laplacian,[br]what do you want?

0:48:24.104,0:48:25.102
What do you need?

0:48:25.102,0:48:28.096
You have F sub x[br]x plus F sub y1.

0:48:28.096,0:48:29.593
So you need C2.

0:48:29.593,0:48:32.587
You work with at least C2.

0:48:32.587,0:48:35.082
Many examples have C infinity.

0:48:35.082,0:48:37.826
That means you're having[br]really beautiful functions that

0:48:37.826,0:48:39.074
are elementary.

0:48:39.074,0:48:41.070
Some of them even[br]polynomial approximations.

0:48:41.070,0:48:43.930
And then you really[br]can differentiate

0:48:43.930,0:48:47.812
them ad infinitum and all[br]the derivatives [INAUDIBLE],

0:48:47.812,0:48:50.292
and then you can[br]call yourself lucky.

0:48:50.292,0:48:54.260
How do you introduce the[br]notion of curl of it?

0:48:54.260,0:48:57.732
And it sounds funny, and this[br]is why they made this fun.

0:48:57.732,0:49:01.204
And my hair used to be[br]curly, but I shaved my head

0:49:01.204,0:49:04.340
over the holiday,[br]and now it's between.

0:49:04.340,0:49:09.180
So curl of F is something[br]that looks horrible

0:49:09.180,0:49:12.145
when you try to memorize it.

0:49:12.145,0:49:15.510
So you say, OK, if I'm going[br]to get this on the final,

0:49:15.510,0:49:18.030
you better wear this T-shirt.

0:49:18.030,0:49:21.510
No, there is something[br]better than that.

0:49:21.510,0:49:24.890
One time I was the wearing-- OK.

0:49:24.890,0:49:29.930
My students got no permission[br]from the [INAUDIBLE]

0:49:29.930,0:49:33.374
to come in with a cheat sheet.

0:49:33.374,0:49:36.265
But I was wearing a T-shirt[br]that had Green's theorem.

0:49:36.265,0:49:37.640
I don't know how[br]many of you have

0:49:37.640,0:49:39.062
heard about Green's theorem.

0:49:39.062,0:49:41.050
We are going to learn[br]it in two weeks.

0:49:41.050,0:49:44.262
And I was wearing that T-shirt.

0:49:44.262,0:49:46.732
And it was by accident, OK?

0:49:46.732,0:49:49.696
I didn't do it on purpose[br]to help my students cheat.

0:49:49.696,0:49:53.318
So one student at some[br]point goes like, well, I

0:49:53.318,0:49:54.636
don't remember Green's theorem.

0:49:54.636,0:49:56.029
And then he looked my T-shirt.

0:49:56.029,0:49:56.612
Oh, all right.

0:49:56.612,0:49:58.094
Never mind.

0:49:58.094,0:50:01.780
So I had Green's theorem[br]on my shirt, [INAUDIBLE].

0:50:01.780,0:50:04.334


0:50:04.334,0:50:08.473
But it's hard to wear like[br]10 T-shirts, one for the-- I

0:50:08.473,0:50:12.126
have one for the formula of the[br]curvature of a curve in space.

0:50:12.126,0:50:14.561
Remember that one,[br]how it is so nasty?

0:50:14.561,0:50:16.022
OK, I have this one.

0:50:16.022,0:50:16.996
I have Green's theorem.

0:50:16.996,0:50:19.410
I have [INAUDIBLE], all the[br]important formulas actually.

0:50:19.410,0:50:20.753
I have 10 T-shirts.

0:50:20.753,0:50:23.218
And then I was[br]thinking, how will I

0:50:23.218,0:50:27.162
be if I were like taking ten[br]T-shirts on top of the other

0:50:27.162,0:50:31.120
and taking them one off at[br]a time during the final.

0:50:31.120,0:50:32.780
There is no cheat sheet.

0:50:32.780,0:50:35.506
There are no formula[br]sheets, no nothing.

0:50:35.506,0:50:38.060
But I would look like[br]Joey from "Friends."

0:50:38.060,0:50:41.950
Remember Joey, when he was[br]dressed in many layers.

0:50:41.950,0:50:47.410
So rather than[br]that, I say ask me.

0:50:47.410,0:50:49.970
Say oh, you know,[br]I'm freaking out.

0:50:49.970,0:50:55.100
I'm taking this final,[br]and I forgot curl.

0:50:55.100,0:50:59.320
Rather than not attempting[br]the complex problem at all,

0:50:59.320,0:51:03.780
ask me before the exam,[br]and I will remind everybody

0:51:03.780,0:51:07.700
how to set up the curl formula.

0:51:07.700,0:51:11.376
So you simply have[br]to think in terms

0:51:11.376,0:51:15.420
of operators-- ddx, ddy, ddz.

0:51:15.420,0:51:16.386
What are these?

0:51:16.386,0:51:22.092
These are derivative operators.

0:51:22.092,0:51:29.510
So if you take this and[br]multiply it by a function,

0:51:29.510,0:51:34.260
that means df, ds-- [INAUDIBLE].

0:51:34.260,0:51:39.714
All right, so in this[br]case, if F is-- I'll

0:51:39.714,0:51:54.050
go by my T-shirt-- PI plus QJ[br]plus RK, where PQ and R are all

0:51:54.050,0:52:00.720
scalar functions of xyz.

0:52:00.720,0:52:07.920


0:52:07.920,0:52:09.840
STUDENT: Then we[br]will not forget it.

0:52:09.840,0:52:11.760
PROFESSOR: Then we are[br]no longer forget it,

0:52:11.760,0:52:15.490
and you'll no longer[br]need my T-shirt.

0:52:15.490,0:52:18.580
All right, so how[br]do you do that?

0:52:18.580,0:52:22.430
You go expand along[br]your first row,

0:52:22.430,0:52:28.780
I times whoever the minor[br]will be, which is this guy.

0:52:28.780,0:52:31.490
How do you do the[br][? cowboy ?] problem?

0:52:31.490,0:52:34.167
These guys multiply each other.

0:52:34.167,0:52:38.530
So you go dr, dy.

0:52:38.530,0:52:39.220
Plus or minus?

0:52:39.220,0:52:45.410
Minus dq, dz.

0:52:45.410,0:52:55.497
Close times I. So the I is[br]the corresponding element

0:52:55.497,0:52:58.395
to the minor that[br]I just completed.

0:52:58.395,0:53:03.650
This minor is the determinant,[br]which is exactly this guy.

0:53:03.650,0:53:07.030
And this is exactly[br]what my T-shirt says.

0:53:07.030,0:53:08.130
Right, precisely.

0:53:08.130,0:53:09.380
OK.

0:53:09.380,0:53:12.895
The second term, if[br]we put the minus-- no,

0:53:12.895,0:53:14.840
they changed the signs.

0:53:14.840,0:53:15.780
That's the thing.

0:53:15.780,0:53:23.970
I would put minus, because I am[br]expanding along the first row.

0:53:23.970,0:53:28.010
And the second that I'm in[br]minus something minor times

0:53:28.010,0:53:29.790
J. Which minor?

0:53:29.790,0:53:34.160
Let me make in the lime.

0:53:34.160,0:53:35.520
Lime is a nice color.

0:53:35.520,0:53:50.990
And then I'll take this,[br]this, this, and that-- dr,

0:53:50.990,0:53:58.210
dx shooting [? cowboys ?][br]there-- minus dq, dz.

0:53:58.210,0:54:10.070
And of course they wrote[br]dq, dz minus dr, dx.

0:54:10.070,0:54:12.684
So I would leave it like that.

0:54:12.684,0:54:13.676
It doesn't matter.

0:54:13.676,0:54:16.160
You can put the[br]minus in if you want.

0:54:16.160,0:54:19.620
Plus the k dot.

0:54:19.620,0:54:22.308
k goes at the end.

0:54:22.308,0:54:24.292
All right, now k[br]goes at the end.

0:54:24.292,0:54:27.280


0:54:27.280,0:54:38.544
And then k multiplies this[br]determinant-- dq, dx minus dp,

0:54:38.544,0:54:39.044
dy.

0:54:39.044,0:54:43.970


0:54:43.970,0:54:45.620
dq, dx minus dp, dy.

0:54:45.620,0:54:48.434


0:54:48.434,0:54:49.341
Is it hard?

0:54:49.341,0:54:49.841
No.

0:54:49.841,0:54:51.700
It is not going to[br]be hard to memorize.

0:54:51.700,0:54:54.014
So then how did we do that?

0:54:54.014,0:54:57.592
We set up the first row to[br]be I, J, K, the second row

0:54:57.592,0:54:59.870
to be ddx, ddy, and ddz.

0:54:59.870,0:55:04.262
And then all in order the[br]components of your vector value

0:55:04.262,0:55:07.190
function in the exact[br]order they are with respect

0:55:07.190,0:55:11.120
to the standard basis i j k.

0:55:11.120,0:55:13.860
All right, now there[br]are other names

0:55:13.860,0:55:18.490
and other symbols for[br]curl of F. They use

0:55:18.490,0:55:21.766
curl because it's in English.

0:55:21.766,0:55:24.110
Well actually, in[br]Great Britain I

0:55:24.110,0:55:30.293
saw that they used [INAUDIBLE],[br]or else they use both.

0:55:30.293,0:55:34.450
In my language, in Romanian,[br]we call it [? rotore. ?]

0:55:34.450,0:55:38.390
And I saw that in French[br]it's very similar.

0:55:38.390,0:55:40.070
They use the same.

0:55:40.070,0:55:45.200
Now in the mechanical[br]engineering notation

0:55:45.200,0:55:46.625
it's funny.

0:55:46.625,0:55:53.644
They use another symbol and a[br]cross [? broad dot ?] symbol F.

0:55:53.644,0:56:00.390
And by that they mean[br]curl F. So if you

0:56:00.390,0:56:03.694
talk to a professor who's[br]in mechanical engineering,

0:56:03.694,0:56:06.260
or fluid mechanics,[br]or something,

0:56:06.260,0:56:10.680
when they talk about curl,[br]they will use this notation.

0:56:10.680,0:56:13.526
When they use this[br]other notation,

0:56:13.526,0:56:16.558
what do you think this is again?

0:56:16.558,0:56:17.743
Divergence, yes.

0:56:17.743,0:56:21.300
I told you last time[br]that is divergence of F.

0:56:21.300,0:56:26.090
So make the distinction[br]between-- again,

0:56:26.090,0:56:28.846
when are you leaving?

0:56:28.846,0:56:29.824
Huh?

0:56:29.824,0:56:34.050
OK, so you have[br]been in [INAUDIBLE].

0:56:34.050,0:56:38.620
And then we have[br]this distinction

0:56:38.620,0:56:40.272
we use here, like[br]for dot product

0:56:40.272,0:56:42.860
and you use here[br]as a cross product.

0:56:42.860,0:56:46.990
Now you have to understand[br]the conceptual difference is

0:56:46.990,0:56:49.280
huge between these guys.

0:56:49.280,0:56:52.210
This is a scalar function.

0:56:52.210,0:56:55.760
This is a vector function--[br]vector, scalar-- vector,

0:56:55.760,0:56:58.220
scalar, vector scalar.

0:56:58.220,0:57:00.188
Because I've had to do[br]it on so [INAUDIBLE].

0:57:00.188,0:57:01.664
It makes [INAUDIBLE].

0:57:01.664,0:57:06.584
And I heard of[br]colleagues complaining

0:57:06.584,0:57:10.028
while grading the final[br]that the students did not

0:57:10.028,0:57:14.990
understand that this is a[br]vector, and this is a scalar.

0:57:14.990,0:57:18.800
OK, a few simple exercises--[br]I'm going to go ahead and do

0:57:18.800,0:57:20.510
some of them.

0:57:20.510,0:57:24.450
We tried to make the[br]data on the final exam

0:57:24.450,0:57:30.314
very accessible and very[br]easy to apply in problems.

0:57:30.314,0:57:36.736
And one of the problems that--[br]we'll start with example 2--

0:57:36.736,0:57:40.700
would be this one.

0:57:40.700,0:57:43.251
And you may think, why?

0:57:43.251,0:57:45.736
Sometimes we put it in disguise.

0:57:45.736,0:57:50.457
And we said assume you[br]have a sphere-- that's

0:57:50.457,0:58:07.590
the unit sphere-- of[br]origin O. And say compute.

0:58:07.590,0:58:10.500


0:58:10.500,0:58:13.842
What is the equation of[br]the unit sphere, guys?

0:58:13.842,0:58:17.720
X squared plus y squared plus[br]z squared equals one, right?

0:58:17.720,0:58:23.749
From [INAUDIBLE], F[br]equals normal-- external

0:58:23.749,0:58:35.270
normal-- to the unit sphere[br]pointing out, [? through ?]

0:58:35.270,0:58:49.770
than N is the same at a[br]different point as the position

0:58:49.770,0:58:50.270
vector.

0:58:50.270,0:58:55.210


0:58:55.210,0:58:57.085
Then compute.

0:58:57.085,0:59:00.963


0:59:00.963,0:59:11.786
[? Now follow. ?] Gradient[br]of F, divergence of F,

0:59:11.786,0:59:15.782
and curl of F. Now that[br]should be a piece of cake.

0:59:15.782,0:59:19.275
Now one is not [INAUDIBLE][br]so much of a piece of cake

0:59:19.275,0:59:23.220
if you don't understand what[br]the problem wants from you.

0:59:23.220,0:59:28.240
It is to actually graph[br]the expression of this one.

0:59:28.240,0:59:32.462
So you're going to[br]say what is the normal

0:59:32.462,0:59:34.910
to a function like that?

0:59:34.910,0:59:37.360
First of all, we just[br]talked today about it.

0:59:37.360,0:59:42.220
If you have a function, even[br]if it's implicitly as F of x,

0:59:42.220,0:59:52.530
y, z equals c, in that case N[br]is your friend from the past.

0:59:52.530,1:00:00.832
If it's a unit normal,[br]unit normal to a surface

1:00:00.832,1:00:03.796
happens all the[br]time in engineering.

1:00:03.796,1:00:06.760
Whether you do solid[br]mechanics or fluid mechanics,

1:00:06.760,1:00:09.970
you always have to[br]complete these things.

1:00:09.970,1:00:11.757
This is going to be hard.

1:00:11.757,1:00:24.177
The gradient of F[br]divided by the length

1:00:24.177,1:00:27.099
of-- but here I have a problem.

1:00:27.099,1:00:32.943
I have to put G here, because[br]G will be my position vector.

1:00:32.943,1:00:35.890
This is the point x,y,z.

1:00:35.890,1:00:38.890
Or you prefer big R. But[br]I think I prefer big G,

1:00:38.890,1:00:41.900
because big R looks[br]like a scalar radius,

1:00:41.900,1:00:43.289
and I don't like that.

1:00:43.289,1:00:47.260
So the position vector[br]will be the circle middle

1:00:47.260,1:00:52.180
that starts at the origin and[br]whose N is on the surface,

1:00:52.180,1:00:53.160
right?

1:00:53.160,1:00:57.080
And this is the equation,[br]xy equals yj plus [? ek1. ?]

1:00:57.080,1:01:02.430
Because my point x,y,z has a[br]corresponding vector xi plus yj

1:01:02.430,1:01:04.550
plus zk-- big deal.

1:01:04.550,1:01:08.130
Now I'm trying to convince[br]you that, for the unit

1:01:08.130,1:01:12.570
normal for the sphere, I[br]have the same kind of thing.

1:01:12.570,1:01:17.330
So how do we compute[br]this normally?

1:01:17.330,1:01:23.092
I take the function F that[br]implicitly defines the surface.

1:01:23.092,1:01:27.020
All right, so in my case[br]F is something else.

1:01:27.020,1:01:29.966
What is it? x squared plus[br]y squared plus z squared.

1:01:29.966,1:01:34.385


1:01:34.385,1:01:37.331
Let's compute it.

1:01:37.331,1:01:40.440
N is going to be [INAUDIBLE].

1:01:40.440,1:01:41.680
It's very nice.

1:01:41.680,1:01:50.740
2x comma 2y comma 2z divided[br]by the square root of the sums.

1:01:50.740,1:01:52.405
Do I like this?

1:01:52.405,1:01:56.285
Uh, no, but I'll have to do[br]it whether I like it or not.

1:01:56.285,1:02:02.590


1:02:02.590,1:02:05.710
I want to simplify[br]up and down via 2.

1:02:05.710,1:02:07.264
Can I do that?

1:02:07.264,1:02:08.080
Of course I can.

1:02:08.080,1:02:15.950
I'm going to get x,y,z divided[br]by square root of x squared

1:02:15.950,1:02:18.110
plus y squared plus z squared.

1:02:18.110,1:02:22.054


1:02:22.054,1:02:23.040
And this was 1.

1:02:23.040,1:02:28.195


1:02:28.195,1:02:29.695
STUDENT: Wouldn't[br]there still be a 2

1:02:29.695,1:02:32.407
there, because it's 2[br]squared [INAUDIBLE]?

1:02:32.407,1:02:33.886
PROFESSOR: No, I pulled it out.

1:02:33.886,1:02:35.650
That's exactly what I said.

1:02:35.650,1:02:37.237
There was a 4 inside.

1:02:37.237,1:02:39.185
I pulled out with the forceps.

1:02:39.185,1:02:41.133
I put it up here,[br]square root of 4.

1:02:41.133,1:02:45.010
And I have a 2 here,[br]and that cancels out.

1:02:45.010,1:02:48.200
So I got something much[br]simpler than you guys

1:02:48.200,1:02:50.370
expected at first.

1:02:50.370,1:02:59.325
I got xi plus yj plus[br]zk as being the normal.

1:02:59.325,1:03:01.273
Did you expect this?

1:03:01.273,1:03:03.708
And you were supposed to[br]expect that this is y,

1:03:03.708,1:03:07.117
because this is the position[br]vector that has one length.

1:03:07.117,1:03:09.990
The length of a[br]root vector is 1,

1:03:09.990,1:03:11.445
and the point is on the sphere.

1:03:11.445,1:03:14.700
The normal will be[br]exactly the continuation.

1:03:14.700,1:03:16.900
Take your root[br]vector, and continue

1:03:16.900,1:03:20.080
in the same direction--[br]this is the beauty

1:03:20.080,1:03:23.840
of the normal to a surface,[br]that it continues the radius.

1:03:23.840,1:03:27.750
It continues the radius of the[br]sphere in the same direction.

1:03:27.750,1:03:30.160
So you copy and paste[br]your vector here.

1:03:30.160,1:03:34.870
Position vector G will be[br]the same as the normal N.

1:03:34.870,1:03:38.056
All you do is you shift,[br]but it's the same vector

1:03:38.056,1:03:40.350
at the different point.

1:03:40.350,1:03:43.530
Instead of starting at[br]O, it starts at P. So

1:03:43.530,1:03:45.546
[? that ?] is the same vector.

1:03:45.546,1:03:48.580
So you take the radius vector[br]from inside the sphere--

1:03:48.580,1:03:51.315
the position vector--[br]and you shift it out,

1:03:51.315,1:03:54.790
and that's the[br]normal to the sphere.

1:03:54.790,1:03:59.290
So the equation is still[br]xi plus yj plus zk.

1:03:59.290,1:03:59.790
Yes, sir.

1:03:59.790,1:04:02.225
STUDENT: Does it remain the[br]same for any other functions,

1:04:02.225,1:04:03.199
like [INAUDIBLE]?

1:04:03.199,1:04:07.810


1:04:07.810,1:04:09.800
PROFESSOR: For the[br]unit sphere, yes it is.

1:04:09.800,1:04:12.800
But for a general sphere, no.

1:04:12.800,1:04:15.760
For example, what[br]if my sphere will be

1:04:15.760,1:04:19.360
of center origin and radius R?

1:04:19.360,1:04:22.610


1:04:22.610,1:04:28.750
And its position vector[br]v is x,y,z-- like that.

1:04:28.750,1:04:31.635


1:04:31.635,1:04:33.380
[INAUDIBLE] I don't know.

1:04:33.380,1:04:34.090
G, right?

1:04:34.090,1:04:35.610
That's the position normal.

1:04:35.610,1:04:40.557
STUDENT: [INAUDIBLE] just[br]divide them by the R.

1:04:40.557,1:04:44.350
PROFESSOR: You just[br]divide by the R.

1:04:44.350,1:04:47.960
So instead of[br]radius being big R,

1:04:47.960,1:04:50.580
your unit vector[br]will be this one.

1:04:50.580,1:04:52.660
And you take this one[br]and shift it here,

1:04:52.660,1:04:53.940
and that's all you have.

1:04:53.940,1:04:55.380
For the sphere, it's beautiful.

1:04:55.380,1:04:58.077
For any surface in general, no.

1:04:58.077,1:04:59.985
Let me show you.

1:04:59.985,1:05:04.278
You have a bunch of [INAUDIBLE],[br]and your position vectors

1:05:04.278,1:05:06.663
look like crazies like that.

1:05:06.663,1:05:11.990
And the normals could[br]be-- they don't have

1:05:11.990,1:05:13.240
to continue their position.

1:05:13.240,1:05:23.220
They could be-- it depends how[br]the tangent planes look like.

1:05:23.220,1:05:31.204
And the tangent[br]plane at the point

1:05:31.204,1:05:33.699
has to be perpendicular[br]to the normal.

1:05:33.699,1:05:37.810
So the normal field is the[br]N of [INAUDIBLE] vectors.

1:05:37.810,1:05:41.200
But the little thingies[br]that look like rectangles

1:05:41.200,1:05:44.024
or whatever they are--[br]those are the tangent planes

1:05:44.024,1:05:45.680
of those points.

1:05:45.680,1:05:48.585
So in general there is[br]no obvious relationship

1:05:48.585,1:05:53.240
between the position and[br]the normal for the surface.

1:05:53.240,1:05:55.430
You are really lucky[br]for this [? field. ?]

1:05:55.430,1:06:00.166
And for many reasons, like[br]how beautiful the sphere is,

1:06:00.166,1:06:04.118
these functions will[br]be easy to compute.

1:06:04.118,1:06:06.588
Can you tell me what they[br]are without computing?

1:06:06.588,1:06:11.034
Because that should[br]be a piece of cake.

1:06:11.034,1:06:14.492
What is the gradient field?

1:06:14.492,1:06:19.432
STUDENT: [INAUDIBLE][br]to that one?

1:06:19.432,1:06:22.235
That's the x, y, and z.

1:06:22.235,1:06:24.890
PROFESSOR: For the sphere.

1:06:24.890,1:06:26.710
STUDENT: 2x, 2y--

1:06:26.710,1:06:36.795
PROFESSOR: Actually, let's do it[br]for both divergence G and curl

1:06:36.795,1:06:46.840
G. And you say wait, they[br]will be-- so gradient-- no,

1:06:46.840,1:06:47.630
I meant here.

1:06:47.630,1:06:50.408
You don't have gradient.

1:06:50.408,1:06:54.840
When F is a scalar function,[br]then you have gradient.

1:06:54.840,1:07:01.270
Then for that gradient you're[br]going to have divergence.

1:07:01.270,1:07:05.330
And for that-- I changed[br]notations, that's shy

1:07:05.330,1:07:06.860
I have to fix it.

1:07:06.860,1:07:10.820
Because F used to be that,[br]and it's not a vector anymore.

1:07:10.820,1:07:12.800
So big F is not[br]a vector anymore.

1:07:12.800,1:07:16.760
It's a scalar function, and now[br]I have to change the problem.

1:07:16.760,1:07:18.245
What is the gradient there?

1:07:18.245,1:07:19.730
What's divergence[br]of the gradient?

1:07:19.730,1:07:24.185
[INAUDIBLE] gradient of F.[br]And for the G that I gave you,

1:07:24.185,1:07:27.660
I want the divergence[br]in the curve?

1:07:27.660,1:07:31.320
So I made the problem[br]fluffier that it was before.

1:07:31.320,1:07:34.590
More things to[br]confuse for practice.

1:07:34.590,1:07:35.480
What's the gradient?

1:07:35.480,1:07:36.600
We did it before.

1:07:36.600,1:07:37.100
2x--

1:07:37.100,1:07:38.260
STUDENT: 2xi, 2--

1:07:38.260,1:07:42.470
PROFESSOR: 2y, 2z-- we are at a[br][? 93 ?] point p on the sphere.

1:07:42.470,1:07:46.195
It could be anywhere--[br]anywhere in space.

1:07:46.195,1:07:49.165
What's the divergence[br]of this individual?

1:07:49.165,1:07:51.640
So remember guys,[br]what I told you?

1:07:51.640,1:07:54.610
First component differentiated[br]with a straight 2x

1:07:54.610,1:07:57.332
plus second component[br]differentiated with respect

1:07:57.332,1:08:01.045
to y plus third[br]component differentiated

1:08:01.045,1:08:04.015
with respect to z.

1:08:04.015,1:08:11.950
2 plus 2 plus 2 equals[br]6-- piece of cake.

1:08:11.950,1:08:18.023
And curl of the gradient[br]of F-- is that hard?

1:08:18.023,1:08:18.856
[? STUDENT: Yeah. ?]

1:08:18.856,1:08:22.307
PROFESSOR: No, but we have[br]to know the definition.

1:08:22.307,1:08:26.743
And without looking at the[br]T-shirt, how do we do that?

1:08:26.743,1:08:30.194


1:08:30.194,1:08:36.899
The determinant-- I, J, K.[br]Operators-- ddx, ddy, and ddz.

1:08:36.899,1:08:40.875


1:08:40.875,1:08:44.850
STUDENT: [INAUDIBLE][br]2x, 2y, 2z, correct?

1:08:44.850,1:08:48.827
PROFESSOR: And we copy and[br]paste the three components.

1:08:48.827,1:08:50.814
[INAUDIBLE] in the trash.

1:08:50.814,1:08:53.830
I'll take the blue.

1:08:53.830,1:08:57.957
So we put 2x, 2y, 2z.

1:08:57.957,1:09:00.790
Do you think it's going[br]to be easy or hard?

1:09:00.790,1:09:02.130
Do you see the answer?

1:09:02.130,1:09:06.189
Some of are very sharp,[br]and you may see the answer.

1:09:06.189,1:09:10.254
For example, when the[br]cowboys shoot at each other

1:09:10.254,1:09:14.413
like this, dz, dy is here.

1:09:14.413,1:09:16.341
dy, dz is here.

1:09:16.341,1:09:23.970
So this, as a minor, is[br]0-- 0I, an eye for an eye.

1:09:23.970,1:09:25.759
And what else?

1:09:25.759,1:09:29.660
dz, dx-- dx dz, same[br]thing, minus 0j.

1:09:29.660,1:09:32.395
Is this meant to say minus 0j?

1:09:32.395,1:09:33.210
Yes it is.

1:09:33.210,1:09:37.715
But I did it because I want[br]you to have the good habit

1:09:37.715,1:09:40.189
of saying plus minus plus.

1:09:40.189,1:09:42.910
And that's finally[br]the same kind of thing

1:09:42.910,1:09:46.883
that'll give you 0k[br]if you think that when

1:09:46.883,1:09:49.313
you do partial derivative of[br]y with respect to [? f ?],

1:09:49.313,1:09:50.770
you get 0.

1:09:50.770,1:09:52.229
You have 0.

1:09:52.229,1:09:57.580
So some student of mine asked,[br]so this is the 0 vector,

1:09:57.580,1:10:01.820
how in the world do I[br]write a 0 vector on short?

1:10:01.820,1:10:03.014
Let me show you how.

1:10:03.014,1:10:04.180
You're going to laugh at me.

1:10:04.180,1:10:08.990
Some people write 0 bar,[br]which means the 0 vector.

1:10:08.990,1:10:11.561
Some other people don't[br]like it, it's silly.

1:10:11.561,1:10:17.642
Some people write O with[br]double like that, meaning that,

1:10:17.642,1:10:21.498
hey, this is a vector element,[br]the vector with its components

1:10:21.498,1:10:25.836
of 0, 0, 0-- to distinguish[br]that vector from the number 0,

1:10:25.836,1:10:33.618
which is not in bold-- So the[br]notations for the vector are 0.

1:10:33.618,1:10:37.546
So I'm going to[br]write here 0, 0, 0.

1:10:37.546,1:10:39.510
How about Mr. G?

1:10:39.510,1:10:42.456
Mr. G will act similarly.

1:10:42.456,1:10:47.250
When you do the divergence[br]it's going to be-- 1

1:10:47.250,1:10:50.315
plus 1 plus 1 equals 3.

1:10:50.315,1:10:53.522


1:10:53.522,1:10:55.982
You should remember this thing.

1:10:55.982,1:10:59.180
We are going to do[br]the divergence 3,

1:10:59.180,1:11:02.870
and they will ask you to do a[br]triple integral of a divergence

1:11:02.870,1:11:04.346
of a vector field.

1:11:04.346,1:11:06.642
And when you do[br]that, you are going

1:11:06.642,1:11:08.780
to get a triple[br]integer of something

1:11:08.780,1:11:13.330
like 3, which is a custom, which[br]will make your life very easy.

1:11:13.330,1:11:16.430
So you will very easily[br]compute those triple integrals

1:11:16.430,1:11:18.150
of constants.

1:11:18.150,1:11:22.635
Curl of G, G being[br]of [? a. ?] OK?

1:11:22.635,1:11:26.274
I should make the distinction[br]between a scalar function

1:11:26.274,1:11:30.603
and a vector function by putting[br]a G bar on the vector function.

1:11:30.603,1:11:34.932
How about this?

1:11:34.932,1:11:35.750
Is it hard?

1:11:35.750,1:11:38.156
No, because it's[br]the same fellow.

1:11:38.156,1:11:40.920
Instead of that, I[br]have just x, y, z.

1:11:40.920,1:11:42.770
The answer will be the same.

1:11:42.770,1:11:48.830
So I still want to get[br]0, 0, 0-- the vector 0.

1:11:48.830,1:11:52.670
So the point was that[br]we will give you enough.

1:11:52.670,1:11:54.848
You may expect them[br]to be very hard,

1:11:54.848,1:11:57.842
but they are not[br]going to be very hard.

1:11:57.842,1:12:02.333
Let's do one more like the[br]ones we have in the book.

1:12:02.333,1:12:06.325
What do you think[br]this one will be?

1:12:06.325,1:12:10.816
I'm making you a new[br]vector value function.

1:12:10.816,1:12:16.305
That's maybe two[br]little exercises

1:12:16.305,1:12:19.922
we can do just working[br]exercise three, four,

1:12:19.922,1:12:21.295
I don't know what they are.

1:12:21.295,1:12:23.902


1:12:23.902,1:12:28.276
Let me give you R[br]vector of x, y, z

1:12:28.276,1:12:38.470
equals yzI plus xzj plus xyk.

1:12:38.470,1:12:41.170
Compute the curl.

1:12:41.170,1:12:47.375
Let me write it like engineers[br]do just for fun-- [INAUDIBLE]

1:12:47.375,1:12:48.373
cross.

1:12:48.373,1:12:54.361
R is the same as[br]curl R, which is I,

1:12:54.361,1:12:59.840
J, K-- oh my god--[br]ddx, ddy, ddz.

1:12:59.840,1:13:02.500


1:13:02.500,1:13:07.820
Why is z-- xz-- xy.

1:13:07.820,1:13:13.490
Are you saying oh, that's[br]not so easy anymore.

1:13:13.490,1:13:17.356
You-- you will see[br]that it becomes easy,

1:13:17.356,1:13:22.160
OK? i times what is the minor?

1:13:22.160,1:13:24.880
This times-- x, right?

1:13:24.880,1:13:31.877
Minus x plus minus j[br]times 1 minus what?

1:13:31.877,1:13:34.811
Minor will be the red thingie.

1:13:34.811,1:13:39.212
And the red thingie[br]is beautiful,

1:13:39.212,1:13:45.430
because it's gonna be y[br]minus y plus k times--

1:13:45.430,1:13:49.810
who do you think it's[br]gonna be? z z minus.

1:13:49.810,1:13:53.220
So it's still 0.

1:13:53.220,1:13:57.460
Do we expect something[br]like that on the final?

1:13:57.460,1:13:58.900
An easy computation.

1:13:58.900,1:14:03.494
Somebody says, find me the[br]curve of this function.

1:14:03.494,1:14:04.868
And the functions[br]usually we give

1:14:04.868,1:14:06.460
you are nice and significant.

1:14:06.460,1:14:11.880
Something where the[br]result will be pretty.

1:14:11.880,1:14:12.380
OK.

1:14:12.380,1:14:15.836


1:14:15.836,1:14:18.954
Let me see what else I wanted.

1:14:18.954,1:14:25.898
I'm gonna-- I have space here.

1:14:25.898,1:14:45.738
So compute the curl and[br]Laplace operator of f of xyz

1:14:45.738,1:14:56.238
equals x squared yzi plus x y[br]squared zj plus xy z squared k.

1:14:56.238,1:15:02.530


1:15:02.530,1:15:04.466
Of divergence.

1:15:04.466,1:15:05.930
Sorry, guys.

1:15:05.930,1:15:09.328
This is not a-- it's[br]not a scalar function.

1:15:09.328,1:15:11.240
I want the divergence[br]and the curl.

1:15:11.240,1:15:13.152
The curl will be a vector.

1:15:13.152,1:15:15.550
The divergence will[br]be a scalar function.

1:15:15.550,1:15:18.190
Later on I'll give you a[br]nice function where you can

1:15:18.190,1:15:20.300
compute the Laplace operator.

1:15:20.300,1:15:23.800
That's gonna have to[br]be a scalar function.

1:15:23.800,1:15:27.302
And although the[br]Laplace operator

1:15:27.302,1:15:29.787
can be generalized[br]to vector functions,

1:15:29.787,1:15:31.775
and I'll tell you later[br]how-- what that is.

1:15:31.775,1:15:32.769
It's very easy.

1:15:32.769,1:15:37.620
It's practically the Laplace[br]operators in every direction.

1:15:37.620,1:15:38.450
OK.

1:15:38.450,1:15:42.276
So let's see the curl.

1:15:42.276,1:15:48.500


1:15:48.500,1:15:56.390
i j k, d dx, d dy, d dz.

1:15:56.390,1:15:58.740
Today I'm gonna cook[br]up the homework.

1:15:58.740,1:16:01.120
And with all the practice[br]that we are doing now,

1:16:01.120,1:16:04.880
you should have absolutely[br]no problem doing the homework

1:16:04.880,1:16:06.840
for the first two sections.

1:16:06.840,1:16:10.330
At least for the section--[br]today's section, 13.1.

1:16:10.330,1:16:15.340
x squared yz, x y[br]squared z, xy z squared.

1:16:15.340,1:16:17.393
You see there is some[br]sort of symmetry.

1:16:17.393,1:16:18.800
I'm playing a game here.

1:16:18.800,1:16:23.940


1:16:23.940,1:16:29.840
So I have i.

1:16:29.840,1:16:33.930
I want you to tell me[br][INAUDIBLE], see now,

1:16:33.930,1:16:35.615
I don't work much in groups.

1:16:35.615,1:16:38.116
I don't make you[br]work in groups, but I

1:16:38.116,1:16:40.690
want you to answer my question.

1:16:40.690,1:16:46.925
So what is going to be this[br]minor that I-- the first thing

1:16:46.925,1:16:47.425
is gonna be?

1:16:47.425,1:16:48.910
STUDENT: x z squared.

1:16:48.910,1:16:49.900
PROFESSOR: Very good.

1:16:49.900,1:16:52.375
STUDENT: Minus x y squared.

1:16:52.375,1:16:57.325


1:16:57.325,1:16:58.315
PROFESSOR: Minus j.

1:16:58.315,1:17:00.295
Potential [? plus or ?] minus.

1:17:00.295,1:17:01.780
OK, what is the next guy?

1:17:01.780,1:17:03.070
STUDENT: y z squared.

1:17:03.070,1:17:07.970
PROFESSOR: y z[br]squared, thank you.

1:17:07.970,1:17:08.950
STUDENT: x squared.

1:17:08.950,1:17:12.870
PROFESSOR: x squared, right?

1:17:12.870,1:17:16.790
Plus k times--

1:17:16.790,1:17:20.710
STUDENT: y squared z.

1:17:20.710,1:17:23.160
PROFESSOR: So, you[br]see what I'm doing?

1:17:23.160,1:17:27.504
I'm doing this from[br]respect to x. y squared z.

1:17:27.504,1:17:28.971
You said it, right.

1:17:28.971,1:17:32.883
Minus this guy, x squared z.

1:17:32.883,1:17:37.630
Can I write it more-- I don't[br]really like the way I wrote it.

1:17:37.630,1:17:38.750
But I'll write like that.

1:17:38.750,1:17:45.402
How about x times z squared[br]minus y squared i minus j.

1:17:45.402,1:17:47.390
Or maybe better plus j.

1:17:47.390,1:17:49.378
I'll change this up.

1:17:49.378,1:17:50.869
Plus j at the end.

1:17:50.869,1:17:55.342
Because it's the[br]vector. y times x

1:17:55.342,1:18:02.797
squared minus z squared[br]j plus-- who gets out?

1:18:02.797,1:18:09.258
z-- z times y squared[br]minus x squared k.

1:18:09.258,1:18:12.737


1:18:12.737,1:18:13.731
OK.

1:18:13.731,1:18:16.713
Good There is some[br]symmetry in there.

1:18:16.713,1:18:19.695
The break in the symmetry[br]is in the middle.

1:18:19.695,1:18:23.174
Because, as you[br]see, x is separate,

1:18:23.174,1:18:26.618
and then z is followed[br]by y, and then

1:18:26.618,1:18:31.770
x squared-- x is followed[br]by z and y is followed by x.

1:18:31.770,1:18:37.098
So I have some-- some[br]symmetry of some sort.

1:18:37.098,1:18:40.080


1:18:40.080,1:18:42.565
What else did I want?

1:18:42.565,1:18:44.056
Divergence operator.

1:18:44.056,1:18:47.535
And that will be the[br]last example of the kind.

1:18:47.535,1:18:50.020
[INAUDIBLE]

1:18:50.020,1:18:51.964
How do you write the divergence?

1:18:51.964,1:18:52.505
Is this hard?

1:18:52.505,1:18:53.005
Very easy?

1:18:53.005,1:18:55.510


1:18:55.510,1:18:58.771
I'm going go ask you to simplify[br]because I don't like it,

1:18:58.771,1:19:01.226
like, as a sum.

1:19:01.226,1:19:02.208
2xyz--

1:19:02.208,1:19:10.560
STUDENT: 2xyz plus[br]2xyz plus 2xyz.

1:19:10.560,1:19:13.411
PROFESSOR: And now you see[br]why I don't like it as a sum.

1:19:13.411,1:19:17.387
Because it's 6xyz and it's[br]very pretty like that.

1:19:17.387,1:19:20.120
I'd like you-- on[br]the exam, I'd like

1:19:20.120,1:19:24.345
you to take the function[br]and box the answer,

1:19:24.345,1:19:28.815
and that's all I want you to do.

1:19:28.815,1:19:29.315
All right.

1:19:29.315,1:19:31.303
I'm gonna go ahead and erase.

1:19:31.303,1:19:49.692


1:19:49.692,1:19:53.171
I'm going to move[br]on to 13.2, but I'd

1:19:53.171,1:19:59.848
like to review some physics[br]a little bit with you

1:19:59.848,1:20:02.800
and see what you[br]remember from physics.

1:20:02.800,1:20:21.496


1:20:21.496,1:20:23.464
It's a little bit messy.

1:20:23.464,1:20:27.892
I'll use this instead because[br]I like the board to be clean.

1:20:27.892,1:20:34.260
If I were to ask you to[br]remember work in physics,

1:20:34.260,1:20:41.708
I would say-- I'm changing a[br]little bit the order in 13.2.

1:20:41.708,1:20:46.688
I'd like you to go[br]back in time and see

1:20:46.688,1:20:48.680
what work was in physics class.

1:20:48.680,1:20:53.660


1:20:53.660,1:20:56.150
STUDENT: [INAUDIBLE] force dx.

1:20:56.150,1:20:59.670


1:20:59.670,1:21:02.090
PROFESSOR: What if you[br]didn't know any calculus?

1:21:02.090,1:21:04.318
Let's go a long time.

1:21:04.318,1:21:10.780
The section is[br]13.2 preliminaries.

1:21:10.780,1:21:13.472
STUDENT: Force[br]multiplied distance.

1:21:13.472,1:21:15.424
PROFESSOR: Very good.

1:21:15.424,1:21:17.376
Preliminary work.

1:21:17.376,1:21:25.498
[INAUDIBLE] The notion of work[br]from physics-- hey, come on.

1:21:25.498,1:21:34.960


1:21:34.960,1:21:39.130
Physics, or engineering,[br]mechanics, whatever you study,

1:21:39.130,1:21:39.800
work.

1:21:39.800,1:21:43.922
Imagine that you're[br]taking a-- this

1:21:43.922,1:21:47.393
is your body that you're playing[br]with-- not your own body,

1:21:47.393,1:21:50.470
but the body you are[br]acting on in physics.

1:21:50.470,1:21:58.480
And you are dragging this[br]object from a place A

1:21:58.480,1:22:00.890
to a place B, another position.

1:22:00.890,1:22:04.264
A B is the distance.

1:22:04.264,1:22:09.190
And the force is parallel[br]to the trajectory.

1:22:09.190,1:22:10.400
This is very important.

1:22:10.400,1:22:12.349
This is a simpler case.

1:22:12.349,1:22:14.225
In general, it's not so simple.

1:22:14.225,1:22:17.840
So this force is acting,[br]and it's a constant force.

1:22:17.840,1:22:21.910
And you pull the object[br]from one place to another.

1:22:21.910,1:22:24.040
That's case one.

1:22:24.040,1:22:28.010
In case two, life is harder.

1:22:28.010,1:22:35.825
You actually pull[br]the poor object

1:22:35.825,1:22:38.272
with the force in[br]this direction.

1:22:38.272,1:22:40.682
Actually, most of[br]us do that, right?

1:22:40.682,1:22:45.090
If I were to have a gliding[br]object on the surface,

1:22:45.090,1:22:48.500
I would actually act on[br]that object in the direction

1:22:48.500,1:22:51.100
of my arm by pulling it.

1:22:51.100,1:22:56.520
So when I displace this body[br]from point a to point b,

1:22:56.520,1:23:01.140
I still travel the[br]distance d, but-- so d

1:23:01.140,1:23:04.230
is a displacement vector[br]that can be written like--

1:23:04.230,1:23:07.200
or it can be drawn like that.

1:23:07.200,1:23:09.550
I have to be smart[br]in both cases,

1:23:09.550,1:23:11.320
figure out what[br]I want from life,

1:23:11.320,1:23:13.530
because it's not so clear.

1:23:13.530,1:23:15.610
When they taught me I[br]think the first time

1:23:15.610,1:23:18.460
I was in-- oh my[br]God-- eighth grade,

1:23:18.460,1:23:20.770
and they-- that was[br]a long time ago.

1:23:20.770,1:23:26.640
In this case, w is[br]gonna be a scalar

1:23:26.640,1:23:32.260
and I'm gonna have the magnitude[br]of the force F. F is a vector,

1:23:32.260,1:23:36.895
but to indicate it in Newtons[br]or whatever I measure it in,

1:23:36.895,1:23:40.850
it's gonna be in magnitude.

1:23:40.850,1:23:44.552
Times the little d,[br]but instead of little d

1:23:44.552,1:23:48.180
I should be a little[br]smarter and say,

1:23:48.180,1:23:51.859
Magdalena, this is the magnitude[br]of the vector A B, which

1:23:51.859,1:23:52.900
is a displacement vector.

1:23:52.900,1:23:55.520


1:23:55.520,1:23:58.590
So this is called work.

1:23:58.590,1:24:03.151
And if the force is 10[br]Newtons, and the distance

1:24:03.151,1:24:07.480
is 10 meters, because we[br]want to go international.

1:24:07.480,1:24:10.972
We want to be global, right,[br]at Texas Tech, so good.

1:24:10.972,1:24:12.180
So we have 100 Newton-meters.

1:24:12.180,1:24:17.450


1:24:17.450,1:24:23.034
Now you can measure-- well,[br]you can have another example.

1:24:23.034,1:24:27.914
I'm thinking gravity and then[br]you can say it in pounds,

1:24:27.914,1:24:29.866
and that measures force.

1:24:29.866,1:24:32.306
And you have other units[br]that are not international.

1:24:32.306,1:24:34.260
I'm not gonna mess up.

1:24:34.260,1:24:38.310
When you have the work[br]in this case, though,

1:24:38.310,1:24:41.430
it's more complicated.

1:24:41.430,1:24:44.100
And I'm not gonna be mad[br]at you. [INAUDIBLE] is

1:24:44.100,1:24:45.805
trying to tell me what it is.

1:24:45.805,1:24:48.940
I'm not going to be mad at[br]the people who don't know

1:24:48.940,1:24:51.340
what the work is in this case.

1:24:51.340,1:24:56.638
Although, I was looking[br]at-- I am the person

1:24:56.638,1:25:01.558
who has run a[br]committee to oversee

1:25:01.558,1:25:04.264
the finals for different[br]math classes, all the math

1:25:04.264,1:25:05.520
classes we offer here.

1:25:05.520,1:25:09.950
And every semester I see the[br][? streak ?] pre-calculus,

1:25:09.950,1:25:11.976
calc 1, calc 2, calc 3.

1:25:11.976,1:25:13.350
In trig and[br]pre-calculus, they'll

1:25:13.350,1:25:15.880
always have a work there.

1:25:15.880,1:25:19.930
And I was wondering how many[br]of you took pre-calculus,

1:25:19.930,1:25:22.520
and how many of you[br]remember that you

1:25:22.520,1:25:26.158
studied this in pre-calculus.

1:25:26.158,1:25:27.820
It's a little bit awkward.

1:25:27.820,1:25:31.400
I'm thinking, how do they do[br]it, but I gave you the formula.

1:25:31.400,1:25:35.770
And they say the force in[br]itself as a vector dot product

1:25:35.770,1:25:38.680
the displacement vector.

1:25:38.680,1:25:41.674
So they are both[br]forces in dot product.

1:25:41.674,1:25:45.482
And I was surprised to see[br]that they gave you [INAUDIBLE]

1:25:45.482,1:25:51.420
If I were to express it,[br]how would I express it?

1:25:51.420,1:25:55.340
I'll say the magnitude[br]of F, of course,

1:25:55.340,1:25:57.790
in Newtons, whatever it[br]is, times the magnitude

1:25:57.790,1:26:00.240
of the displacement vector--

1:26:00.240,1:26:01.720
STUDENT: Multiply those cosines.

1:26:01.720,1:26:04.580
PROFESSOR: Cosine of[br]the angle between.

1:26:04.580,1:26:06.580
And I'm too lazy, I[br]don't know, theta.

1:26:06.580,1:26:08.980
Let's call it angle theta.

1:26:08.980,1:26:16.200
Because I don't want to[br]include that in locations, OK?

1:26:16.200,1:26:18.460
It really doesn't matter[br]in which direction

1:26:18.460,1:26:23.340
I'm going, because cosine theta,[br]thank God, is an even function.

1:26:23.340,1:26:25.790
It's equal to cosine[br]of minus theta.

1:26:25.790,1:26:29.710
So whether I go this way or[br]that way, it's the same cosine.

1:26:29.710,1:26:30.690
All right.

1:26:30.690,1:26:33.630
So the cosine of the[br]angle between the two.

1:26:33.630,1:26:37.060
It's very easy when you[br]don't need calculus.

1:26:37.060,1:26:41.960
But when you use calculus,[br]because your trajectory is

1:26:41.960,1:26:45.880
no longer a line, life is[br]becoming more complicated.

1:26:45.880,1:26:49.733
So we have to come up[br]with a different formula,

1:26:49.733,1:26:53.581
with a different notion of work.

1:26:53.581,1:26:56.510
I'm gonna erase-- are[br]you guys done with that?

1:26:56.510,1:26:58.200
Is it visible?

1:26:58.200,1:26:59.192
You're done.

1:26:59.192,1:27:08.116


1:27:08.116,1:27:08.616
OK.

1:27:08.616,1:27:14.100


1:27:14.100,1:27:18.680
So again, life is not so[br]easy in reality anymore.

1:27:18.680,1:27:28.500
I have a particle in physics-- a[br]photon enters a four-star hotel

1:27:28.500,1:27:32.838
and says-- talks to the[br]bellboy, and the bellboy,

1:27:32.838,1:27:34.780
can I help you[br]with your luggage.

1:27:34.780,1:27:36.732
No, I'm traveling light.

1:27:36.732,1:27:37.694
[LAUGHTER]

1:27:37.694,1:27:41.390
So the particle, the[br]photon-- whatever.

1:27:41.390,1:27:45.190
A particle is moving-- is[br]moving on a trajectory.

1:27:45.190,1:27:47.790
Suppose that[br]trajectory is planar,

1:27:47.790,1:27:50.690
just to make your[br]life easier at first.

1:27:50.690,1:27:53.426
It's in the plane x y.

1:27:53.426,1:27:56.842
And this is the little[br]particle that's moving.

1:27:56.842,1:28:04.162
And this is R. And[br]that is x i plus yj.

1:28:04.162,1:28:08.554


1:28:08.554,1:28:11.482
Good.

1:28:11.482,1:28:13.900
And this is the point x, y.

1:28:13.900,1:28:16.350
And that's the position,[br]the current position

1:28:16.350,1:28:18.320
of my particle, right now.

1:28:18.320,1:28:20.650
Not in the past,[br]not in the future.

1:28:20.650,1:28:23.697
My particle is moving,[br]and this is now.

1:28:23.697,1:28:26.132
Suppose time doesn't even exist.

1:28:26.132,1:28:30.220
We think of the movies[br]that we saw lately,

1:28:30.220,1:28:33.670
in The Theory of Everything.

1:28:33.670,1:28:38.740
So then, they say OK, we only[br]care about now. x, y is now

1:28:38.740,1:28:43.740
and that is the current[br]position vector.

1:28:43.740,1:28:49.600
Well, what would be[br]the work between now--

1:28:49.600,1:28:55.480
whatever now-- and the next,[br]let's say, this is gonna be x1,

1:28:55.480,1:28:57.254
y1.

1:28:57.254,1:28:59.222
And this is x0, y0.

1:28:59.222,1:29:04.634


1:29:04.634,1:29:14.000
That's the general formula,[br]will be x i plus yj.

1:29:14.000,1:29:19.150
So I actually cannot[br]forget about time.

1:29:19.150,1:29:21.070
Not as much as I want.

1:29:21.070,1:29:27.500
So x and y-- x and y are[br]both changing in time.

1:29:27.500,1:29:30.965
We're gonna have x equals[br]x sub t, y equals y sub t.

1:29:30.965,1:29:34.925
Do you guys remember what we[br]call that kind of equation

1:29:34.925,1:29:39.352
for a curve from here to here?

1:29:39.352,1:29:40.275
Para--

1:29:40.275,1:29:41.316
STUDENT: Parametrization.

1:29:41.316,1:29:45.735
PROFESSOR: Parametrization,[br]or parametric equations.

1:29:45.735,1:29:50.154
Parametric equations.

1:29:50.154,1:29:57.028


1:29:57.028,1:29:59.020
Good.

1:29:59.020,1:30:04.330
So I have may the[br]force be with you.

1:30:04.330,1:30:06.780
I have a force.

1:30:06.780,1:30:10.890
I have a force, and I have[br]some sort of displacement.

1:30:10.890,1:30:15.330
But I cannot express that[br]displacement linearly anymore.

1:30:15.330,1:30:18.630
I'm moving along a[br][INAUDIBLE] of a curve.

1:30:18.630,1:30:22.572
So I have to think differently.

1:30:22.572,1:30:27.810
And the work will be defined,[br]whether you like it or not,

1:30:27.810,1:30:35.760
as F vector field, dot[br]product with dR over c.

1:30:35.760,1:30:40.080
And you say, what in[br]the world is that?

1:30:40.080,1:30:43.535
What would c be?

1:30:43.535,1:30:46.430
How do I integrate along a path?

1:30:46.430,1:30:50.336
And I will tell you in a[br]second what we mean by that.

1:30:50.336,1:30:53.230


1:30:53.230,1:30:58.596
Meaning that-- this is by[br]definition if you want.

1:30:58.596,1:31:03.490
This is like an[br]application of calculus 1.

1:31:03.490,1:31:07.699
It can be proved, but we[br]don't-- we do a rigorous job

1:31:07.699,1:31:12.316
in the book about introducing[br]and proving that along

1:31:12.316,1:31:13.774
a curvilinear path.

1:31:13.774,1:31:17.662
This is gonna be-- where[br]am I here, at time t0?

1:31:17.662,1:31:19.710
And this is at time t1.

1:31:19.710,1:31:23.370
That means x0 is x of t0.

1:31:23.370,1:31:26.580
y0 is y of t0.

1:31:26.580,1:31:31.530
And at the finish point,[br]I'm at x1, which is x of t1,

1:31:31.530,1:31:34.880
and y1 equals y of t1.

1:31:34.880,1:31:40.170
So between t0 and[br]t1, I have traveled

1:31:40.170,1:31:46.130
and I have F where[br]measure at x of t y of t,

1:31:46.130,1:31:50.262
where t is between--[br]moving between t0 and t1.

1:31:50.262,1:31:53.040
I'm done with this[br]is the F part.

1:31:53.040,1:31:54.981
What is the dR?

1:31:54.981,1:31:57.869
Now you guys know[br]about differential.

1:31:57.869,1:32:00.364
Thank God you know[br]about differential,

1:32:00.364,1:32:06.851
because this is gonna[br]help you very much.

1:32:06.851,1:32:07.849
OK.

1:32:07.849,1:32:18.100
So instead of dR, I'm going[br]to write dot, and let's

1:32:18.100,1:32:24.295
see how I write-- what I write[br]in terms of dR. You may say,

1:32:24.295,1:32:28.020
well, what does she mean?

1:32:28.020,1:32:34.617
dR was dxi plus dyj.

1:32:34.617,1:32:36.281
And you say, why is that?

1:32:36.281,1:32:37.072
I don't understand.

1:32:37.072,1:32:41.491
Because R itself is x i plus yj.

1:32:41.491,1:32:46.638
And x is a function of t,[br]and y is a function of t.

1:32:46.638,1:32:49.910
That means that when you[br]apply the differential,

1:32:49.910,1:32:53.566
you are gonna apply the[br]differentials to dx and dy,

1:32:53.566,1:32:57.003
and these are gonna be[br]infinitesimal displacement.

1:32:57.003,1:33:00.440
Infinitesimal displacement.

1:33:00.440,1:33:07.805


1:33:07.805,1:33:09.769
Infinitesimal displacement.

1:33:09.769,1:33:14.188
What is an infinitesimal[br]displacement in terms of time?

1:33:14.188,1:33:16.590
Well, we have our[br]parametric equations.

1:33:16.590,1:33:21.280
So Mr. dx as a differential[br]is just x prime dt.

1:33:21.280,1:33:24.060
It's like in the[br][INAUDIBLE] substitution.

1:33:24.060,1:33:26.162
dy is just y prime dt.

1:33:26.162,1:33:28.617
So let me write this down again.

1:33:28.617,1:33:44.554
This is x prime of t i plus[br]y prime of t j times dt.

1:33:44.554,1:33:48.033
So Mr. dt is like[br]a common factor.

1:33:48.033,1:33:50.021
If he wants to go[br]out for a walk,

1:33:50.021,1:33:52.177
he says, I'm gonna[br]go out for a walk.

1:33:52.177,1:33:53.010
I go out for a walk.

1:33:53.010,1:33:58.420
So dR is actually x[br]prime of t times i

1:33:58.420,1:34:02.972
plus y prime of t times j dt.

1:34:02.972,1:34:06.850


1:34:06.850,1:34:14.650
And this will represent[br]the derivative of R

1:34:14.650,1:34:16.054
with respect to pi.

1:34:16.054,1:34:18.030
So that will be what?

1:34:18.030,1:34:19.512
The differential.

1:34:19.512,1:34:26.428
Differential of R[br]with respect to pi.

1:34:26.428,1:34:31.370


1:34:31.370,1:34:34.550
This is the same as[br]writing dx i plus dy j.

1:34:34.550,1:34:37.690


1:34:37.690,1:34:44.112
And it's the same as writing[br]dR. Why is this happening?

1:34:44.112,1:34:47.330
Because it's [INAUDIBLE][br]Because x and y

1:34:47.330,1:34:52.536
themselves are functions of one[br]variable only, which is time.

1:34:52.536,1:34:56.488
This is why it happens.

1:34:56.488,1:34:59.946
Oh, so we will[br]simply have to do--

1:34:59.946,1:35:02.910
to learn new things, right?

1:35:02.910,1:35:05.380
We are gonna have[br]to learn new things,

1:35:05.380,1:35:10.080
like integral from a time--[br]fixed time 0 to t1, which

1:35:10.080,1:35:15.457
is 10 seconds, of a dot product[br]between a certain vector that

1:35:15.457,1:35:18.610
depends on time and another[br]vector that depends on time,

1:35:18.610,1:35:20.552
and dt.

1:35:20.552,1:35:23.468
So we are gonna[br]have to learn how

1:35:23.468,1:35:28.000
to compute the work through this[br]type of curvilinear integral.

1:35:28.000,1:35:32.492
And this is-- this is called[br]either path integral-- path

1:35:32.492,1:35:50.907
integral along the curve c, or[br]curvilinear integral along c.

1:35:50.907,1:35:53.853
Yes.

1:35:53.853,1:35:56.799
STUDENT: Let's say if I[br]move the force this is

1:35:56.799,1:35:58.763
[INAUDIBLE] function, correct?

1:35:58.763,1:36:01.954
So if I can find[br][? arc length ?] that

1:36:01.954,1:36:03.720
is between the x--

1:36:03.720,1:36:04.570
PROFESSOR: Yeah--

1:36:04.570,1:36:05.736
STUDENT: [INAUDIBLE] points.

1:36:05.736,1:36:08.870
PROFESSOR: Yeah, we will do[br]the one with that length next.

1:36:08.870,1:36:10.914
The [? reason ?] so--

1:36:10.914,1:36:11.830
STUDENT: Is it harder?

1:36:11.830,1:36:12.790
PROFESSOR: No.

1:36:12.790,1:36:15.064
No, you can pass[br]through a plane.

1:36:15.064,1:36:17.499
And you can-- we'll[br]do that next time.

1:36:17.499,1:36:20.908
You will have an integral[br]with respect to S.

1:36:20.908,1:36:23.343
So the integration will[br]be with respect to dS,

1:36:23.343,1:36:24.317
they are correct.

1:36:24.317,1:36:27.726
And then you will have a[br]function that depends on S

1:36:27.726,1:36:31.340
[INAUDIBLE] So I'll-- for[br]today, I'll only teach you that.

1:36:31.340,1:36:33.360
Next time I'll teach[br]you that with respect

1:36:33.360,1:36:36.766
to arc length, which is also[br]very-- it's not hard at all.

1:36:36.766,1:36:37.266
STUDENT: OK.

1:36:37.266,1:36:41.532
PROFESSOR: So I will work[br]with you on [INAUDIBLE]

1:36:41.532,1:36:48.773
Now assume that we have-- I[br]will spray all this thing.

1:36:48.773,1:36:55.156


1:36:55.156,1:36:58.102
Assume that I have a problem.

1:36:58.102,1:37:02.670
I have a parabola-- arc[br]of a parabola, all right?

1:37:02.670,1:37:08.504
Between-- let's[br]say the parabola is

1:37:08.504,1:37:16.440
y equals x squared[br]between two points.

1:37:16.440,1:37:19.416


1:37:19.416,1:37:21.896
And I'll ask you to[br]compute some work,

1:37:21.896,1:37:25.368
and I'll tell you in a second[br]what [INAUDIBLE] to do.

1:37:25.368,1:37:38.264


1:37:38.264,1:37:52.648
So exercise-- assume[br]the parabola y

1:37:52.648,1:38:03.340
equals x squared between[br]points A of coordinates 0, 0

1:38:03.340,1:38:07.040
and point B of coordinates 1, 1.

1:38:07.040,1:38:15.826
a, Parametrized this parabola[br]in the simplest way you can.

1:38:15.826,1:38:31.070


1:38:31.070,1:38:41.360
And b, compute the work[br]along this arc of a parabola,

1:38:41.360,1:38:44.316
arc AB of this parabola.

1:38:44.316,1:38:50.268


1:38:50.268,1:39:04.246
For [INAUDIBLE] the[br]function big F of t,

1:39:04.246,1:39:09.922
you see that-- I'm going to say,[br]no, big F of the point x, y,

1:39:09.922,1:39:15.131
because you haven't parametrized[br]that yet, big F of x,

1:39:15.131,1:39:22.310
y being xi plus yg.

1:39:22.310,1:39:27.722


1:39:27.722,1:39:30.182
So you say, OK, wait a minute.

1:39:30.182,1:39:34.610
W will be integral over[br]the arc of a parabola.

1:39:34.610,1:39:37.040
Do you want to draw that first?

1:39:37.040,1:39:39.160
Yes, I need to draw that first.

1:39:39.160,1:39:43.998
So I have this parabola from A[br]to B. A is of coordinates 0, 0.

1:39:43.998,1:39:45.962
B is of coordinates 1, 1.

1:39:45.962,1:39:48.908
And this is y equals x squared.

1:39:48.908,1:39:52.099
So what kind of[br]parametrization is

1:39:52.099,1:39:55.291
the simplest one, the[br]regular one that people take?

1:39:55.291,1:39:57.255
Take x to be t?

1:39:57.255,1:40:00.390
And of course, take y in that[br]case. y will be t squared.

1:40:00.390,1:40:01.553
And for 1 you have 1.

1:40:01.553,1:40:04.018
For 0 you have 0.

1:40:04.018,1:40:07.469
When you have that work by[br]definition, what was that?

1:40:07.469,1:40:12.247
It was written as integral[br]or on the graph C. Let's call

1:40:12.247,1:40:14.227
this path C a curvilinear path.

1:40:14.227,1:40:16.455
Look, script C-- so beautiful.

1:40:16.455,1:40:25.365
Let me [INAUDIBLE] red and[br]draw the C of what is work?

1:40:25.365,1:40:36.616
F force-- may the force be with[br]us-- dot dR. All righty, that's

1:40:36.616,1:40:39.050
a little bit of a headache.

1:40:39.050,1:40:45.910
This F is going to be-- can I[br]write an alternative formula

1:40:45.910,1:40:50.810
that I have not written yet[br]but I will write in a second?

1:40:50.810,1:40:56.060
dR will be dxi plus dyj.

1:40:56.060,1:41:01.628
So I can also[br]write that F dot dR

1:41:01.628,1:41:04.556
as the dot product will seem to[br]be-- what was the dot product

1:41:04.556,1:41:06.020
guys, do you remember?

1:41:06.020,1:41:10.430
First component times[br]first component, F1dx

1:41:10.430,1:41:14.931
plus second scalar component[br]times second scalar component,

1:41:14.931,1:41:15.431
F2dy.

1:41:15.431,1:41:20.170


1:41:20.170,1:41:21.510
I'll write it down.

1:41:21.510,1:41:26.382
Along the path C I'll[br]have F1dx plus F2dy.

1:41:26.382,1:41:29.890
But god knows what it's[br]going to be in terms of time.

1:41:29.890,1:41:33.140
So I have to change[br]variable thinking.

1:41:33.140,1:41:36.690
Okey-dokey, Mr.[br]dx by substitution

1:41:36.690,1:41:41.260
was x prime to T. Mr. dy by[br]substitution was y prime dt.

1:41:41.260,1:41:46.010
So I'd rather write[br]this in a simpler way.

1:41:46.010,1:41:50.206
This is a new object,[br]path integral.

1:41:50.206,1:41:52.980
But we know this[br]object from Calc I

1:41:52.980,1:41:56.830
as being a simple[br]integral from time t0--

1:41:56.830,1:42:04.070
I'll write it down-- time[br]t1, F1x prime of t plus F2y

1:42:04.070,1:42:05.911
prime of t.

1:42:05.911,1:42:09.760
This is the integral dt.

1:42:09.760,1:42:12.927
This would be a[br]piece of cake for us

1:42:12.927,1:42:16.067
to apply in this problem.

1:42:16.067,1:42:19.931
Equals-- now you tell me[br]what I'm supposed to write.

1:42:19.931,1:42:23.260
Because if you don't, I'm[br]going to not write anything.

1:42:23.260,1:42:26.400
t0 for me is what time?

1:42:26.400,1:42:28.370
When did we leave this?

1:42:28.370,1:42:29.346
0.

1:42:29.346,1:42:31.251
And when did we arrive?

1:42:31.251,1:42:32.604
At 1 o'clock.

1:42:32.604,1:42:38.150
We arrived when t is 1, or[br]every one second or whatever

1:42:38.150,1:42:40.287
depending on [INAUDIBLE].

1:42:40.287,1:42:45.780
OK, from 0 to 1, now who is F1?

1:42:45.780,1:42:48.365
F1 is this.

1:42:48.365,1:42:49.350
But it drives me crazy.

1:42:49.350,1:42:52.800
Because I need this[br]to be expressed in t.

1:42:52.800,1:42:55.810
So I think of x and[br]y as functions of t.

1:42:55.810,1:42:59.524
So if 1 is not x,[br]not [INAUDIBLE]

1:42:59.524,1:43:04.484
right here, but t, which is the[br]same thing in parametrization--

1:43:04.484,1:43:07.956
this is t, t times.

1:43:07.956,1:43:10.436
Who is x prime?

1:43:10.436,1:43:11.924
1, thank god.

1:43:11.924,1:43:17.380
That is easy, times 1, plus F2.

1:43:17.380,1:43:19.860
Who is F2?

1:43:19.860,1:43:20.360
t squared.

1:43:20.360,1:43:22.640
I'll have to write it down.

1:43:22.640,1:43:25.420
Times who is y prime?

1:43:25.420,1:43:26.230
2t.

1:43:26.230,1:43:27.925
y prime is t2.

1:43:27.925,1:43:34.140
So I write it down-- 2t, dt.

1:43:34.140,1:43:37.980


1:43:37.980,1:43:40.970
So that's how I compute[br]this integral back.

1:43:40.970,1:43:41.820
Is it hard?

1:43:41.820,1:43:47.370
No, because it's just a simple[br]integral from Calculus I.

1:43:47.370,1:43:49.900
So I have to integrate[br]what function?

1:43:49.900,1:43:55.690
A polynomial, 2t cubed[br]plus t with respect

1:43:55.690,1:43:59.720
to t between 0,[br]time 0 and time 1.

1:43:59.720,1:44:02.630


1:44:02.630,1:44:05.060
Good, let's do it.

1:44:05.060,1:44:09.620
Because that's a piece of[br]cake-- 2 times t to the 4 over 4

1:44:09.620,1:44:12.390
plus t squared over 2.

1:44:12.390,1:44:16.782
I take the whole thing between,[br]I apply the fundamental theorem

1:44:16.782,1:44:21.590
of calculus, and I have between[br]t equals 1 up and t equals 0

1:44:21.590,1:44:22.360
down.

1:44:22.360,1:44:26.114
What's the final answer?

1:44:26.114,1:44:27.600
It's a single final answer.

1:44:27.600,1:44:30.430
And again, on the[br]exam, on the final,

1:44:30.430,1:44:32.576
do not expect a[br]headache computation.

1:44:32.576,1:44:34.730
Do expect something[br]simple like that

1:44:34.730,1:44:36.706
where you don't[br]need a calculator.

1:44:36.706,1:44:40.510
You just have either integers[br]only or simple fractions

1:44:40.510,1:44:42.130
to add, and you[br]should get the answer.

1:44:42.130,1:44:44.340
What is the answer, guys?

1:44:44.340,1:44:47.760
1-- 1/2 plus 1/2 equals 1.

1:44:47.760,1:44:52.390
So 1 is the value[br]of the work in what?

1:44:52.390,1:44:54.940
Measured in newtons[br]times meters,

1:44:54.940,1:44:57.815
whatever your units are.

1:44:57.815,1:45:01.630
When you drag the[br]object from this point

1:45:01.630,1:45:06.050
to this point, on which the[br]acting force is the only

1:45:06.050,1:45:08.030
acting force-- it[br]could be the result

1:45:08.030,1:45:09.880
that there are several forces.

1:45:09.880,1:45:13.318
That is that force[br]that you have here.

1:45:13.318,1:45:15.274
Is it useful?

1:45:15.274,1:45:17.230
It's very useful for engineers.

1:45:17.230,1:45:18.697
It's very useful for physicists.

1:45:18.697,1:45:21.300
It's very useful for[br]anybody who works

1:45:21.300,1:45:27.227
in applied mathematics, this[br]notion of work given like that.

1:45:27.227,1:45:29.620
I'm going to go ahead and erase.

1:45:29.620,1:45:34.860
And I'll ask you one[br]thing here that is not

1:45:34.860,1:45:38.812
in the book I think[br]as far as I remember.

1:45:38.812,1:45:45.350
Can you guys prove that this[br]sophisticated formula becomes

1:45:45.350,1:45:48.030
your formula of the[br]one you claimed,

1:45:48.030,1:45:52.060
the first formula you gave me?

1:45:52.060,1:45:53.052
Is it hard?

1:45:53.052,1:45:58.012
Do you think it's[br]hard to prove this?

1:45:58.012,1:46:01.484
OK, what if we have the[br]simplest possible case.

1:46:01.484,1:46:03.468
Let's think of--

1:46:03.468,1:46:06.940
STUDENT: [INAUDIBLE]

1:46:06.940,1:46:09.916


1:46:09.916,1:46:11.652
PROFESSOR: Yeah,[br]I'm thinking maybe I

1:46:11.652,1:46:22.316
should do, well, A to B, right?

1:46:22.316,1:46:33.228
AB, what kind of expression[br]do I have [INAUDIBLE]?

1:46:33.228,1:46:38.188
If I take this to be-- I[br]could have any line, right?

1:46:38.188,1:46:39.854
I could have any line.

1:46:39.854,1:46:43.500
But if I have any line,[br]I can pick my frame

1:46:43.500,1:46:46.930
according to my preference.

1:46:46.930,1:46:49.420
Nobody's going to[br]tell me, well, you

1:46:49.420,1:46:51.430
have to take the[br]frame like that,

1:46:51.430,1:46:55.990
and then your line will be of[br]the form ax plus by equals.

1:46:55.990,1:47:04.760
No, I'll just take the[br]frame to be this one, where

1:47:04.760,1:47:09.846
AB will be x axis, and A[br]will be of coordinates 0, 0

1:47:09.846,1:47:14.540
and B will be of[br]coordinates B and 0.

1:47:14.540,1:47:18.550
And this is just my line.

1:47:18.550,1:47:28.100
So x will be moving between[br]0 and B. And y is 0, right?

1:47:28.100,1:47:31.390
It should be, at least.

1:47:31.390,1:47:43.729
And then F, let's say, should[br]be this function, this.

1:47:43.729,1:47:48.530


1:47:48.530,1:47:50.570
I'll assume the[br]angle is constant,

1:47:50.570,1:47:53.398
just like I had it with theta.

1:47:53.398,1:47:57.009
And then it's acting all[br]the way on your object.

1:47:57.009,1:47:58.965
You have the same[br]angle here always.

1:47:58.965,1:48:02.877


1:48:02.877,1:48:07.820
So F is F1i plus F2j.

1:48:07.820,1:48:12.820


1:48:12.820,1:48:22.561
dR will be dxi plus dyj.

1:48:22.561,1:48:29.766


1:48:29.766,1:48:32.390
But then you say, wait a minute,[br]but didn't you say, Magdalena,

1:48:32.390,1:48:34.400
that you are along this line?

1:48:34.400,1:48:36.740
Didn't you say that y is 0?

1:48:36.740,1:48:37.630
So which y?

1:48:37.630,1:48:38.450
So there is no y.

1:48:38.450,1:48:41.690
So this is 0, right?

1:48:41.690,1:48:50.068
OK, Mr. x, I want to[br]parametrize my trajectory.

1:48:50.068,1:48:53.393
How do I parametrize[br]it the simplest way?

1:48:53.393,1:48:55.858
I'll take x to be t.

1:48:55.858,1:49:00.788
And time will be[br]exactly between 0 and d.

1:49:00.788,1:49:02.267
And y will be 0.

1:49:02.267,1:49:06.211
And thank you god,[br]because that's easy.

1:49:06.211,1:49:10.180
And so all you[br]need to give me is

1:49:10.180,1:49:17.024
W is integral of F[br]dR C in that case.

1:49:17.024,1:49:20.398
So what am I going[br]to have in that case?

1:49:20.398,1:49:21.844
I'll have this formula.

1:49:21.844,1:49:26.440
I'll skip a step, and[br]I'll have that formula.

1:49:26.440,1:49:30.280
And that means I have integral[br]from t0 equals 0 to t1

1:49:30.280,1:49:35.736
equals B.

1:49:35.736,1:49:39.208
F1-- now you have to tell[br]me what F1 will be. x prime

1:49:39.208,1:49:40.696
[? noted ?] is 1.

1:49:40.696,1:49:45.656
The second guy is 0, thank you[br]very much, and [INAUDIBLE].

1:49:45.656,1:49:49.128


1:49:49.128,1:49:52.970
F1 will be what?

1:49:52.970,1:49:56.230
Well, life is nice.

1:49:56.230,1:50:01.480
F1 will be the projection of[br]the vector F on my x-axis.

1:50:01.480,1:50:06.406
So F1 is the length of[br]this blue vector, I'll say.

1:50:06.406,1:50:09.328
So F1 is a scalar.

1:50:09.328,1:50:12.250
Let's say F1 is a[br]scalar component.

1:50:12.250,1:50:16.650
That means it's F[br]length cosine theta.

1:50:16.650,1:50:19.710
Because it's hypotenuse[br]times cosine theta.

1:50:19.710,1:50:20.610
So it's easy.

1:50:20.610,1:50:25.730
So you have length[br]of F, how much it is,

1:50:25.730,1:50:29.890
how big this vector is, times[br]cosine theta, times what

1:50:29.890,1:50:31.060
when you integrate it, guys?

1:50:31.060,1:50:34.676
When you integrate 1 with[br]respect to t, what do you get?

1:50:34.676,1:50:36.940
t between d and 0.

1:50:36.940,1:50:41.170
So you have t between d and 0.

1:50:41.170,1:50:42.924
We got the formula.

1:50:42.924,1:50:48.043
So we got that F length[br]times [INAUDIBLE]

1:50:48.043,1:50:50.550
times cosine theta times d,[br]this is the displacement.

1:50:50.550,1:50:52.478
This is the cosine.

1:50:52.478,1:50:57.418
This is the magnitude of[br]the force that I'm-- look,

1:50:57.418,1:50:59.394
this is the force.

1:50:59.394,1:51:01.864
My force is along my arm.

1:51:01.864,1:51:04.828
I'm just dragging[br]this poor object.

1:51:04.828,1:51:07.298
The force I'm[br]acting with, suppose

1:51:07.298,1:51:11.744
it's always the same parallel[br]to that that I can feel.

1:51:11.744,1:51:16.172
So that's what I have, F[br]cosine theta, and it was easy.

1:51:16.172,1:51:21.100
So as a particular case[br]of this nasty integral,

1:51:21.100,1:51:26.940
I have my old work from[br]school that I had to believe.

1:51:26.940,1:51:30.230
I tell you guys, I did[br]not believe a word.

1:51:30.230,1:51:32.980
Because my teacher[br]in eighth grade

1:51:32.980,1:51:35.660
came up with this[br]out of nothing,

1:51:35.660,1:51:38.810
and we were supposed to[br]be good students preparing

1:51:38.810,1:51:42.970
for a high school like this[br]kind of scientific-- back home,

1:51:42.970,1:51:45.130
there are different[br]kinds of high school.

1:51:45.130,1:51:47.400
There is scientific high[br]school with emphasis

1:51:47.400,1:51:48.233
in math and physics.

1:51:48.233,1:51:49.950
There is one for[br]chemistry/biology.

1:51:49.950,1:51:54.780
There is one for language,[br]linguistics, [INAUDIBLE],

1:51:54.780,1:51:55.746
and so on.

1:51:55.746,1:51:59.610
And I was for the[br]math and physics one.

1:51:59.610,1:52:03.980
And I had to solve this formula[br]without understanding it.

1:52:03.980,1:52:07.980
And it took me many[br]other years to understand

1:52:07.980,1:52:10.930
that it's just a little[br]piece of a big picture,

1:52:10.930,1:52:16.270
and that there's[br]something bigger than what

1:52:16.270,1:52:18.350
we were taught in eighth grade.

1:52:18.350,1:52:28.440


1:52:28.440,1:52:31.736
STUDENT: [INAUDIBLE]

1:52:31.736,1:52:39.672


1:52:39.672,1:52:43.640
PROFESSOR: Yeah,[br]yeah, it's true.

1:52:43.640,1:52:47.112
Now I want to ask[br]you a question.

1:52:47.112,1:52:58.725
So do you think that I would get[br]any kind of conservation laws

1:52:58.725,1:53:03.108
in physics that[br]apply to calculus?

1:53:03.108,1:53:10.360
I mean, how hard is it[br]really to compute the work?

1:53:10.360,1:53:15.012


1:53:15.012,1:53:18.180
And I'm making an[br]announcement now.

1:53:18.180,1:53:22.572
Since I have not[br]given you a break,

1:53:22.572,1:53:26.964
I have to let you[br]go in a few minutes.

1:53:26.964,1:53:29.910


1:53:29.910,1:53:32.390
But I'm making a[br]big announcement

1:53:32.390,1:53:33.845
without proving it.

1:53:33.845,1:53:42.100


1:53:42.100,1:53:46.122
So we will, in about[br]one week at the maximum,

1:53:46.122,1:54:11.529
in maximum one week, study the[br]independence of path of work

1:54:11.529,1:54:24.347
if that work is performed[br]by a conservative force.

1:54:24.347,1:54:31.742


1:54:31.742,1:54:33.730
And you're going to[br]say, wait a minute,

1:54:33.730,1:54:37.618
what the heck is a conservative[br]force and what does she mean?

1:54:37.618,1:54:42.478
Well, I just showed you that[br]the work is a path integral.

1:54:42.478,1:54:44.908
We don't know what that is.

1:54:44.908,1:54:46.360
I'll introduce more.

1:54:46.360,1:54:50.468
I just introduced the definition[br]of a path integral with respect

1:54:50.468,1:54:54.241
to parametrization, general[br]parametrization with respect

1:54:54.241,1:54:54.741
to t.

1:54:54.741,1:54:57.699
So that becomes an integral[br]with respect to dt,

1:54:57.699,1:55:01.020
like the one in[br]Calc I. This is how

1:55:01.020,1:55:02.662
you have to view it at first.

1:55:02.662,1:55:08.185
But guys, if this force[br]is not just any force,

1:55:08.185,1:55:26.841
it's something magic, if F comes[br]from a scalar potential that

1:55:26.841,1:55:33.140
is F represents the gradient[br]of a scalar function F--

1:55:33.140,1:55:44.352
this is called scalar[br]potential-- then

1:55:44.352,1:55:51.946
F is called-- now let's[br]see how much money I

1:55:51.946,1:55:56.906
have for just the last two or[br]three minutes that I have left.

1:55:56.906,1:55:59.386
I don't have money[br]or I have money?

1:55:59.386,1:56:00.900
Come on, big money.

1:56:00.900,1:56:03.605


1:56:03.605,1:56:07.028
No, I have $5.

1:56:07.028,1:56:10.766
I was looking for $1.

1:56:10.766,1:56:14.682
Here, I'll give you $5[br]if you give me $4 back

1:56:14.682,1:56:18.168
if you guess-- I don't know.

1:56:18.168,1:56:21.156
So maybe in your[br]engineering courses-- maybe

1:56:21.156,1:56:23.148
I give you some candy instead.

1:56:23.148,1:56:30.640


1:56:30.640,1:56:38.540
So if there is a scalar function[br]little f of coordinates x,

1:56:38.540,1:56:41.180
y, whatever you[br]have in the problem,

1:56:41.180,1:56:44.430
so that big F will be the nabla.

1:56:44.430,1:56:47.070
F nabla means the gradient.

1:56:47.070,1:56:50.030
We say that F comes[br]from a scalar potential.

1:56:50.030,1:57:00.627
But it has also a name,[br]which is called-- god.

1:57:00.627,1:57:06.110
It starts with a C, ends[br]with an E. In that case,

1:57:06.110,1:57:12.100
if this is going to be equal[br]to nabla F, in that case,

1:57:12.100,1:57:14.842
there is a magic theorem[br]that I'm anticipating.

1:57:14.842,1:57:16.660
I'm not proving.

1:57:16.660,1:57:18.550
I'm doing exercises right now.

1:57:18.550,1:57:22.244
We'll see it in two sections,[br]that the work does not depend

1:57:22.244,1:57:23.940
on the path you are taking.

1:57:23.940,1:57:27.241
So you can go from A to B like[br]that, or you can go like this.

1:57:27.241,1:57:28.116
You can go like this.

1:57:28.116,1:57:28.612
You can go like this.

1:57:28.612,1:57:29.487
You can go like that.

1:57:29.487,1:57:33.030
You can go on a parabola,[br]on a line, on anything.

1:57:33.030,1:57:34.749
The result is always the same.

1:57:34.749,1:57:36.290
And it's like the[br]fundamental theorem

1:57:36.290,1:57:39.020
of Calc III in[br]plane for the work.

1:57:39.020,1:57:42.851
So you have little f endpoint.

1:57:42.851,1:57:45.940
STUDENT: Is that [INAUDIBLE].

1:57:45.940,1:57:47.980
PROFESSOR: Little[br]f of [INAUDIBLE].

1:57:47.980,1:57:51.660
So all that matters is[br]computing this scalar potential

1:57:51.660,1:57:53.520
here and here, making[br]the difference,

1:57:53.520,1:57:55.008
and that will be your work.

1:57:55.008,1:57:56.496
It's a magic thing.

1:57:56.496,1:57:58.976
In mechanical[br]engineering maybe you

1:57:58.976,1:58:03.440
met it, in physics-- in[br]mechanical engineering,

1:58:03.440,1:58:06.664
because that's where you guys[br]drag all sorts of objects

1:58:06.664,1:58:09.392
around.

1:58:09.392,1:58:10.880
STUDENT: Conservative.

1:58:10.880,1:58:12.864
PROFESSOR: Ah, thank god.

1:58:12.864,1:58:16.336
Rachel, you're a[br]math major I think.

1:58:16.336,1:58:18.320
You're an engineering major.

1:58:18.320,1:58:20.800
STUDENT: [INAUDIBLE]

1:58:20.800,1:58:24.272
PROFESSOR: Wow, OK, and[br]who else said conservative?

1:58:24.272,1:58:28.160
And were there other people[br]who said conservative?

1:58:28.160,1:58:29.926
I'm sorry I don't have.

1:58:29.926,1:58:32.480
Well, next time I'll[br]bring a bunch of dollars,

1:58:32.480,1:58:35.740
and I'll start giving[br]prizes as dollar bills

1:58:35.740,1:58:37.957
like I used to give in[br]differential equations.

1:58:37.957,1:58:39.865
Everybody was so[br]happy in my class.

1:58:39.865,1:58:42.250
Because for everything that[br]they got quickly and right,

1:58:42.250,1:58:44.160
they got $1.

1:58:44.160,1:58:47.026
So conservative-- very good.

1:58:47.026,1:58:52.160
Remember that for[br]the next few lessons.

1:58:52.160,1:58:57.475
We will show that when this f is[br]magical, that is conservative,

1:58:57.475,1:59:02.730
you guys don't have to[br]compute the integral at all.

1:59:02.730,1:59:05.020
There's no parametrization,[br]no nothing.

1:59:05.020,1:59:07.192
It really doesn't depend[br]on what path you take.

1:59:07.192,1:59:11.128
All you would need is to figure[br]who this little f will be,

1:59:11.128,1:59:12.604
this scalar potential.

1:59:12.604,1:59:14.080
Our future work can do that.

1:59:14.080,1:59:18.370
And then you compute the values[br]of that scalar potential here

1:59:18.370,1:59:19.890
and here, make the difference.

1:59:19.890,1:59:23.600
And for sure you'll have[br]such a problem in the final.

1:59:23.600,1:59:26.190
So I'm just anticipating[br]it, because I

1:59:26.190,1:59:32.600
want this to be absorbed[br]in time into your system.

1:59:32.600,1:59:34.780
When we will do the[br]final exam review,

1:59:34.780,1:59:36.813
you should be baptized[br]in this kind of problem

1:59:36.813,1:59:41.766
so that everybody will get[br]100% on that for the final.

1:59:41.766,1:59:43.740
OK, now I'll let you go.

1:59:43.740,1:59:45.240
Sorry I didn't give you a break.

1:59:45.240,1:59:47.848
But now I give you more time.

1:59:47.848,1:59:49.844
And enjoy the day.

1:59:49.844,1:59:51.341
I'll see you Thursday.

1:59:51.341,1:59:58.824


1:59:58.824,1:59:59.824
I'm moving to my office.

1:59:59.824,2:00:02.818
If you have questions,[br]you can come to my office.

2:00:02.818,2:00:06.311


2:00:06.311,2:00:09.305
Maybe you were getting close.

2:00:09.305,2:00:13.796
How did-- did you know,[br]or it just came to you?

2:00:13.796,2:00:17.788
[BACKGROUND CHATTER]

2:00:17.788,2:01:03.524


2:01:03.524,2:01:05.440
STUDENT: Do you know[br]what section it would be?

2:01:05.440,2:01:09.470
Because I don't even think[br]he's listed or anything.

2:01:09.470,2:01:12.374
PROFESSOR: Send me an email[br]if you don't figure it out.

2:01:12.374,2:01:15.350
But for sure [INAUDIBLE].

2:01:15.350,2:01:17.830
STUDENT: OK, because I was[br]going to do the honors,

2:01:17.830,2:01:19.318
but it was with [INAUDIBLE].

2:01:19.318,2:01:20.310
I don't know if he's[br]good, or she's good.

2:01:20.310,2:01:21.226
PROFESSOR: She's good.

2:01:21.226,2:01:25.370
But he's fantastic in the[br]sense that he will help you

2:01:25.370,2:01:27.160
whenever you stumble.

2:01:27.160,2:01:29.860
He's an extremely good teacher.

2:01:29.860,2:01:31.660
He explains really well.

2:01:31.660,2:01:32.860
He has a talent.

2:01:32.860,2:01:36.235


2:01:36.235,2:01:37.360
STUDENT: I'll look for him.

2:01:37.360,2:01:38.260
Thank you.

2:01:38.260,2:01:40.360
PROFESSOR: And if[br]you don't get him,

2:01:40.360,2:01:43.360
she is good as well-- not[br]exceptional like he is.

2:01:43.360,2:01:44.860
He's an exceptional teacher.

2:01:44.860,2:01:48.760


2:01:48.760,2:01:50.560
STUDENT: I'll go to the office.

2:01:50.560,2:01:53.310
PROFESSOR: Yes,[br]yes, [INAUDIBLE].

2:01:53.310,2:01:54.911