0:00:00.000,0:00:05.247


0:00:05.247,0:00:07.850
PROFESSOR: I know you have[br]encountered difficulties

0:00:07.850,0:00:12.672
on the last few problems,[br]maybe four, maybe five, maybe

0:00:12.672,0:00:14.600
the last 10, I don't know.

0:00:14.600,0:00:19.100
But today, I want to--[br]we have plenty of time.

0:00:19.100,0:00:22.580
We still have time[br]for chapter 13,

0:00:22.580,0:00:25.050
and plenty of time[br]for the final review.

0:00:25.050,0:00:29.078
I can afford to spend two or[br]three hours just reviewing

0:00:29.078,0:00:31.870
chapter 12, if I wanted to.

0:00:31.870,0:00:37.814
All right, so I have this[br]question from one of you saying

0:00:37.814,0:00:43.736
what part of the problem is[br]that in terms of a two point

0:00:43.736,0:00:44.236
integral.

0:00:44.236,0:00:50.658
We have a solid[br]bounded by z equals 3x,

0:00:50.658,0:00:56.586
and z equals x squared,[br]and is a plane.

0:00:56.586,0:00:59.730
And can anybody tell[br]me what this is?

0:00:59.730,0:01:04.028
Just out of curiosity,[br]you don't have to know.

0:01:04.028,0:01:05.069
STUDENT: It's a parabola.

0:01:05.069,0:01:06.485
PROFESSOR: It would[br]be a parabola,

0:01:06.485,0:01:08.214
if we were in[br][INTERPOSING VOICES]

0:01:08.214,0:01:12.166
if we were in 2D.

0:01:12.166,0:01:17.450
So the parabola is missing[br]the y, and y could be anybody.

0:01:17.450,0:01:21.130
So it's a parabola[br]that's shifted along y.

0:01:21.130,0:01:24.220
It's going to give you[br]a cylindrical surface.

0:01:24.220,0:01:29.422
It's like something used[br]for drainage, I don't know.

0:01:29.422,0:01:31.286
Water, like a valve.

0:01:31.286,0:01:34.090


0:01:34.090,0:01:38.213
So this is what it is,[br]a cylindrical surface.

0:01:38.213,0:01:42.440


0:01:42.440,0:01:52.690
And you know that z must be[br]between x squared and 3x.

0:01:52.690,0:01:58.620
How do you know which one is[br]bigger, which one is smaller?

0:01:58.620,0:02:01.230
You should think about it.

0:02:01.230,0:02:04.040


0:02:04.040,0:02:08.859
When you draw, you[br]draw like that.

0:02:08.859,0:02:18.200


0:02:18.200,0:02:19.570
Do these guys intersect?

0:02:19.570,0:02:22.750


0:02:22.750,0:02:24.590
We are in the xz plane.

0:02:24.590,0:02:26.160
Do these guys intersect?

0:02:26.160,0:02:32.120
x squared equals[br]3x intersect where?

0:02:32.120,0:02:37.340
They intersect at 0, and at 3.

0:02:37.340,0:02:40.190
x1 is 0, and x2 is 3.

0:02:40.190,0:02:45.540
So when I want to draw this,[br]I would say that indeed, it's

0:02:45.540,0:02:48.360
a bounded domain.

0:02:48.360,0:02:51.500
If it where unbounded, it[br]wouldn't ask for the volume,

0:02:51.500,0:02:54.170
because the volume[br]would be nothing.

0:02:54.170,0:02:59.330
So this thing must[br]be a bounded domain.

0:02:59.330,0:03:03.480
x cannot go on, this[br]is the infinite part.

0:03:03.480,0:03:09.810
So we are thinking of just this[br]striped piece of a domain x

0:03:09.810,0:03:16.102
here, where this piece is[br]between z equals x squared,

0:03:16.102,0:03:19.006
then z equals 3x.

0:03:19.006,0:03:23.070
This is 0 origin, and this is 3.

0:03:23.070,0:03:26.670


0:03:26.670,0:03:30.740
So at 3, they meet again.

0:03:30.740,0:03:31.970
Are you guys with me?

0:03:31.970,0:03:34.710
At 3 o'clock they meet again.

0:03:34.710,0:03:37.190
I'm just kidding, x[br]doesn't have to be time.

0:03:37.190,0:03:39.220
It's a special coordinate.

0:03:39.220,0:03:42.680


0:03:42.680,0:03:48.886
And y is looking at you,[br]and is going towards you.

0:03:48.886,0:03:52.880
Well, if it's toward like[br]that, it's probably not

0:03:52.880,0:03:53.780
positively oriented.

0:03:53.780,0:03:57.830
So y should come from you,[br]and go into the board,

0:03:57.830,0:04:00.780
and then keep going in that[br]direction for the frame

0:04:00.780,0:04:03.380
to be positive oriented.

0:04:03.380,0:04:07.290
Positively oriented means x like[br]that, y like this, z like that.

0:04:07.290,0:04:11.200
So k must be the[br]crossproduct between i and j.

0:04:11.200,0:04:12.960
i cross j must be k.

0:04:12.960,0:04:16.329
If I use the right hand[br]rule, and I go y like that,

0:04:16.329,0:04:18.290
that means I changed[br]the orientation.

0:04:18.290,0:04:23.390
So the y, you have to imagine[br]the y coming from you, going

0:04:23.390,0:04:26.700
perpendicular to[br]the board, then keep

0:04:26.700,0:04:30.170
going inside the[br]board infinitely much.

0:04:30.170,0:04:34.840
Now, if we were to[br]play with Play-Doh,

0:04:34.840,0:04:42.030
and we were on the other side of[br]that, like Alice in the mirror,

0:04:42.030,0:04:47.310
we would have the y in the[br]mirror world, going between 0

0:04:47.310,0:04:50.532
and 2, inside the board.

0:04:50.532,0:04:53.410
If I were to draw[br]this piece of cake,

0:04:53.410,0:04:55.670
I start dreaming[br]again, I apologize.

0:04:55.670,0:05:05.530
But I'm dreaming of very[br]nice bounded pieces of solids

0:05:05.530,0:05:07.130
that would be made of cheese.

0:05:07.130,0:05:12.882
This is a perfect example where[br]you have something like curve

0:05:12.882,0:05:16.150
or linear shape, and you[br]kind of slice the cheese,

0:05:16.150,0:05:17.707
and that's a piece[br]of the Parmesan.

0:05:17.707,0:05:20.690


0:05:20.690,0:05:21.650
OK.

0:05:21.650,0:05:27.490
So the y here is[br]going from 0 to 2.

0:05:27.490,0:05:29.370
It's sort of the altitude.

0:05:29.370,0:05:32.340
And this is the piece[br]of cake that you

0:05:32.340,0:05:37.210
were looking-- or the cheese,[br]or whatever you're looking at.

0:05:37.210,0:05:39.160
So what do you put here?

0:05:39.160,0:05:44.656
You put z between[br]x squared and 3x.

0:05:44.656,0:05:56.560
You put y between--[br]y is between 0 and 2.

0:05:56.560,0:06:00.840
And you Mr. x as the last[br]of them, he can go from-- he

0:06:00.840,0:06:04.520
goes from 0 to 3.

0:06:04.520,0:06:08.920
So x has the freedom[br]to go from 0 to 3.

0:06:08.920,0:06:13.506
y has the freedom to go[br]from flat line to flat line,

0:06:13.506,0:06:17.645
from between to flat planes--[br]from two horizontal planes.

0:06:17.645,0:06:24.730
But Mr. z is married to[br]x, he cannot escape this

0:06:24.730,0:06:25.520
relationship.

0:06:25.520,0:06:30.690
So we can only take this z[br]with respect between these two

0:06:30.690,0:06:34.150
values that depend on x.

0:06:34.150,0:06:35.050
That's all.

0:06:35.050,0:06:38.760
Now why would we have 1?

0:06:38.760,0:06:41.360
Because by definition,[br]if you remember

0:06:41.360,0:06:43.790
the volume was the[br]triple integral

0:06:43.790,0:06:48.690
of 1 dv over any[br]solid value domain.

0:06:48.690,0:06:50.910
Right?

0:06:50.910,0:06:53.350
So whenever you see[br]a problem like that,

0:06:53.350,0:06:55.273
you know how to start it.

0:06:55.273,0:06:58.290
One triple integrate,[br]and that's going to work.

0:06:58.290,0:07:02.160
Something else that[br]gave you a big headache

0:07:02.160,0:07:04.560
was the ice cream cone.

0:07:04.560,0:07:09.270
The ice cream cone problem[br]gave a big headache

0:07:09.270,0:07:13.230
to most of my students[br]over the past 14 years

0:07:13.230,0:07:15.770
that I've been teaching Cal 3.

0:07:15.770,0:07:17.246
It's a beautiful problem.

0:07:17.246,0:07:18.880
It's one of my[br]favorites problems,

0:07:18.880,0:07:21.550
because it makes me[br]think of food again.

0:07:21.550,0:07:27.860
And not just any food, but[br]some nice ice cream cone

0:07:27.860,0:07:34.820
that's original ice cream, not[br]the one you find in a box like

0:07:34.820,0:07:36.941
Blue Bell or Ben and Jerry's.

0:07:36.941,0:07:39.700


0:07:39.700,0:07:43.440
All right, so how is the[br]ice cream cone problem

0:07:43.440,0:07:49.640
that-- he showed it to me, but[br]I forgot the problem number.

0:07:49.640,0:07:50.510
It was--

0:07:50.510,0:07:51.910
STUDENT: It's number 20.

0:07:51.910,0:07:52.660
PROFESSOR: Number?

0:07:52.660,0:07:53.390
STUDENT: 20.

0:07:53.390,0:07:55.160
PROFESSOR: Number 20, thanks.

0:07:55.160,0:07:57.940
And I want the data [INAUDIBLE].

0:07:57.940,0:08:01.120
I want to test my memory,[br]see how many neurons

0:08:01.120,0:08:03.130
died since last time.

0:08:03.130,0:08:04.930
Don't tell me.

0:08:04.930,0:08:11.030
So I think the sphere was[br]a radius 2, and the cone

0:08:11.030,0:08:13.990
that we picked for you,[br]we picked it on purpose.

0:08:13.990,0:08:18.240
So that the results that[br]come up for the ice cream

0:08:18.240,0:08:21.990
cone boundaries will[br]be nice and workable.

0:08:21.990,0:08:25.850
So we can propose some data[br]where the ice cream cone will

0:08:25.850,0:08:28.210
give you really nasty radii.

0:08:28.210,0:08:29.350
Can I draw it?

0:08:29.350,0:08:30.660
Hopefully.

0:08:30.660,0:08:35.140
This ice cream cone is[br]based off the waffle cone.

0:08:35.140,0:08:40.159
I don't like the waffle[br]cone, because I'm dreaming.

0:08:40.159,0:08:45.956
But the problem is, the waffle[br]cone is not a finite surface.

0:08:45.956,0:08:49.350
It's infinite,[br]it's a double cone.

0:08:49.350,0:08:51.410
It's the dream of every binger.

0:08:51.410,0:08:55.930
So it goes to infinity,[br]and to negative infinity,

0:08:55.930,0:08:57.910
and that's not my problem.

0:08:57.910,0:09:02.820
My problem is to intersect[br]this cone with the sphere,

0:09:02.820,0:09:04.800
and make it finite.

0:09:04.800,0:09:07.680
So to make it the[br]true waffle cone,

0:09:07.680,0:09:14.245
I would have to draw a[br]sphere of what radius?

0:09:14.245,0:09:17.085
Root 2.

0:09:17.085,0:09:20.016
I'll try to draw a[br]sphere of root 2,

0:09:20.016,0:09:21.390
but I cannot[br]predict the results.

0:09:21.390,0:09:26.720
Now I'm going to only[br]look at this v1 cone.

0:09:26.720,0:09:31.294
I don't know what the problem[br]wanted, but I'm looking at,

0:09:31.294,0:09:35.700
do they say in what domain?

0:09:35.700,0:09:37.825
Above the plane?

0:09:37.825,0:09:40.640
STUDENT: It just says[br]lies above the cone.

0:09:40.640,0:09:43.390
PROFESSOR: That[br]lies above the cone.

0:09:43.390,0:09:48.300
But look, if I turn my head like[br]this, depending on my reference

0:09:48.300,0:09:51.180
frame, I have[br]cervical spondylosis,

0:09:51.180,0:09:53.640
this is also lying[br]above the cone.

0:09:53.640,0:09:56.400
So the problem is[br]a little bit silly,

0:09:56.400,0:09:58.450
that whoever wrote[br]it should have

0:09:58.450,0:10:07.240
said the sphere lies above a[br]cone, for z greater than 0.

0:10:07.240,0:10:10.355
In the basement,[br]it can continue--

0:10:10.355,0:10:13.920
that's for z-- for z[br]less than 0 can continue

0:10:13.920,0:10:18.682
upside down, and then between[br]the sphere and the cone,

0:10:18.682,0:10:22.360
you'll have another ice[br]cream cone outside that.

0:10:22.360,0:10:24.510
But practically what[br]they mean is just do v1

0:10:24.510,0:10:26.990
and forget about this one.

0:10:26.990,0:10:28.640
It's not very nicely phrased.

0:10:28.640,0:10:29.540
Above, beyond.

0:10:29.540,0:10:34.130
Are we above Australia?

0:10:34.130,0:10:35.990
That's stupid, right?

0:10:35.990,0:10:39.680
Because they may say, oh no,[br]depends on where you are.

0:10:39.680,0:10:41.540
We are above you guys.

0:10:41.540,0:10:43.920
You think you're better[br]than us because you

0:10:43.920,0:10:46.040
are closer to the North Pole.

0:10:46.040,0:10:50.470
But who made that rule that[br]t the North Pole is superior?

0:10:50.470,0:10:55.280
If you look at the universe,[br]who is above, who is beyond?

0:10:55.280,0:10:56.960
There is no direction.

0:10:56.960,0:10:58.845
So they would be very offended.

0:10:58.845,0:11:00.790
I have a friend who[br]works in Sydney.

0:11:00.790,0:11:04.250
She is a brilliant geometer.

0:11:04.250,0:11:07.330
And I bet if I asked[br]her, she would say

0:11:07.330,0:11:10.048
who says you guys are above?

0:11:10.048,0:11:14.360
Because it depends on where your[br]head is and how you look at it.

0:11:14.360,0:11:16.680
The planet is the[br]same, so would you

0:11:16.680,0:11:19.070
say that the people[br]who are walking closer

0:11:19.070,0:11:22.380
to the North Pole have[br]their body upside down?

0:11:22.380,0:11:26.324


0:11:26.324,0:11:31.250
So it really matters how[br]you look, what's above.

0:11:31.250,0:11:36.410
So assume that above means--[br]the word above means positive.

0:11:36.410,0:11:40.900
And this is the ice cream cone.

0:11:40.900,0:11:43.750
Now how do I find out[br]where to cut the waffle?

0:11:43.750,0:11:46.030
Because this is the question.

0:11:46.030,0:11:50.710
I need to know where the[br]boundary of the waffle is.

0:11:50.710,0:11:53.680
I'm not allowed to eat[br]anything above that.

0:11:53.680,0:11:56.860
So that's going[br]to be the waffle.

0:11:56.860,0:12:01.730
And for that, any ideas-- how do[br]I get to see what the circle--

0:12:01.730,0:12:03.934
where the circle will be?

0:12:03.934,0:12:05.482
STUDENT: Do they[br]meet each other?

0:12:05.482,0:12:06.565
PROFESSOR: They intersect.

0:12:06.565,0:12:07.356
Excellent, Matthew.

0:12:07.356,0:12:08.000
Thank you.

0:12:08.000,0:12:14.010
So intersect the two surfaces[br]by setting up a system to solve.

0:12:14.010,0:12:17.350
Solve the system.

0:12:17.350,0:12:26.308
And the intersection of the two[br]is Mr. z, which is Mr. circle.

0:12:26.308,0:12:26.808
All right.

0:12:26.808,0:12:29.290
So what do I do?

0:12:29.290,0:12:32.330
I'm going to say I have to[br]be smart about that one.

0:12:32.330,0:12:36.780
So if z squared from[br]here plugging in,

0:12:36.780,0:12:41.300
substitute, is the same as[br]x squared plus y squared.

0:12:41.300,0:12:49.980
So that means this is if[br]and only if 2x squared

0:12:49.980,0:12:51.630
plus y squared.

0:12:51.630,0:12:54.450
2 times x squared[br]plus y squared.

0:12:54.450,0:12:56.330
We have an x squared[br]plus y squared,

0:12:56.330,0:12:59.426
and another x squared[br]plus y squared equals 2.

0:12:59.426,0:13:02.460
You see how nicely the[br]problem was picked?

0:13:02.460,0:13:05.800
It was picked it's going[br]to give you some nice data.

0:13:05.800,0:13:10.060
z equals-- z[br]squared equals that.

0:13:10.060,0:13:12.240
Keep going with if and only if.

0:13:12.240,0:13:14.490
If you're a math major,[br]you will understand

0:13:14.490,0:13:18.005
why x squared plus[br]y squared equals 1.

0:13:18.005,0:13:21.100


0:13:21.100,0:13:23.095
And z squared must be 1.

0:13:23.095,0:13:26.090


0:13:26.090,0:13:28.940
Well so, we really[br]get two solutions,

0:13:28.940,0:13:32.420
the one close to the South[br]Pole, and the one close

0:13:32.420,0:13:35.700
to the North Pole,[br]because I'm going

0:13:35.700,0:13:39.248
to have z equals plus minus 1.

0:13:39.248,0:13:40.709
Where's the North Pole?

0:13:40.709,0:13:44.118
The North Pole would be[br]0, 0, square root of 2.

0:13:44.118,0:13:48.370
The South Pole would be 0,[br]0, minus square root of 2.

0:13:48.370,0:13:52.990
And my plain here is[br]cut at which altitude?

0:13:52.990,0:13:54.550
z equals 1.

0:13:54.550,0:13:59.330
And I have another plane here,[br]and an imaginary intersection

0:13:59.330,0:14:02.610
that I'm not going to talk[br]about today. z equals minus y.

0:14:02.610,0:14:08.062
I don't care about the mirror[br]image of the-- with respect

0:14:08.062,0:14:11.305
to the equator of the cone.

0:14:11.305,0:14:12.796
All right, good.

0:14:12.796,0:14:15.860
So we know who this[br]guy is, we know

0:14:15.860,0:14:22.421
that he is-- I have a[br]red marker, and a green,

0:14:22.421,0:14:25.360
and a blue.

0:14:25.360,0:14:27.924
I cannot live without colors.

0:14:27.924,0:14:29.800
Life is ugly enough.

0:14:29.800,0:14:31.930
Let's try to make it colorful.

0:14:31.930,0:14:35.825
x squared plus y[br]squared, equals 1.

0:14:35.825,0:14:39.860


0:14:39.860,0:14:44.910
We are happy, because[br]that's a nice, simple circle

0:14:44.910,0:14:47.380
of radius 1.

0:14:47.380,0:14:49.560
Now you have to think[br]in which coordinate

0:14:49.560,0:14:51.650
you can write this problem.

0:14:51.650,0:14:54.570
And I'm going to[br]beg of you to help

0:14:54.570,0:14:59.037
me review the material for[br]the final and for the midterm.

0:14:59.037,0:15:01.620
Then again, on the midterm, I'm[br]not going to put this problem.

0:15:01.620,0:15:04.820


0:15:04.820,0:15:11.078
So for the final, do[br]expect something like that.

0:15:11.078,0:15:14.410
We may have, instead of[br]the cone in the book,

0:15:14.410,0:15:17.773
you'll have a paraboloid[br]and a sphere intersecting.

0:15:17.773,0:15:21.810
It's sort of the same thing, but[br]instead of the ice cream cone,

0:15:21.810,0:15:25.160
you have the valley[br]full of cream.

0:15:25.160,0:15:27.970


0:15:27.970,0:15:30.140
I cannot stop, right?

0:15:30.140,0:15:34.045
So if you do the volume[br]like I told you before,

0:15:34.045,0:15:38.410
you simply have[br][INTERPOSING VOICES]

0:15:38.410,0:15:40.180
No, I'll do the volume first.

0:15:40.180,0:15:41.710
I know you have[br]the surface there,

0:15:41.710,0:15:46.300
but what-- I'm doing the[br]volume because I have plans,

0:15:46.300,0:15:49.130
and I didn't want[br]to say what plans.

0:15:49.130,0:15:50.765
You forgot what I said, right?

0:15:50.765,0:15:53.768


0:15:53.768,0:15:58.250
So suppose somebody's[br]asking you for the volume.

0:15:58.250,0:16:01.360
The volume-- how much[br]ice cream you have inside

0:16:01.360,0:16:03.740
depends on that very much.

0:16:03.740,0:16:07.090
And I'd like you to[br]remember that the v is

0:16:07.090,0:16:11.673
piratically against[br]the ybc, right?

0:16:11.673,0:16:17.000
Right until the Cartesian[br]coordinates would be a killer,

0:16:17.000,0:16:19.550
we try to write it[br]in either cylindrical

0:16:19.550,0:16:23.250
or spherical to make[br]our life easier.

0:16:23.250,0:16:27.100
If I want to make my life[br]easier, first in cylindrical

0:16:27.100,0:16:31.400
coordinates, and then in[br]spherical coordinates.

0:16:31.400,0:16:34.220


0:16:34.220,0:16:37.805
Could you help me[br]find the limit points?

0:16:37.805,0:16:41.301


0:16:41.301,0:16:44.840
And then we'll do the surface.

0:16:44.840,0:16:47.546
Just remind me, OK?

0:16:47.546,0:16:55.365
[INTERPOSING VOICES][br]The volume of the-- OK.

0:16:55.365,0:16:58.290
The volume occupied[br]by the ice cream.

0:16:58.290,0:17:01.532
The ice cream is between[br]this plastic cap,

0:17:01.532,0:17:03.280
that is the sphere.

0:17:03.280,0:17:05.165
We cover it for[br]hygiene purposes.

0:17:05.165,0:17:10.858


0:17:10.858,0:17:18.521
So for cylindrical coordinates,[br]rho and theta are really nice.

0:17:18.521,0:17:20.970
We don't worry about them yet.

0:17:20.970,0:17:24.140
But z should be between?

0:17:24.140,0:17:26.875
OK.

0:17:26.875,0:17:27.800
Really, 0?

0:17:27.800,0:17:30.590


0:17:30.590,0:17:34.940
So because it's[br]between the ice cream

0:17:34.940,0:17:37.700
the cone-- do you[br]think the waffle

0:17:37.700,0:17:40.830
cone-- what's the equation[br]of the waffle cone?

0:17:40.830,0:17:43.945
And how do you get to the[br]equation of the waffle cone?

0:17:43.945,0:17:47.930
The waffle cone meant--[br]oh guys, help me.

0:17:47.930,0:17:50.777
z equals-- now you[br]take the square root

0:17:50.777,0:17:55.500
of x squared plus y squared,[br]because the other one would

0:17:55.500,0:17:56.010
be here.

0:17:56.010,0:17:58.051
The imaginary one, z equals[br]minus the square root

0:17:58.051,0:17:59.850
of x squared plus y squared.

0:17:59.850,0:18:04.680
Forget about the[br]world in the basement.

0:18:04.680,0:18:06.955
So you take z to[br]the square root of,

0:18:06.955,0:18:09.520
with a plus, plus y squared.

0:18:09.520,0:18:14.560
In cylindrical coordinates,[br]what does this mean?

0:18:14.560,0:18:16.010
In cylindrical coordinates?

0:18:16.010,0:18:18.191
STUDENT: It equals r.

0:18:18.191,0:18:19.690
PROFESSOR: This[br]equals r, very good.

0:18:19.690,0:18:21.175
He's thinking faster.

0:18:21.175,0:18:23.390
Do you guys[br]understand why he said

0:18:23.390,0:18:25.245
x squared plus y[br]squared, if we work

0:18:25.245,0:18:27.120
with polar coordinates--[br]which is cylindrical

0:18:27.120,0:18:29.250
coordinates it's the same[br]thing-- polar coordinates

0:18:29.250,0:18:30.416
and cylindrical coordinates.

0:18:30.416,0:18:33.570
x squared plus y squared[br]would be little r squared.

0:18:33.570,0:18:36.030
Under the square[br]root would be r.

0:18:36.030,0:18:39.100
So you'll see, we're between r.

0:18:39.100,0:18:42.310
And now, another hard part.

0:18:42.310,0:18:53.350
What is the z equals plus square[br]root of-- What was it, guys?

0:18:53.350,0:18:55.972
STUDENT: So are we taking[br]the volume of all of the ice

0:18:55.972,0:18:57.820
cream inside the cone?

0:18:57.820,0:19:02.530
PROFESSOR: So between the[br]cone, ice cream lies here.

0:19:02.530,0:19:05.020
Ice cream chips,[br]chocolate chips,

0:19:05.020,0:19:06.514
between the cone and the sphere.

0:19:06.514,0:19:09.150


0:19:09.150,0:19:13.335
The sphere is the bottom[br]function, the lower function.

0:19:13.335,0:19:14.436
Is that good?

0:19:14.436,0:19:15.560
STUDENT: 2 minus r squared.

0:19:15.560,0:19:20.593
PROFESSOR: So I have the square[br]root of 2 minus r squared.

0:19:20.593,0:19:23.630
I don't like the[br]square root, I hate it.

0:19:23.630,0:19:25.380
Can I do it with it?

0:19:25.380,0:19:26.560
Yes, I can.

0:19:26.560,0:19:29.130
Maybe I can apply the[br]use of solution later.

0:19:29.130,0:19:31.210
Don't worry about[br]me, I'll make it.

0:19:31.210,0:19:34.990
I will live better if I didn't[br]have any ugly things like that.

0:19:34.990,0:19:36.220
So let's see what we have.

0:19:36.220,0:19:37.510
Theta.

0:19:37.510,0:19:41.230
Your cone is not just sliced[br]cone, it's all the cone.

0:19:41.230,0:19:43.774
So you have 0 to 2 pi.

0:19:43.774,0:19:46.140
One revolution,[br]complete revolution.

0:19:46.140,0:19:49.540
How about rho?

0:19:49.540,0:19:50.550
Rho is limited.

0:19:50.550,0:19:53.325
Rho is what lies in the[br]plane in terms of radius.

0:19:53.325,0:19:56.780


0:19:56.780,0:19:58.085
0, 2?

0:19:58.085,0:20:01.430
How much is from here to here?

0:20:01.430,0:20:04.790
How much is from here to here?

0:20:04.790,0:20:07.730
Didn't we do it x squared[br]plus y squared equals 1?

0:20:07.730,0:20:10.290
So what is this radius[br]from here to here?

0:20:10.290,0:20:10.790
1.

0:20:10.790,0:20:12.921
And what is this radius[br]from here to here?

0:20:12.921,0:20:13.420
1.

0:20:13.420,0:20:15.260
So the projection[br]of this ice cream

0:20:15.260,0:20:19.225
cone-- if you had the[br]eye of god is here,

0:20:19.225,0:20:22.350
sun, you have a[br]shadow on the ground,

0:20:22.350,0:20:25.850
coming from your ice cream[br]cone, and this is the shadow.

0:20:25.850,0:20:29.480
Your shadow is simply[br]a disk of radius 1.

0:20:29.480,0:20:31.430
Good, that's the[br]projection you have.

0:20:31.430,0:20:34.065
So Mr. rho is between 0 and 1.

0:20:34.065,0:20:38.785
For polar coordinates,[br]it's not so ugly actually.

0:20:38.785,0:20:40.490
0 to 2 pi.

0:20:40.490,0:20:42.820
0 to 1.

0:20:42.820,0:20:46.760
r to square root of[br]2 minus r squared.

0:20:46.760,0:20:53.140
Instead of r, it's[br]OK to react with rho.

0:20:53.140,0:20:55.060
And here's the[br]big j coordinates.

0:20:55.060,0:20:57.670
I'm going to erase[br]j, don't write j.

0:20:57.670,0:20:59.790
Jacobian in general is Jacobian.

0:20:59.790,0:21:02.560
The one that does the[br]transformation between

0:21:02.560,0:21:04.470
coordinates to[br]other coordinates.

0:21:04.470,0:21:06.250
Let me finish on that.

0:21:06.250,0:21:08.725
How about this one?

0:21:08.725,0:21:11.635
This is simply r, very good.

0:21:11.635,0:21:12.590
Your old friend.

0:21:12.590,0:21:17.110
So you [INAUDIBLE][br]the dz d, theta,

0:21:17.110,0:21:24.640
dr, d is your [INAUDIBLE][br]It's not easy there.

0:21:24.640,0:21:28.100
If I were to continue--[br]maybe on the final-- OK

0:21:28.100,0:21:30.330
I'm talking too much, as usual.

0:21:30.330,0:21:34.390
Maybe on whatever test[br]you're going to have,

0:21:34.390,0:21:39.850
this kind of stuff, with[br]a formulation saying you

0:21:39.850,0:21:42.600
do not have to compute it.

0:21:42.600,0:21:46.290
But if you wanted to compute it,[br]would it be hard from 0 to 1,

0:21:46.290,0:21:52.160
from 0 to 2 pi, and say[br]forgot the stinky pi?

0:21:52.160,0:21:56.000
Take the 2 pi out to[br]make your life easier.

0:21:56.000,0:21:59.370
Because the theta isn't[br]depending from-- there

0:21:59.370,0:22:01.480
is no theta inside.

0:22:01.480,0:22:03.770
So take the 2 pi[br]out, and then you

0:22:03.770,0:22:09.550
have an integral from 0 to[br]1, and integral from-- now.

0:22:09.550,0:22:11.452
R got out for a walk.

0:22:11.452,0:22:13.560
This is r going out for a walk.

0:22:13.560,0:22:17.060
Integral of 1 with[br]respect to dz.

0:22:17.060,0:22:21.620
So z is taken between r and[br]root of 2 minus r squared.

0:22:21.620,0:22:22.510
Right?

0:22:22.510,0:22:27.483
So I would have to write[br]here the 1 on top minus the 1

0:22:27.483,0:22:30.417
on the bottom, which is a[br]little bit of a headache for me.

0:22:30.417,0:22:35.190
I'm looking at it,[br]I'm getting angry.

0:22:35.190,0:22:39.010
Now, times the r that[br]went out for a walk.

0:22:39.010,0:22:42.470


0:22:42.470,0:22:45.300
So practically, the[br]0 is solved, and I

0:22:45.300,0:22:49.915
have the dr. And from 0 to 1.

0:22:49.915,0:22:53.310


0:22:53.310,0:22:54.866
So I took care or who?

0:22:54.866,0:22:57.770
I took care of the[br]integral with respect to z.

0:22:57.770,0:23:00.030
This is r here.

0:23:00.030,0:23:03.040
This was done first.

0:23:03.040,0:23:06.320
And you gave me that[br]between those two.

0:23:06.320,0:23:07.430
So I got that.

0:23:07.430,0:23:12.430
Times the r, between 0 and 1,[br]with respect to r, and then

0:23:12.430,0:23:14.270
the 2 pi gets outside.

0:23:14.270,0:23:16.880
Now, if I split this[br]into two integrals,

0:23:16.880,0:23:18.380
it's going to be easy, right?

0:23:18.380,0:23:20.810
Because I go--[br]the first integral

0:23:20.810,0:23:23.690
will be r times square[br]root of 2 minus r squared.

0:23:23.690,0:23:26.320
How can you do such an integral?

0:23:26.320,0:23:27.400
We do substitution.

0:23:27.400,0:23:32.800
For example, your u would[br]b 2 minus r squared.

0:23:32.800,0:23:34.460
Can you keep going?

0:23:34.460,0:23:37.440
The second integral[br]is a piece of cake.

0:23:37.440,0:23:38.730
A piece of ice cream.

0:23:38.730,0:23:40.580
The integral of r squared.

0:23:40.580,0:23:44.260
r cubed over 3,[br]between 0 and 1, 1/3.

0:23:44.260,0:23:49.920
So we can still solve the ice[br]cream cone volume like that.

0:23:49.920,0:23:50.933
Do I like it?

0:23:50.933,0:23:51.432
No.

0:23:51.432,0:23:55.830
Can you suspect why[br]I don't like it?

0:23:55.830,0:23:56.933
Oh, by the way.

0:23:56.933,0:23:59.960
Suppose you got to[br]this on the final,

0:23:59.960,0:24:05.290
how much do you get[br]for-- you mess up

0:24:05.290,0:24:07.508
the algebra, how[br]much do you get?

0:24:07.508,0:24:17.940


0:24:17.940,0:24:21.250
You say, I can do[br]that in my sleep,

0:24:21.250,0:24:27.950
u equals 2 minus r squared, u[br]equals minus 2r, I can go on.

0:24:27.950,0:24:32.142
Even if you mess up the[br]algebra, you get most of it.

0:24:32.142,0:24:33.531
Why don't I like it?

0:24:33.531,0:24:36.190
Because it involves[br]work, and I'm lazy.

0:24:36.190,0:24:42.800
So can I find a[br]better way to do it?

0:24:42.800,0:24:45.780
Can I get use[br]spherical coordinates?

0:24:45.780,0:24:47.950
And how do I use[br]spherical coordinates?

0:24:47.950,0:24:50.750
So let me see how I do that.

0:24:50.750,0:24:53.720
In spherical coordinates,[br]it should be easier.

0:24:53.720,0:25:00.155


0:25:00.155,0:25:03.930
Remember that for[br]mathematicians, they

0:25:03.930,0:25:07.400
include this course Cal[br]3 multivariable calculus.

0:25:07.400,0:25:10.000
We are not studying geography.

0:25:10.000,0:25:14.690
So for us, a lot can happen[br]between minus 90 and plus 90

0:25:14.690,0:25:19.860
degrees, but it measures[br]from the North Pole,

0:25:19.860,0:25:22.430
because we believe[br]in Santa Clause.

0:25:22.430,0:25:23.880
Always remember that.

0:25:23.880,0:25:29.310
So we go all the way from[br]0 degrees to 180 degrees.

0:25:29.310,0:25:34.290
So your-- in principle--[br]your latitude

0:25:34.290,0:25:39.595
will go from 0 to[br]all the way to pi.

0:25:39.595,0:25:43.204
But it doesn't, because[br]it gets stuck here.

0:25:43.204,0:25:46.390
What is the latitude[br]of the ice cream cone?

0:25:46.390,0:25:50.860
So what is the pi angle[br]for this ice cream cone?

0:25:50.860,0:25:53.800


0:25:53.800,0:25:56.070
It's a 45 degree angle.

0:25:56.070,0:25:58.620
That is true.

0:25:58.620,0:26:05.150
For anything like that-- I'm[br]looking again at this cone.

0:26:05.150,0:26:07.051
z squared equals x[br]squared plus y squared.

0:26:07.051,0:26:10.890


0:26:10.890,0:26:13.510
I just want to talk a[br]little bit about that.

0:26:13.510,0:26:23.270
So if you have x and[br]y, this is the x.

0:26:23.270,0:26:26.470
This is x, you have to use[br]your imagination on me.

0:26:26.470,0:26:33.290
And the hypotenuse would be[br]x squared plus y squared.

0:26:33.290,0:26:36.166
And this is the z.

0:26:36.166,0:26:44.520
And then, I draw[br]what is in between.

0:26:44.520,0:26:47.800
This has to be 45 degrees.

0:26:47.800,0:26:50.270
Can you see what's going on?

0:26:50.270,0:26:57.990
So theta has to be[br]between 0 and what?

0:26:57.990,0:26:59.804
STUDENT: 2 pi.

0:26:59.804,0:27:01.470
PROFESSOR: Yes, you[br]are smarter than me.

0:27:01.470,0:27:02.630
That was the longitude.

0:27:02.630,0:27:03.290
Thank you.

0:27:03.290,0:27:05.930
I'm sorry, I meant to[br]write the latitude.

0:27:05.930,0:27:10.600
Phi is between 0 and pi/4.

0:27:10.600,0:27:14.120
How about the radius?

0:27:14.120,0:27:17.150
Are you afraid of the radius?

0:27:17.150,0:27:18.260
No.

0:27:18.260,0:27:19.420
Why?

0:27:19.420,0:27:21.014
The radius is your friend.

0:27:21.014,0:27:23.810
It was not your friend before.

0:27:23.810,0:27:25.290
Look how wobbly it is.

0:27:25.290,0:27:30.070
But in this case, the radius[br]goes all the way from 0

0:27:30.070,0:27:33.805
to a finite value,[br]which is exactly

0:27:33.805,0:27:37.500
the radius of the sphere.

0:27:37.500,0:27:43.330
Because you have rays of light[br]coming from the source origin,

0:27:43.330,0:27:48.870
and they bounce[br]against this profile,

0:27:48.870,0:27:51.920
which is the profile[br]of the sphere, which

0:27:51.920,0:27:53.930
has radius square root of 2.

0:27:53.930,0:27:58.440
So life is good for[br]you in this case.

0:27:58.440,0:28:00.620
Are you guys with me?

0:28:00.620,0:28:01.710
Should it be easy?

0:28:01.710,0:28:05.960
Yes, it should be easy to[br]write that in the integral,

0:28:05.960,0:28:08.300
if you know how to write it.

0:28:08.300,0:28:09.460
So you have.

0:28:09.460,0:28:10.050
OK.

0:28:10.050,0:28:11.340
What do you want to do first?

0:28:11.340,0:28:14.180
It doesn't matter that you[br]apply Fubini's theorem.

0:28:14.180,0:28:19.220
You have fixed limits.

0:28:19.220,0:28:27.970
You have 0 to 2 pi, 0 to[br]pi/4, 0 to square root 2.

0:28:27.970,0:28:32.030
Inside, there must be a[br]Jacobian that you know by heart,

0:28:32.030,0:28:35.810
and I'm asking you to learn[br]it by heart before the final,

0:28:35.810,0:28:38.220
if not for now, but[br]maybe before the final.

0:28:38.220,0:28:42.740
But by now, you should[br]know it by heart.

0:28:42.740,0:28:44.200
Thank you so much, Matthew.

0:28:44.200,0:28:45.440
Yes.

0:28:45.440,0:28:48.660
You don't have much to memorize,[br]but this is one of the things

0:28:48.660,0:28:51.910
that I told you I did not[br]memorize it, I was a freshman,

0:28:51.910,0:28:55.330
I was stubborn and silly.

0:28:55.330,0:28:57.900
So I have to compute what?

0:28:57.900,0:28:59.590
I have to compute the Jacobian.

0:28:59.590,0:29:05.060
Imagine what work you have[br]when you're limited in time.

0:29:05.060,0:29:09.090
dx, dr. dx, d theta.

0:29:09.090,0:29:10.980
dx, d phi.

0:29:10.980,0:29:13.340
I thought I was[br]about to kill myself.

0:29:13.340,0:29:23.360
dy, dr. dy, d theta. dy, d[br]phi, and finally, dz, dr. dz,

0:29:23.360,0:29:24.830
d theta.

0:29:24.830,0:29:27.280
dz d phi.

0:29:27.280,0:29:28.760
And I did this.

0:29:28.760,0:29:32.211
And I thought I was about to[br]just collapse and not finish

0:29:32.211,0:29:32.710
my exam.

0:29:32.710,0:29:37.374
I finished my exam, but since[br]then, I didn't remember that.

0:29:37.374,0:29:38.900
I had to compute it.

0:29:38.900,0:29:42.260
It took me 10 minutes[br]to compute the Jacobian.

0:29:42.260,0:29:45.528
So this is r squared, psi, phi.

0:29:45.528,0:29:49.870
If you have nothing better[br]to do, you can do that.

0:29:49.870,0:29:52.460


0:29:52.460,0:29:56.680
Do you remember what the[br]spherical coordinates were, out

0:29:56.680,0:29:57.835
of curiosity?

0:29:57.835,0:29:59.715
Who remembers that?

0:29:59.715,0:30:02.565
There are some[br]pre-med majors here,

0:30:02.565,0:30:04.950
who probably remember that.

0:30:04.950,0:30:10.430
So when you have a phi here,[br]you have r-- sine or cosine?

0:30:10.430,0:30:11.330
Cosine.

0:30:11.330,0:30:12.730
r cosine phi.

0:30:12.730,0:30:18.050
And then r sine phi[br]for both times what?

0:30:18.050,0:30:20.620
The first one comes[br]from theta, like that.

0:30:20.620,0:30:23.560
It's going to be cosine[br]theta, and sine theta.

0:30:23.560,0:30:29.560
Well imagine me taking these[br]functions and differentiating,

0:30:29.560,0:30:31.460
partial derivatives.

0:30:31.460,0:30:37.830
And after I differentiated down,[br]compute the 3 by 3 determining.

0:30:37.830,0:30:42.340
It's an error, no matter how[br]good you are at computing.

0:30:42.340,0:30:45.690
So don't do that,[br]just memorize it.

0:30:45.690,0:30:47.570
Don't do like I did.

0:30:47.570,0:30:49.890
And then you have d what?

0:30:49.890,0:30:54.110
dr, d phi, d theta.

0:30:54.110,0:30:57.470


0:30:57.470,0:31:03.040
Now what is the volume[br]of the ice cream cone?

0:31:03.040,0:31:05.620
Let me erase.

0:31:05.620,0:31:07.270
This shouldn't be hard.

0:31:07.270,0:31:10.456
This is the type[br]of problem where

0:31:10.456,0:31:14.665
you have a product of[br]functions of several variables.

0:31:14.665,0:31:21.976
You can separate as a product[br]of three independent integrals

0:31:21.976,0:31:25.448
as a consequence of[br]Fubini's theorem.

0:31:25.448,0:31:28.920
So you have integral from,[br]integral from, integral from.

0:31:28.920,0:31:29.709
Who's your friend?

0:31:29.709,0:31:30.750
Who do you like the most?

0:31:30.750,0:31:31.420
STUDENT: Theta.

0:31:31.420,0:31:33.790
PROFESSOR: You like[br]theta the most?

0:31:33.790,0:31:37.300
Because it comes[br]from Santa Clause?

0:31:37.300,0:31:38.470
No, the theta doesn't.

0:31:38.470,0:31:42.720
This is the easiest step.

0:31:42.720,0:31:44.910
So that's why you[br]like it, because it's

0:31:44.910,0:31:46.490
the easiest to deal with.

0:31:46.490,0:31:48.010
How about phi?

0:31:48.010,0:31:51.274
Sine phi, d phi.

0:31:51.274,0:31:55.520
I agree with you,[br]it's not so easy,

0:31:55.520,0:31:58.310
but it's going to be a[br]piece of cake anyway.

0:31:58.310,0:32:03.660
How about this one, 0[br]to root 2 r squared dr.

0:32:03.660,0:32:06.600
Is this guy hard to do?

0:32:06.600,0:32:13.800
r cubed over 3 will give[br]me root 2 cubed over 3.

0:32:13.800,0:32:16.080
How much is that, by the way?

0:32:16.080,0:32:18.740
2 root 2 over 3.

0:32:18.740,0:32:22.238
Oh bless your heart,[br]that's not so hard.

0:32:22.238,0:32:24.234
This is not a problem.

0:32:24.234,0:32:25.731
How about that?

0:32:25.731,0:32:28.600


0:32:28.600,0:32:29.430
What do you have?

0:32:29.430,0:32:32.590
What is the integral of sine?

0:32:32.590,0:32:33.860
Negative cosine.

0:32:33.860,0:32:42.370
So you have minus cosine phi[br]between pi/4 up and 0 down.

0:32:42.370,0:32:43.700
Good luck to you.

0:32:43.700,0:32:48.170
Well, the first guy.

0:32:48.170,0:32:49.990
Good, minus root 2 over 2.

0:32:49.990,0:32:53.260
Minus, second guy?

0:32:53.260,0:32:55.890
Minus, minus 1.

0:32:55.890,0:32:57.360
Don't fall into the trap.

0:32:57.360,0:32:59.000
Pay attention to the signs.

0:32:59.000,0:33:00.820
Don't mess up,[br]because that's where

0:33:00.820,0:33:06.850
you can hurt your grade by[br]messing up with minus signs.

0:33:06.850,0:33:15.968
So this is 1 plus 1,[br]1 minus root 2 over 2.

0:33:15.968,0:33:22.730


0:33:22.730,0:33:25.560
And finally, let's see what[br]that is, the whole thing being.

0:33:25.560,0:33:28.920


0:33:28.920,0:33:31.340
Can we write it nicely?

0:33:31.340,0:33:34.300
What's 2 times-- 4.

0:33:34.300,0:33:40.620
4, root 2 over 3 pi.

0:33:40.620,0:33:41.960
The first and the last.

0:33:41.960,0:33:47.140
4 times 2, pi/3 times this[br]nasty guy 1 minus root 2.

0:33:47.140,0:33:51.640
I don't like it, let's[br]make it look better.

0:33:51.640,0:33:53.310
Well OK, you can[br]give me this answer,

0:33:53.310,0:33:56.060
of course you'll get 100%.

0:33:56.060,0:33:57.610
But am I happy with it?

0:33:57.610,0:34:00.180
If you were to publish[br]this in a journal,

0:34:00.180,0:34:02.430
how would you simplify?

0:34:02.430,0:34:04.310
This is dry.

0:34:04.310,0:34:07.830
OK so what do you have?

0:34:07.830,0:34:11.310
1 is 2/2.

0:34:11.310,0:34:21.560
2 minus root 2 pi,[br]4 and 2 simplify.

0:34:21.560,0:34:23.003
Are you guys with me?

0:34:23.003,0:34:26.851
There is a 2 down, and a 4[br]up, so I'm going to have a 2

0:34:26.851,0:34:29.525
and another 2, all over 3.

0:34:29.525,0:34:33.190


0:34:33.190,0:34:35.500
So I have 2 and 2 pi,[br]times 2 minus root 2.

0:34:35.500,0:34:37.670
Do you like it like that?

0:34:37.670,0:34:38.350
I don't.

0:34:38.350,0:34:39.489
So what do you do next?

0:34:39.489,0:34:43.710


0:34:43.710,0:34:46.330
I can even pull[br]the root 2 inside.

0:34:46.330,0:34:51.670
So I go 4 root 2, minus what?

0:34:51.670,0:34:56.909
Minus 4, because this is[br]2 times 3 is 4, pi over--

0:34:56.909,0:34:57.985
Do you like it?

0:34:57.985,0:35:01.680
Still I don't,[br]because I'm stubborn.

0:35:01.680,0:35:06.530
4 root 2 minus 1 over 3 is[br]the most beautiful form.

0:35:06.530,0:35:09.940
So I'll try to brush[br]it up, and put it

0:35:09.940,0:35:12.640
in the most elegant form.

0:35:12.640,0:35:15.840


0:35:15.840,0:35:17.710
It doesn't matter.

0:35:17.710,0:35:21.920
If you want to give[br]me a correct answer,

0:35:21.920,0:35:23.222
any form it would be OK.

0:35:23.222,0:35:23.722
Yes?

0:35:23.722,0:35:25.305
STUDENT: If it was[br]slightly different,

0:35:25.305,0:35:30.940
how would we find phi for the[br]limits in the second part?

0:35:30.940,0:35:32.350
PROFESSOR: If you have a what?

0:35:32.350,0:35:35.840
STUDENT: How would we find[br]phi if it wasn't obvious,

0:35:35.840,0:35:37.920
if it wasn't x[br]squared, or c squared?

0:35:37.920,0:35:40.742
PROFESSOR: If it wasn't[br]a 45 degree angle?

0:35:40.742,0:35:41.242
[INAUDIBLE]

0:35:41.242,0:35:44.630


0:35:44.630,0:35:47.710
It's not so bad, you[br]need a calculator.

0:35:47.710,0:35:55.550
Assume that I would have given[br]you the sphere of radius 7,

0:35:55.550,0:36:00.120
or square root of 7,[br]intersecting with this cone.

0:36:00.120,0:36:03.150
Then to compute[br]that phi, you would

0:36:03.150,0:36:06.287
have needed to intersect[br]the two surfaces

0:36:06.287,0:36:10.170
and then compute it, maybe[br]look at tangent inverse.

0:36:10.170,0:36:13.534
Compute phi with[br]tangent inverse.

0:36:13.534,0:36:15.450
And you will have tangent[br]inverse of a number.

0:36:15.450,0:36:19.095
Well, you cannot put tangent[br]inverse of a number everywhere,

0:36:19.095,0:36:20.870
it's not nice.

0:36:20.870,0:36:23.900
So what you would[br]do is in the end,

0:36:23.900,0:36:26.260
you would do it with[br]a calculator, come up

0:36:26.260,0:36:30.155
with a nice truncated[br]result with 5 decimals,

0:36:30.155,0:36:34.431
or 10 decimals, whatever the[br]calculator will give you.

0:36:34.431,0:36:34.930
OK?

0:36:34.930,0:36:37.530


0:36:37.530,0:36:39.000
Or, you can do it with MathLab.

0:36:39.000,0:36:42.166


0:36:42.166,0:36:46.870
You can do it with scientific[br]software, for sure.

0:36:46.870,0:36:52.328
Let's do what I-- Ryan[br]you said this was a what?

0:36:52.328,0:36:53.952
STUDENT: Number 20[br]is for surface area.

0:36:53.952,0:36:54.535
PROFESSOR: OK.

0:36:54.535,0:36:58.315
So, it's-- read it to me again.

0:36:58.315,0:37:02.050
What does it say?

0:37:02.050,0:37:03.262
I'm coming to you.

0:37:03.262,0:37:05.860


0:37:05.860,0:37:10.160
It says, find the surface area[br]of the part of the sphere that

0:37:10.160,0:37:14.140
lies where you have 64.

0:37:14.140,0:37:18.764
This is all because[br]of [INAUDIBLE]

0:37:18.764,0:37:21.259
But yours is not[br]very even, right?

0:37:21.259,0:37:26.748


0:37:26.748,0:37:29.250
You shouldn't have bad results.

0:37:29.250,0:37:30.410
And guess what?

0:37:30.410,0:37:33.750
If you do, you use[br]your calculator

0:37:33.750,0:37:37.990
to find out the upper limit[br]of the angle for the volume.

0:37:37.990,0:37:39.910
OK.

0:37:39.910,0:37:47.280
So now, you say oh[br]my god, this is ugly.

0:37:47.280,0:37:48.866
I agree with you, it's not nice.

0:37:48.866,0:37:56.120


0:37:56.120,0:38:03.210
You have square root of 2 minus[br]x squared minus y squared.

0:38:03.210,0:38:10.140
And when you compute the[br]surface area of the cap-- cup,

0:38:10.140,0:38:13.050
cap means spherical cap.

0:38:13.050,0:38:15.760
A little hat that[br]looks like this?

0:38:15.760,0:38:18.200
That's why it's called cap.

0:38:18.200,0:38:20.615
That will integrate[br]over the disk d.

0:38:20.615,0:38:35.330


0:38:35.330,0:38:40.200
Square root of 1[br]plus f of x squared

0:38:40.200,0:38:49.390
plus f of y squared, dx da.

0:38:49.390,0:38:52.370
Is that the only[br]way you can do this?

0:38:52.370,0:38:53.140
No.

0:38:53.140,0:38:57.307
You can actually do it with[br]parametrization of a sphere,

0:38:57.307,0:39:01.070
and you have the[br]element limit over here.

0:39:01.070,0:39:05.450
So that might be easier.

0:39:05.450,0:39:08.020


0:39:08.020,0:39:08.530
Yeah.

0:39:08.530,0:39:10.360
You can also do it in homework.

0:39:10.360,0:39:14.970
But what if you went up there--[br]let's see, how hard is life?

0:39:14.970,0:39:18.422
How hard would it be[br]to do it like this?

0:39:18.422,0:39:21.220


0:39:21.220,0:39:22.640
That's good.

0:39:22.640,0:39:24.510
First of all, let's[br]think everything

0:39:24.510,0:39:27.090
that's under the square root.

0:39:27.090,0:39:28.260
And write it down.

0:39:28.260,0:39:29.970
1 plus.

0:39:29.970,0:39:33.300
Now, computing this problem[br]with respect to x and you say,

0:39:33.300,0:39:35.010
oh my god, that's hard.

0:39:35.010,0:39:36.450
No, it's not.

0:39:36.450,0:39:38.905
If you want to do the[br]hard one, and most of you

0:39:38.905,0:39:41.250
were, and you have[br]that professors who

0:39:41.250,0:39:44.660
gave you enough practice,[br]what did you have done?

0:39:44.660,0:39:47.110
Chain rule.

0:39:47.110,0:39:51.790
On the bottom, you have[br]this nasty guy twice.

0:39:51.790,0:39:56.310
But on the top, your minus 2x.

0:39:56.310,0:39:59.864
So when you simplify[br]your life becomes easier.

0:39:59.864,0:40:02.760


0:40:02.760,0:40:06.230
And you will square it.

0:40:06.230,0:40:09.910
Are you guys with me,[br]have I lost you yet?

0:40:09.910,0:40:12.720
And then the same thing in y.

0:40:12.720,0:40:19.230
Minus 2y, over 2 square root 2[br]minus x square minus y squared,

0:40:19.230,0:40:19.820
square it.

0:40:19.820,0:40:24.910


0:40:24.910,0:40:28.000
Some things cancel out.

0:40:28.000,0:40:30.670
So let's be patient[br]and see what we have.

0:40:30.670,0:40:34.070


0:40:34.070,0:40:37.940
First of all, 1 is not going to[br]give you trouble, because let

0:40:37.940,0:40:42.440
write 1 as this over itself.

0:40:42.440,0:40:45.640
Plus, minus squared[br]is plus, thank god.

0:40:45.640,0:40:51.030
x squared over 2 minus x squared[br]minus y squared plus y squared

0:40:51.030,0:40:57.500
over 2 minus x squared[br]minus y squared.

0:40:57.500,0:41:01.310
And these guys go for a walk.

0:41:01.310,0:41:08.648
Minus x squared, minus y[br]squared, plus y squared.

0:41:08.648,0:41:12.400


0:41:12.400,0:41:14.650
They disappear[br]together in the dark.

0:41:14.650,0:41:19.744
So you have 2 over 2 minus[br]x squared minus y squared.

0:41:19.744,0:41:29.010


0:41:29.010,0:41:31.000
OK let's try to do that.

0:41:31.000,0:41:33.570


0:41:33.570,0:41:35.580
Guys, I have to erase.

0:41:35.580,0:41:36.976
I will erase.

0:41:36.976,0:41:43.330


0:41:43.330,0:41:46.970
So what you see[br]here, some people

0:41:46.970,0:41:51.180
call it ds, and use the[br]element of area on the surface.

0:41:51.180,0:41:57.834


0:41:57.834,0:42:03.875
It's like the area of[br]a small surface patch.

0:42:03.875,0:42:07.730


0:42:07.730,0:42:09.050
So the curve linear squared.

0:42:09.050,0:42:14.220


0:42:14.220,0:42:15.410
Alright.

0:42:15.410,0:42:22.220
So area of the cup will be--[br]now you say, well over the d,

0:42:22.220,0:42:23.610
let me think.

0:42:23.610,0:42:27.890
d represents those[br]xy's with a property

0:42:27.890,0:42:36.420
that x squared plus y squared[br]was between what and what?

0:42:36.420,0:42:41.610
0 and 1, because[br]that was our, the

0:42:41.610,0:42:46.810
predicted domain on the[br]shadow on the ground.

0:42:46.810,0:42:48.345
OK, that was this.

0:42:48.345,0:42:51.650


0:42:51.650,0:42:56.400
And as you look at it, I have[br]to put it on the square root.

0:42:56.400,0:42:58.710
Don't be afraid of[br]it, because it's not

0:42:58.710,0:43:02.730
much up here than you thought.

0:43:02.730,0:43:05.090
And let's solve this together.

0:43:05.090,0:43:10.630
What is your luck that this is[br]a symmetric polynomial index,

0:43:10.630,0:43:14.505
and why x squared plus y squared[br]that you can rewrite as r

0:43:14.505,0:43:17.120
squared, polar coordinates?

0:43:17.120,0:43:19.840
And Ryan asked, can I[br]do polar coordinates?

0:43:19.840,0:43:22.250
That's exactly what[br]you're going to do.

0:43:22.250,0:43:25.060
You didn't know, unless[br]your intuition is strong.

0:43:25.060,0:43:26.750
Yes?

0:43:26.750,0:43:27.417
Alex tell me.

0:43:27.417,0:43:29.750
STUDENT: I was going to ask,[br]if you could have done that

0:43:29.750,0:43:35.108
by taking the r plane and[br]multiplying that by 2 pi r?

0:43:35.108,0:43:36.554
PROFESSOR: Yeah,[br]you can do that.

0:43:36.554,0:43:39.270


0:43:39.270,0:43:40.780
Well, that is a way to do that.

0:43:40.780,0:43:43.480
So practically, he's[br]asking-- I don't

0:43:43.480,0:43:47.990
know if you guys[br]remember, in Cal 2,

0:43:47.990,0:43:50.860
you have the surface[br]of revolution, right?

0:43:50.860,0:43:53.610
And if you knew the[br]length of an arc,

0:43:53.610,0:43:56.780
you would be able[br]to revolve that arc.

0:43:56.780,0:43:58.110
This is the cap.

0:43:58.110,0:44:01.360
And you take one of the[br]meridians of the hat,

0:44:01.360,0:44:03.970
and revolve it, can[br]redo with a form,

0:44:03.970,0:44:08.970
like you did the washer[br]and dryer method.

0:44:08.970,0:44:10.370
It always amuses me.

0:44:10.370,0:44:14.760
Yes, you could have[br]done that from Cal 2.

0:44:14.760,0:44:17.490
Computing the area of[br]the cap as a surface

0:44:17.490,0:44:23.380
of revolution, chapter-- c'mon,[br]I'm a co-author of this book.

0:44:23.380,0:44:27.600
Chapter 7?

0:44:27.600,0:44:28.820
What chapter?

0:44:28.820,0:44:30.030
Chapter 6?

0:44:30.030,0:44:31.851
No.

0:44:31.851,0:44:33.510
The washer and dryer?

0:44:33.510,0:44:35.630
Chapter 6, right?

0:44:35.630,0:44:36.390
OK.

0:44:36.390,0:44:38.510
But now we already[br]have three, and we

0:44:38.510,0:44:41.534
don't want to remember Cal 2[br]because it was a nightmare.

0:44:41.534,0:44:44.250
Several of you told[br]me that this is

0:44:44.250,0:44:48.190
easier, these things are[br]generally easier than Cal 2,

0:44:48.190,0:44:50.620
because Cal 2 was headache.

0:44:50.620,0:44:55.530
And what seemed to be giving[br]you most of the headache

0:44:55.530,0:44:59.020
was a salad of[br]different ingredients

0:44:59.020,0:45:01.380
that seemed to be unrelated.

0:45:01.380,0:45:03.060
Which I agree.

0:45:03.060,0:45:11.535
You have arcing, washer,[br]slices, then Greek substitution,

0:45:11.535,0:45:13.155
the partial fractions.

0:45:13.155,0:45:15.030
All sorts of things and[br]series and sequences.

0:45:15.030,0:45:19.150
And they are little[br]things that don't quite

0:45:19.150,0:45:22.150
follow one from another.

0:45:22.150,0:45:24.650
They are a little bit unrelated.

0:45:24.650,0:45:26.650
OK, how do you do that?

0:45:26.650,0:45:30.830
You have to help me because[br]that was the idea, that now you

0:45:30.830,0:45:32.476
can help me, right?

0:45:32.476,0:45:34.448
Square root of

0:45:34.448,0:45:38.885
STUDENT: 2 over 2 minus r[br]squared, times r, dr d theta.

0:45:38.885,0:45:44.310


0:45:44.310,0:45:47.290
PROFESSOR: And do we like it?

0:45:47.290,0:45:50.590
No, but we have to continue.

0:45:50.590,0:45:55.080
0 to 1, this is 0 to 2 pi.

0:45:55.080,0:46:03.450
I can get rid of the 2 pi,[br]and put it here and say, OK.

0:46:03.450,0:46:12.410
I should be as good as taking[br]out square root of 2 from here.

0:46:12.410,0:46:15.310


0:46:15.310,0:46:17.760
He goes out for a walk.

0:46:17.760,0:46:24.698
And then I have integral 1 over[br]this long line of fraction.

0:46:24.698,0:46:30.564


0:46:30.564,0:46:35.510
STUDENT: And that would be 2r[br]so that the r will cancel out.

0:46:35.510,0:46:41.920
PROFESSOR: So r dr, if[br]u is 2 minus r squared,

0:46:41.920,0:46:43.130
the u is minus.

0:46:43.130,0:46:48.030
I have to pay attention, so[br]I don't mess up the signs.

0:46:48.030,0:46:50.840
So rdr is a block.

0:46:50.840,0:46:55.350
And this block is[br]simply minus du/2.

0:46:55.350,0:47:02.670
So I write it here,[br]minus 1/2, du/2.

0:47:02.670,0:47:07.760


0:47:07.760,0:47:10.810
Don't be nervous[br]about this minus,

0:47:10.810,0:47:13.230
because it's not going to[br]give me a minus result,

0:47:13.230,0:47:15.480
a negative result.[br]If it did, that means

0:47:15.480,0:47:18.040
that I was drunk when I[br]did it, because I will

0:47:18.040,0:47:21.070
get the area of the cap as[br]a negative number, which

0:47:21.070,0:47:22.880
is impossible.

0:47:22.880,0:47:25.550
But it's going to happen[br]when I change the limits.

0:47:25.550,0:47:26.380
Yes?

0:47:26.380,0:47:28.270
STUDENT: Where did that[br]last one come from?

0:47:28.270,0:47:29.952
PROFESSOR: From this one.

0:47:29.952,0:47:33.800
[INTERPOSING VOICES][br]Oh, I put too many.

0:47:33.800,0:47:38.090
So this guy is[br]this guy, which is

0:47:38.090,0:47:39.835
this guy minus the [INAUDIBLE].

0:47:39.835,0:47:42.912


0:47:42.912,0:47:48.730
Now, do you want me to go[br]ahead and cancel this out?

0:47:48.730,0:47:50.120
Right?

0:47:50.120,0:47:51.030
OK.

0:47:51.030,0:47:54.576
I have squared 2 pi.

0:47:54.576,0:47:56.680
I did not get the[br]endpoints, you have

0:47:56.680,0:47:58.295
to help me put the endpoints.

0:47:58.295,0:48:01.640


0:48:01.640,0:48:05.140
From 2 down.

0:48:05.140,0:48:06.280
To 1.

0:48:06.280,0:48:07.390
Which is crazy, right?

0:48:07.390,0:48:08.830
Because 2 is bigger than 1.

0:48:08.830,0:48:13.090
That's exactly where the next[br]minus is going to come from.

0:48:13.090,0:48:18.380
So integral from 2 to 1 is[br]minus integral from 1 to 2.

0:48:18.380,0:48:20.360
So I shouldn't be[br]worried, because I already

0:48:20.360,0:48:23.820
have the minus out, with the[br]minus that's going to come out,

0:48:23.820,0:48:25.450
I'm going to have a[br]positive variable.

0:48:25.450,0:48:28.280


0:48:28.280,0:48:29.315
Square root 2 pi.

0:48:29.315,0:48:32.740


0:48:32.740,0:48:37.150
Somebody was smarter than[br]me and said Magdalene,

0:48:37.150,0:48:39.110
I think Alex-- was it you?

0:48:39.110,0:48:43.520
You said, why don't you[br]take advantage of the fact

0:48:43.520,0:48:48.240
that you already have 1 over[br]2 square root u and integrate?

0:48:48.240,0:48:51.370
And that is going[br]to be squared u.

0:48:51.370,0:48:52.790
Can you understand?

0:48:52.790,0:48:53.890
Who said that?

0:48:53.890,0:48:56.550
I heard a voice, it[br]was not in my head.

0:48:56.550,0:48:57.780
I'm Innocent.

0:48:57.780,0:49:02.230
I heard a voice that told[br]me, if you are smart,

0:49:02.230,0:49:04.743
you would understand[br]to pull out the minus.

0:49:04.743,0:49:09.580
You would understand[br]that this exactly

0:49:09.580,0:49:12.750
is the derivative[br]of square root of u.

0:49:12.750,0:49:16.400
And I will be faster than[br]you, because you have just

0:49:16.400,0:49:20.600
computed made between 1 and 2.

0:49:20.600,0:49:22.040
Some people aren't too smart.

0:49:22.040,0:49:23.250
I didn't think of that.

0:49:23.250,0:49:27.070
Now I've been thing about that.

0:49:27.070,0:49:30.950
Why cancel out the 2[br]when you can [INAUDIBLE].

0:49:30.950,0:49:35.420
So you have a minus out.

0:49:35.420,0:49:38.850
STUDENT: There still needs to[br]be a 2 on the outside, right?

0:49:38.850,0:49:40.810
PROFESSOR: Yes, I have[br]to put 2 together.

0:49:40.810,0:49:42.780
Minus 2 and 2.

0:49:42.780,0:49:45.730
2 pi, that's a collective thing.

0:49:45.730,0:49:50.250
Squared and cubed[br]between 1 and 2.

0:49:50.250,0:49:51.100
Do I like this?

0:49:51.100,0:49:55.480
No, but you tell me[br]what is in the bracket.

0:49:55.480,0:49:59.400
How much u minus 12, 13.

0:49:59.400,0:50:02.530
1 minus square root of[br]2 is a negative number.

0:50:02.530,0:50:05.988
But with the minus outside,[br]I"m going to fix it.

0:50:05.988,0:50:08.478
And I'm going to get[br]something really ugly.

0:50:08.478,0:50:15.950


0:50:15.950,0:50:17.660
Yeah.

0:50:17.660,0:50:21.480
So when I'm multiplying sides,[br]this by that, I get 4, right?

0:50:21.480,0:50:22.770
Guys?

0:50:22.770,0:50:23.600
I get a 4.

0:50:23.600,0:50:29.030
This guy and this guy, minus,[br]minus, plus 2, 2 times 2 is 4.

0:50:29.030,0:50:35.270
And then minus, to make[br]it look better, 2 root 2.

0:50:35.270,0:50:37.904
And multiply out.

0:50:37.904,0:50:40.140
Minus root 2 and the pi.

0:50:40.140,0:50:43.540
Now, I don't care[br]where you stop.

0:50:43.540,0:50:47.260
I swear that if you stop[br]here, you'll still get 100%.

0:50:47.260,0:50:50.580
Because what I care about is[br]not to see a nice simplified

0:50:50.580,0:50:51.980
result, so much.

0:50:51.980,0:50:54.110
I won't go over your work.

0:50:54.110,0:50:56.850
But to see that[br]you understood how

0:50:56.850,0:50:59.890
you solve this kind of problem.

0:50:59.890,0:51:02.350
It's not the sign[br]of intelligence

0:51:02.350,0:51:07.120
being able to simplify[br]answers very much.

0:51:07.120,0:51:12.490
But the method in itself, why[br]and how, what the steps are,

0:51:12.490,0:51:15.126
that shows knowledge[br]and intelligence.

0:51:15.126,0:51:18.830


0:51:18.830,0:51:19.830
Have I mess up?

0:51:19.830,0:51:21.094
I don't think so.

0:51:21.094,0:51:24.980
STUDENT: Does the order matter,[br]of the dr d theta, or dx/dy,

0:51:24.980,0:51:28.620
does that matter which[br]order you put them in?

0:51:28.620,0:51:30.025
PROFESSOR: In this case, no.

0:51:30.025,0:51:31.320
STUDENT: Or over here?

0:51:31.320,0:51:33.270
PROFESSOR: In this[br]case, no again.

0:51:33.270,0:51:36.360
But if you were[br]to swap them, you

0:51:36.360,0:51:39.510
would have to swap[br]the values as well.

0:51:39.510,0:51:41.170
Why is that?

0:51:41.170,0:51:42.710
That's a very good question.

0:51:42.710,0:51:45.507
He's right, but why is that?

0:51:45.507,0:51:49.000
It doesn't matter, why?

0:51:49.000,0:51:50.456
In general, it matters.

0:51:50.456,0:51:53.880


0:51:53.880,0:52:01.700
They have to be from a given[br]number to a given number.

0:52:01.700,0:52:05.340
It's not like reversing--[br]when you reverse

0:52:05.340,0:52:08.360
the order of integrals,[br]it's usually harder,

0:52:08.360,0:52:11.180
because you have to draw[br]the vertical strips.

0:52:11.180,0:52:12.860
And you have it[br]between two functions.

0:52:12.860,0:52:16.125
And then from vertical strips,[br]you go to horizontal strips,

0:52:16.125,0:52:17.500
and you have other[br]two functions.

0:52:17.500,0:52:20.980
So you always have to think[br]how to change the function.

0:52:20.980,0:52:23.820
Here, you don't have[br]to think at all.

0:52:23.820,0:52:27.510
You have a function[br]that depends on r only.

0:52:27.510,0:52:29.260
There is no theta[br]in the picture.

0:52:29.260,0:52:31.880
Plus these two[br]are fixed numbers.

0:52:31.880,0:52:35.040
You can reverse the[br]integration in your sleep.

0:52:35.040,0:52:37.123
OK, you get the same thing.

0:52:37.123,0:52:40.081
All you have to do is[br]swap these two guys,

0:52:40.081,0:52:42.053
and swap-- the 0 is the same.

0:52:42.053,0:52:44.518
So I swap these two guys.

0:52:44.518,0:52:48.960
STUDENT: How did you[br]take the 2 pi out?

0:52:48.960,0:52:50.970
PROFESSOR: What did I do?

0:52:50.970,0:52:52.810
How did I take this out?

0:52:52.810,0:52:55.680
STUDENT: No, the 2 pi.

0:52:55.680,0:52:57.040
PROFESSOR: Oh, the 2 pi?

0:52:57.040,0:52:57.860
OK.

0:52:57.860,0:52:59.950
Let me show you it better[br]here, because we've

0:52:59.950,0:53:03.390
discussed about this before.

0:53:03.390,0:53:06.995
When you have[br]integral from a to b,

0:53:06.995,0:53:11.396
or integral from c to[br]d of a function or r

0:53:11.396,0:53:18.740
and a function of[br]theta, what do you go?

0:53:18.740,0:53:20.250
There is a theorem[br]that says that--

0:53:20.250,0:53:22.290
and thanks for this[br]theorem and the fact

0:53:22.290,0:53:23.430
that they're separable.

0:53:23.430,0:53:26.260
The variables are[br]separated in this product.

0:53:26.260,0:53:31.400
This is the product between[br]integral from a to b,

0:53:31.400,0:53:34.190
here of theta to[br]theta, and integral

0:53:34.190,0:53:38.150
from c to d, f of r/dr.[br]They are nothing to one

0:53:38.150,0:53:39.890
another but a product.

0:53:39.890,0:53:41.190
So what do you do?

0:53:41.190,0:53:46.980
You say this is integral[br]of 1, from 0 to pi d theta,

0:53:46.980,0:53:49.082
times the other guy.

0:53:49.082,0:53:53.220
So this is 2 pi.

0:53:53.220,0:53:57.150
When theta doesn't appear[br]inside, it's a blessing.

0:53:57.150,0:53:59.650
But if and if there[br]is, I have a question.

0:53:59.650,0:54:00.910
What if theta appeared inside?

0:54:00.910,0:54:03.820


0:54:03.820,0:54:06.350
Theta doesn't appear[br]inside by himself.

0:54:06.350,0:54:09.310
He appears inside[br]of a trig function.

0:54:09.310,0:54:13.840
So assume you have cosine[br]theta here times r.

0:54:13.840,0:54:17.860
You would have pulled[br]cosine theta out,

0:54:17.860,0:54:21.000
and integrated cosine[br]theta, that would be easy.

0:54:21.000,0:54:25.950
And if you have a problem like[br]that, you would have gotten 0.

0:54:25.950,0:54:28.580
Because integral of[br]course of cosine theta

0:54:28.580,0:54:32.960
would be sine of theta, and[br]then theta between 0 and 2

0:54:32.960,0:54:36.935
pi is sine of theta between,[br]which would give you 0.

0:54:36.935,0:54:40.171
It happened to me,[br]many times in the exam.

0:54:40.171,0:54:40.920
It was a blessing.

0:54:40.920,0:54:44.580
I was 19, and I was so happy.

0:54:44.580,0:54:48.030
Professors wanted to[br]see only the answer.

0:54:48.030,0:54:52.451
Because in Romania,[br]it's different.

0:54:52.451,0:54:55.420
You come take a written[br]exam, and the professor

0:54:55.420,0:54:58.310
has five hours to grade it.

0:54:58.310,0:55:02.350
The same day, two hours later,[br]you have the oral examination.

0:55:02.350,0:55:04.510
You pick up a ticket,[br]on the ticket,

0:55:04.510,0:55:07.550
you see three things to[br]solve, four things to solve.

0:55:07.550,0:55:09.120
You go take a seat.

0:55:09.120,0:55:12.895
And while the professor and[br]the assistant grade the exam,

0:55:12.895,0:55:16.540
you actually think[br]of your oral exam.

0:55:16.540,0:55:20.670
When you come and present[br]your results on the board,

0:55:20.670,0:55:23.250
they tell you,[br]you messed up, you

0:55:23.250,0:55:27.122
got a 60% on this sticking exam.

0:55:27.122,0:55:28.300
This is how it goes.

0:55:28.300,0:55:32.310
Or, on the contrary, hey,[br]listen, you got a 95%

0:55:32.310,0:55:33.970
on the written part, OK?

0:55:33.970,0:55:36.572
I don't want to see[br]what you have there,

0:55:36.572,0:55:37.970
it really looks good.

0:55:37.970,0:55:39.660
I don't want to see[br]it, it's clear to me

0:55:39.660,0:55:41.490
that you know what you're doing.

0:55:41.490,0:55:44.692
So it's a different[br]kind of examination.

0:55:44.692,0:55:47.644
I hear that Princeton[br]does that, I wonder

0:55:47.644,0:55:49.120
how are all the exams here.

0:55:49.120,0:55:51.870
I don't think people[br]are ready for them yet.

0:55:51.870,0:55:54.420
But at Princeton they[br]do a lot, all the same.

0:55:54.420,0:55:59.190
They make a hat, and take a[br][INAUDIBLE], put tickets in it.

0:55:59.190,0:56:02.100
And the teacher comes,[br]and closes his eyes

0:56:02.100,0:56:06.400
and picks a ticket,[br]and says oh my god, I

0:56:06.400,0:56:08.010
got proof of Fubini's theorem.

0:56:08.010,0:56:11.160
And do these three[br]triple integrals.

0:56:11.160,0:56:15.130
This is a type of oral[br]exam that you would have.

0:56:15.130,0:56:18.880
But if you know that,[br]because you studied,

0:56:18.880,0:56:22.980
you're not afraid[br]to present them.

0:56:22.980,0:56:26.240
But you have to present them,[br]and you have a limited time.

0:56:26.240,0:56:30.290
Because there are other 30[br]students in the classroom.

0:56:30.290,0:56:32.040
You only have five minutes.

0:56:32.040,0:56:36.576
And I only pick-- I want to[br]see your work on all of them.

0:56:36.576,0:56:37.950
And I'll teach[br]you how to present

0:56:37.950,0:56:39.520
on this problem on the board.

0:56:39.520,0:56:42.290
And then you have five[br]minutes to present.

0:56:42.290,0:56:46.880
If you are really[br]embarrassed and you

0:56:46.880,0:56:50.150
don't want to speak in public,[br]then you have a problem.

0:56:50.150,0:56:53.360
I've had many fears--[br]and in other countries--

0:56:53.360,0:56:55.690
I heard that in England,[br]they have the same system.

0:56:55.690,0:56:59.460
There are people who are too[br]shy to show their results,

0:56:59.460,0:57:01.560
or too shy to talk.

0:57:01.560,0:57:05.370
And then they start stuttering.

0:57:05.370,0:57:07.080
But they have to do it.

0:57:07.080,0:57:10.140
There is no excuse,[br]they don't care if you

0:57:10.140,0:57:13.450
have problems with your speech.

0:57:13.450,0:57:18.590
So I asked the people I[br]knew and I went to London,

0:57:18.590,0:57:22.245
and they said most people[br]will stutter in there.

0:57:22.245,0:57:24.130
I was so scared.

0:57:24.130,0:57:26.487
Most people who stutter[br]in our oral exams

0:57:26.487,0:57:28.900
are people who[br]spend too much time

0:57:28.900,0:57:31.840
in the pub the previous day.

0:57:31.840,0:57:36.410
Pubs were everywhere and I saw[br]lots of students in the pubs.

0:57:36.410,0:57:40.300
I went to University of Durham--[br]this is where Harry Potter was

0:57:40.300,0:57:42.320
filmed, by the way.

0:57:42.320,0:57:45.476
I saw the castle, which[br]is a student dorm.

0:57:45.476,0:57:47.970
You pay something[br]like 500 pounds.

0:57:47.970,0:57:51.590
Which would be like $100?

0:57:51.590,0:57:53.830
$1,000?

0:57:53.830,0:58:00.930
Less, because I think it's[br]7.50, something like $800.

0:58:00.930,0:58:04.279
It used to be that the[br]pound was double the dollar.

0:58:04.279,0:58:05.112
[INTERPOSING VOICES]

0:58:05.112,0:58:10.500


0:58:10.500,0:58:16.380
So you could stay in that[br]dorm for $800 per month.

0:58:16.380,0:58:21.060
And you've got the same table[br]where they ate in the movie.

0:58:21.060,0:58:22.510
It was really nice.

0:58:22.510,0:58:26.910
But the University of[br]Durham is a isolated castle,

0:58:26.910,0:58:29.605
the cathedral,[br]everything is very old,

0:58:29.605,0:58:32.170
from the 11th[br]century, 12th century.

0:58:32.170,0:58:37.229
But if you go into the[br]city, it's full of pubs.

0:58:37.229,0:58:38.920
Who is in the pubs?

0:58:38.920,0:58:40.852
The calculus students.

0:58:40.852,0:58:43.270
This is where they[br]do their homework.

0:58:43.270,0:58:47.630
And it amazes me how[br]they don't get drunk.

0:58:47.630,0:58:50.280
I'm not used to alcohol,[br]because I don't drink.

0:58:50.280,0:58:52.030
Well they are used to it.

0:58:52.030,0:58:56.530
So they may nicely can do[br]their homework, beautifully,

0:58:56.530,0:59:01.060
next to a big draft[br]of Guinness like that.

0:59:01.060,0:59:04.970
And still makes sense when[br]they write the solution.

0:59:04.970,0:59:07.160
They don't miss a minus[br]sign, they're amazing.

0:59:07.160,0:59:10.103


0:59:10.103,0:59:12.990
Alright, is this hard?

0:59:12.990,0:59:17.632
If you are interested, you[br]can ask about study abroad.

0:59:17.632,0:59:20.430
We don't have big[br]business with England,

0:59:20.430,0:59:22.480
but you could go to Seville.

0:59:22.480,0:59:25.960
There are some programs[br]in the summer where

0:59:25.960,0:59:29.110
one of our professors teaches[br]differential equations like I

0:59:29.110,0:59:30.860
told you about.

0:59:30.860,0:59:33.590
He teaches differential[br]equations this summer

0:59:33.590,0:59:34.335
in Seville.

0:59:34.335,0:59:39.520


0:59:39.520,0:59:44.280
I think you can still[br]add in the next two days.

0:59:44.280,0:59:47.390
Some of you did,[br]some of you didn't.

0:59:47.390,0:59:48.870
All right.

0:59:48.870,0:59:53.210
Any questions about[br]other problems?

0:59:53.210,0:59:57.880
I have to apologize,[br]I played the game

0:59:57.880,1:00:01.470
without telling you the truth.

1:00:01.470,1:00:05.230
[INAUDIBLE] he came to[br]me last time and said,

1:00:05.230,1:00:09.080
you never showed[br]us this notation.

1:00:09.080,1:00:16.970
So what if one gives you x[br]of u, v equals 2x minus y.

1:00:16.970,1:00:22.160
y of u, v equals 3x plus y.

1:00:22.160,1:00:24.560
What the heck is that?

1:00:24.560,1:00:27.030
He didn't say heck, because[br]he's a gentlemen, right?

1:00:27.030,1:00:35.930
But he said this is the[br]notation used in web work,

1:00:35.930,1:00:40.336
and the book is actually not[br]emphasizing it, which is true.

1:00:40.336,1:00:45.650
The book is emphasizing the[br]Jacobian in section 12.8

1:00:45.650,1:00:50.340
only, which is not covered,[br]it's not part of the menu.

1:00:50.340,1:00:53.800
But the definition, you[br]should at least know it.

1:00:53.800,1:00:56.566
So what would be the[br]definition of this animal?

1:00:56.566,1:01:00.635
You see that we have to take[br]the partial derivative of x

1:01:00.635,1:01:04.272
with respect to u, the partial[br]derivative of x with respect

1:01:04.272,1:01:08.395
to v, the partial derivative[br]of y with respect to u,

1:01:08.395,1:01:10.530
and the partial derivative[br]of y with respect to v.

1:01:10.530,1:01:13.660
And that's exactly what[br]it is, indeterminate.

1:01:13.660,1:01:15.000
Not matrix, but indeterminate.

1:01:15.000,1:01:17.730


1:01:17.730,1:01:20.510
So do I bother to write it down?

1:01:20.510,1:01:22.796
If I wanted to write[br]down what it is,

1:01:22.796,1:01:27.450
of course I would write[br]it down like that.

1:01:27.450,1:01:31.350
I don't want to spend[br]all my time doing that,

1:01:31.350,1:01:34.490
because it's such[br]an easy problem.

1:01:34.490,1:01:35.580
What do you have to do?

1:01:35.580,1:01:39.680
Just compute for such a simple[br]transformation in plane.

1:01:39.680,1:01:42.370


1:01:42.370,1:01:45.470
Actually, if you[br]took linear-- again,

1:01:45.470,1:01:49.530
who is enrolled[br]in linear algebra?

1:01:49.530,1:01:50.540
Only 1, 2, 3?

1:01:50.540,1:01:52.490
Thought there were only 2.

1:01:52.490,1:01:53.463
OK.

1:01:53.463,1:01:58.690
In linear algebra, you[br]wrote this differently.

1:01:58.690,1:02:06.138
You wrote it like this. x and[br]y equals matrix multiplication.

1:02:06.138,1:02:10.540
You have 2, minus[br]1, 3, 1, by the way

1:02:10.540,1:02:12.380
it's obvious the[br]determinate of this matrix

1:02:12.380,1:02:13.640
is different from 0.

1:02:13.640,1:02:18.680
This is the linear map that you[br]are applying to the vector xy.

1:02:18.680,1:02:25.580
And in your algebra book,[br]you're using Larson, am I right?

1:02:25.580,1:02:28.580


1:02:28.580,1:02:29.640
Larson's book?

1:02:29.640,1:02:30.880
It's a good book.

1:02:30.880,1:02:35.922
So you have a of the vector x.

1:02:35.922,1:02:42.850
a of the vector x[br]is the vector v.

1:02:42.850,1:02:46.330
When you all get to[br]see linear algebra,

1:02:46.330,1:02:52.710
you'll like it more than Cal[br]3, because it's more fun.

1:02:52.710,1:02:55.595
So how do you do this[br]matrix multiplication?

1:02:55.595,1:02:56.220
It's very easy.

1:02:56.220,1:02:58.580
This time that, minus[br]this times this.

1:02:58.580,1:03:04.367


1:03:04.367,1:03:07.290
So can computers do that?

1:03:07.290,1:03:11.140
Yes, computers can, if you[br]have the right program.

1:03:11.140,1:03:14.170
And this is the first[br]program I learned in C++.

1:03:14.170,1:03:16.200
No, it was the second program.

1:03:16.200,1:03:21.040
How to write a little program[br]for multiplication of two

1:03:21.040,1:03:23.260
matrices.

1:03:23.260,1:03:28.150
The first program I[br]had, I learned in C++.

1:03:28.150,1:03:30.960
It was to build an ATM machine.

1:03:30.960,1:03:36.920
I hated that, because every time[br]I went under 0 with my balance,

1:03:36.920,1:03:40.040
I would have new word under 0.

1:03:40.040,1:03:43.180
So I would have to prepare[br]for all the possible cases

1:03:43.180,1:03:44.638
and save.

1:03:44.638,1:03:47.691
If you don't have[br]enough money, whatever.

1:03:47.691,1:03:51.620
So that was the first[br]program we wrote.

1:03:51.620,1:03:54.980
OK so, what do we have?

1:03:54.980,1:04:01.700
2 minus 1, 3 and 1.

1:04:01.700,1:04:03.670
What is the Jacobian[br]in this case?

1:04:03.670,1:04:05.980
It's 2 plus 3, 5.

1:04:05.980,1:04:07.420
Different from 0.

1:04:07.420,1:04:10.780


1:04:10.780,1:04:14.166
You have one or two[br]problems like that.

1:04:14.166,1:04:14.790
Three problems.

1:04:14.790,1:04:16.690
I was really mean.

1:04:16.690,1:04:17.620
I apologize.

1:04:17.620,1:04:19.695
But you still have time[br]to do those problems

1:04:19.695,1:04:21.307
in case of the review.

1:04:21.307,1:04:24.290
STUDENT: So we just take[br]the determinate of it?

1:04:24.290,1:04:28.410
PROFESSOR: And you take the[br]determinate of the matrix.

1:04:28.410,1:04:30.248
And that's you Jacobian.

1:04:30.248,1:04:31.580
STUDENT: What number is that?

1:04:31.580,1:04:34.640
PROFESSOR: I don't remember.

1:04:34.640,1:04:37.005
STUDENT: What if[br]it's the u and the v

1:04:37.005,1:04:41.502
is at the top and x and[br]the y at the bottom?

1:04:41.502,1:04:43.460
PROFESSOR: So the[br]determinate will be the same.

1:04:43.460,1:04:45.680
This is a very good question.

1:04:45.680,1:04:46.990
Are you guys with me?

1:04:46.990,1:04:51.530
So he said, what if you have[br]your first equation's name

1:04:51.530,1:04:53.015
would be this one.

1:04:53.015,1:04:59.240
And you have your equations[br]written like that.

1:04:59.240,1:05:01.160
Right?

1:05:01.160,1:05:06.100
And so, when you[br]look at this, you

1:05:06.100,1:05:11.810
will go-- it depends how you--[br]in which order you do that.

1:05:11.810,1:05:19.200
I wrote u, v. Sorry.

1:05:19.200,1:05:24.040
u and v, but you[br]understood what I meant.

1:05:24.040,1:05:24.760
Right?

1:05:24.760,1:05:26.424
u and v.

1:05:26.424,1:05:28.684
STUDENT: Can you[br]do number three?

1:05:28.684,1:05:30.026
It was a hard one.

1:05:30.026,1:05:31.400
PROFESSOR: I will,[br]just a second.

1:05:31.400,1:05:38.270
So d y, x with respect to u,[br]v. What would happen, I just I

1:05:38.270,1:05:40.744
would flip the x and y.

1:05:40.744,1:05:41.750
What will happen?

1:05:41.750,1:05:46.770
I get 3, 1, it's still[br]the same function.

1:05:46.770,1:05:49.110
2, and minus 1.

1:05:49.110,1:05:51.800
Why do I get minus 5?

1:05:51.800,1:05:58.940


1:05:58.940,1:06:03.070
So imagine guys, what happens[br]when you have x and y?

1:06:03.070,1:06:06.910
If you rotate, you don't[br]change the sign of your matrix,

1:06:06.910,1:06:07.525
or notation.

1:06:07.525,1:06:11.740
Matrix notation will[br]always have [INAUDIBLE].

1:06:11.740,1:06:16.760
But if you flip it,[br]if you swap x and y,

1:06:16.760,1:06:22.120
you are actually changing[br]the sign of the Jacobian,

1:06:22.120,1:06:24.450
the sign of the matrix.

1:06:24.450,1:06:28.160
You are changing[br]your orientation.

1:06:28.160,1:06:31.170
That would be a[br]hypothetical situation.

1:06:31.170,1:06:33.450
You are changing[br]your orientation.

1:06:33.450,1:06:37.030
Do you have a number, Ryan?

1:06:37.030,1:06:38.000
Is it hard?

1:06:38.000,1:06:39.256
Why is it hard?

1:06:39.256,1:06:43.160


1:06:43.160,1:06:45.600
Yeah, let me do that.

1:06:45.600,1:06:55.360


1:06:55.360,1:06:56.466
It's hard enough.

1:06:56.466,1:06:59.938


1:06:59.938,1:07:00.930
It's computation.

1:07:00.930,1:07:26.230


1:07:26.230,1:07:27.430
Were you able to do it?

1:07:27.430,1:07:32.240


1:07:32.240,1:07:33.760
Not yet, right?

1:07:33.760,1:07:40.350
So this is x, not-- OK.

1:07:40.350,1:07:46.790
So you can write this also,[br]differently, except the y sub

1:07:46.790,1:07:52.110
u, y sub v. Who[br]can tell me-- there

1:07:52.110,1:07:55.960
are ways to do it[br]in a simpler way.

1:07:55.960,1:07:58.800
But I don't want to tell[br]you yet what that way is.

1:07:58.800,1:08:01.250
And I'll show you next time.

1:08:01.250,1:08:02.666
What is x sub u?

1:08:02.666,1:08:06.400


1:08:06.400,1:08:10.812
It shouldn't be so hard[br]because it's the quotient rule.

1:08:10.812,1:08:15.738
You have 4 times u squared[br]plus v squared minus

1:08:15.738,1:08:24.680
[INAUDIBLE] minus 2u the[br]derivative of this times 4u

1:08:24.680,1:08:29.529
divided by the square of that.

1:08:29.529,1:08:34.029
Did I go too fast?

1:08:34.029,1:08:38.760
So what you have is 4u[br]squared minus 8u squared

1:08:38.760,1:08:45.729
equals minus 4u squared plus[br]4v squared divided by that.

1:08:45.729,1:08:54.100


1:08:54.100,1:08:54.825
v squared.

1:08:54.825,1:08:55.540
Squared, sorry.

1:08:55.540,1:08:58.149


1:08:58.149,1:09:01.560
x of v, that should be easier.

1:09:01.560,1:09:02.779
Why is it easy?

1:09:02.779,1:09:13.611
The first guy prime[br]minus the second guy

1:09:13.611,1:09:15.950
So the first primes,[br]second not prime.

1:09:15.950,1:09:20.890
Minus second prime, straight to[br]v, times the first not prime.

1:09:20.890,1:09:25.390


1:09:25.390,1:09:29.390
Divided by u squared.

1:09:29.390,1:09:34.890
Which is minus 8uv over that.

1:09:34.890,1:09:41.189
Is this one of those that you[br]said you couldn't do it yet?

1:09:41.189,1:09:41.689
You?

1:09:41.689,1:09:43.279
Both?

1:09:43.279,1:09:44.180
You did this one?

1:09:44.180,1:09:47.769
You got the right answer, good.

1:09:47.769,1:10:01.900
y sub v. Y sub u, it's OK[br]to have a minus 0 times

1:10:01.900,1:10:04.700
the second one.

1:10:04.700,1:10:13.633
Minus this prime with[br]respect to u, times

1:10:13.633,1:10:21.870
6v over the square of that.

1:10:21.870,1:10:30.110
And finally, y sub v[br]equals minus the derivative

1:10:30.110,1:10:34.660
of the top, with respect to[br]6v times u squared plus v

1:10:34.660,1:10:38.590
squared minus the derivative[br]of the bottom with respect

1:10:38.590,1:10:51.750
to v. v times 6v divided[br]by the whole shebang.

1:10:51.750,1:10:53.260
Now is it simplified?

1:10:53.260,1:10:55.650
No, I will simplify in a second.

1:10:55.650,1:10:59.480
You get minus 12uv.

1:10:59.480,1:11:03.280
I'm not going to finish[br]it, but we are almost done.

1:11:03.280,1:11:04.430
Why are we almost done?

1:11:04.430,1:11:06.470
This is very easy.

1:11:06.470,1:11:09.220
I mean, not very[br]easy, but doable.

1:11:09.220,1:11:10.831
How about this guy?

1:11:10.831,1:11:13.040
What do you get?

1:11:13.040,1:11:17.730
A 6u squared, a 6v squared,[br]a minus 12v squared.

1:11:17.730,1:11:19.470
It's not that bad.

1:11:19.470,1:11:24.560
So you have 6u squared[br]minus 6v squared,

1:11:24.560,1:11:27.476
over u squared plus v squared.

1:11:27.476,1:11:32.336


1:11:32.336,1:11:33.794
What did I do?

1:11:33.794,1:11:37.690


1:11:37.690,1:11:39.265
Add a minus in front.

1:11:39.265,1:11:40.940
I didn't copy.

1:11:40.940,1:11:43.488
Let me make room[br]for that, thank you.

1:11:43.488,1:11:45.933
STUDENT: It's also the 12.

1:11:45.933,1:11:49.845
It's 12uv, because there's[br]a negative in front of it.

1:11:49.845,1:11:52.300
It's minus times minus.

1:11:52.300,1:11:53.100
PROFESSOR: Here?

1:11:53.100,1:11:54.245
STUDENT: No, y sub u.

1:11:54.245,1:11:55.310
The third one.

1:11:55.310,1:11:57.990
PROFESSOR: Minus, minus,[br]plus, that's good.

1:11:57.990,1:12:00.110
Thanks for observing things.

1:12:00.110,1:12:03.150
Anything else that's fishy?

1:12:03.150,1:12:04.120
Minus, minus, plus.

1:12:04.120,1:12:06.630


1:12:06.630,1:12:07.600
OK that's better.

1:12:07.600,1:12:09.110
Change the signs.

1:12:09.110,1:12:13.850
When I move onto this one,[br]remind me to change the signs.

1:12:13.850,1:12:15.010
So what is the Jacobian?

1:12:15.010,1:12:18.120
I'm too lazy to[br]write this thing.

1:12:18.120,1:12:21.618
I'm going to have-- so, x sub u.

1:12:21.618,1:12:25.520


1:12:25.520,1:12:32.120
4 times v squared[br]minus u squared.

1:12:32.120,1:12:37.140
Let's me count the OK.

1:12:37.140,1:12:39.010
Let's do it over a.

1:12:39.010,1:12:42.090


1:12:42.090,1:12:43.780
I'll show you what happens.

1:12:43.780,1:12:45.670
Maybe you don't know[br]yet what happens,

1:12:45.670,1:12:49.130
but I'll show you what happens.

1:12:49.130,1:12:58.190
Then the next one is going to[br]be x sub v minus 8, uv over a.

1:12:58.190,1:13:04.830
y sub u, 12.

1:13:04.830,1:13:12.720
uv over a, and last,[br]with your help.

1:13:12.720,1:13:18.380
It's plus this was[br]my-- so 6 times v

1:13:18.380,1:13:21.260
squared minus u squared over a.

1:13:21.260,1:13:24.150


1:13:24.150,1:13:26.412
OK OK, let me erase.

1:13:26.412,1:13:30.140


1:13:30.140,1:13:32.670
So you guys know[br]what happens when

1:13:32.670,1:13:35.770
you have something like that?

1:13:35.770,1:13:40.940
A determinate has one[br]line multiplied or column

1:13:40.940,1:13:43.220
multiplied by a number.

1:13:43.220,1:13:50.220
If you have alpha a,[br]alpha b, alpha c and d.

1:13:50.220,1:13:55.234
The determinate of that[br]is alpha aut, a, b, c, d.

1:13:55.234,1:13:56.900
I assume you know[br]this from high school,

1:13:56.900,1:14:00.366
but I know very well[br]that many of you don't.

1:14:00.366,1:14:02.118
How do you prove this?

1:14:02.118,1:14:02.900
Very easily.

1:14:02.900,1:14:05.910
This times that would be[br]an alpha out, minus this,

1:14:05.910,1:14:07.750
and alpha out.

1:14:07.750,1:14:10.550
It's very easy to prove.

1:14:10.550,1:14:16.580
So when you have one line or one[br]column multiplied by an alpha,

1:14:16.580,1:14:19.170
that alpha gets out.

1:14:19.170,1:14:23.460
So if you have two lines[br]multiplied by an alpha,

1:14:23.460,1:14:27.930
or two rows, alpha[br]squared, excellent.

1:14:27.930,1:14:29.260
So who gets out?

1:14:29.260,1:14:37.330
1 over a squared, which[br]means this guy to the fourth.

1:14:37.330,1:14:39.580
Sorry that this is so long.

1:14:39.580,1:14:43.440
I don't like this problem,[br]because of this computation

1:14:43.440,1:14:45.259
you have to go through here.

1:14:45.259,1:14:51.680


1:14:51.680,1:14:56.712
So I would simplify[br]it as much as I could.

1:14:56.712,1:15:02.310
Let's see, before I[br]missed my a group.

1:15:02.310,1:15:12.160
So you have 24 times v squared[br]minus u squared, plus 96,

1:15:12.160,1:15:14.040
am I right?

1:15:14.040,1:15:18.910
v squared divided by all[br]this ugly guy which I hate,

1:15:18.910,1:15:19.580
to the fourth.

1:15:19.580,1:15:22.667


1:15:22.667,1:15:27.160
Fortunately, everybody's[br]a multiple of 24.

1:15:27.160,1:15:30.572
So we can pull a 24 out.

1:15:30.572,1:15:35.100
and get it out of our life,[br]because it drives us crazy.

1:15:35.100,1:15:39.690
And then you have v to the[br]fourth plus u to the 4,

1:15:39.690,1:15:40.190
minus twice.

1:15:40.190,1:15:42.180
Was that the binomial format?

1:15:42.180,1:15:45.330
Minus 2 us squared, v squared.

1:15:45.330,1:15:48.030
What was left when[br]I pull this out?

1:15:48.030,1:15:51.870


1:15:51.870,1:15:55.972
I pulled 24 out, 96 is what?

1:15:55.972,1:15:56.472
4.

1:15:56.472,1:16:01.950
So I have a 4 left.

1:16:01.950,1:16:04.380
So I would put that down.

1:16:04.380,1:16:09.070
4u squared, v squared over[br]the-- it looks symmetric

1:16:09.070,1:16:11.420
but-- that's OK.

1:16:11.420,1:16:13.845
It's not so bad.

1:16:13.845,1:16:15.300
So can you write this better?

1:16:15.300,1:16:17.725
Look at it.

1:16:17.725,1:16:19.665
Do you like it?

1:16:19.665,1:16:22.090
There is a 3.

1:16:22.090,1:16:26.822
The 4 the 2, 4[br]minus 2 is a plus 2.

1:16:26.822,1:16:30.240
Just like when we did those[br]tricky things in high school.

1:16:30.240,1:16:33.370
That would be, again,[br]the binomial formula.

1:16:33.370,1:16:35.150
u squared plus v squared.

1:16:35.150,1:16:38.832


1:16:38.832,1:16:40.630
Are you guys with me?

1:16:40.630,1:16:44.190
Because minus 2[br]plus 4 is plus 2.

1:16:44.190,1:16:47.210
This is exactly the[br]same thing as that.

1:16:47.210,1:16:51.450
Over u squared plus v[br]squared to the fourth.

1:16:51.450,1:16:53.170
If you have problems[br]computing that,

1:16:53.170,1:16:56.400
send me some emails from[br]WebWork, because I'm

1:16:56.400,1:16:59.400
going to help you do that, OK.

1:16:59.400,1:17:03.140
24 divided by what?

1:17:03.140,1:17:04.800
Yes?

1:17:04.800,1:17:09.980
u squared plus v[br]squared squared.

1:17:09.980,1:17:12.670
Oh my god.

1:17:12.670,1:17:13.300
All right.

1:17:13.300,1:17:17.300
So, I'm not going to[br]think lesser of you

1:17:17.300,1:17:20.920
if you don't put[br]all of this here.

1:17:20.920,1:17:25.774
Therefore, if you[br]get in trouble,

1:17:25.774,1:17:30.760
click from the expression from[br]the whatever you got, and say,

1:17:30.760,1:17:33.890
this horrible problem gives[br]me a headache, help me.

1:17:33.890,1:17:37.720
And I'm going to help you with[br]that simple computation that

1:17:37.720,1:17:39.680
is just algebra.

1:17:39.680,1:17:42.840
That's not going to teach you[br]anything more about Cal 3.

1:17:42.840,1:17:44.250
That's why I'm[br]going to help you.

1:17:44.250,1:17:47.040
I'll help you with[br]the answers on those.

1:17:47.040,1:17:50.160
Just send me an email.

1:17:50.160,1:17:57.210
I'm planning on still[br]reviewing even on Tuesday.

1:17:57.210,1:18:00.085
I don't want to teach anything[br]new, because I'm tired

1:18:00.085,1:18:01.598
and-- I'm just kidding.

1:18:01.598,1:18:05.020
I don't want to teach[br]anything new on Tuesday,

1:18:05.020,1:18:09.550
because I want you to be very[br]well prepared for the midterms.

1:18:09.550,1:18:12.580
So I'll do a general[br]review again,

1:18:12.580,1:18:15.530
and I'll go over some[br]homework like problems,

1:18:15.530,1:18:20.640
but mostly over[br]exam like problems.

1:18:20.640,1:18:26.386
So I want everybody to succeed,[br]to get very high scores.

1:18:26.386,1:18:28.676
But we need to practice,[br]practice, practice.

1:18:28.676,1:18:30.910
It's like you did[br]before your SATs.

1:18:30.910,1:18:34.375


1:18:34.375,1:18:39.785
It's not that much, I mean what[br]happens if you don't do great

1:18:39.785,1:18:40.410
on the midterm?

1:18:40.410,1:18:47.060
Well the midterms is[br]a portion the final.

1:18:47.060,1:18:50.070
But what I am trying[br]to do by reviewing

1:18:50.070,1:18:52.770
so much for the[br]midterm is also trying

1:18:52.770,1:18:54.530
to help you for the final.

1:18:54.530,1:18:57.280
Because on the final,[br]half of the problems

1:18:57.280,1:19:00.190
will be just like the ones[br]on the midterm Emphasizing

1:19:00.190,1:19:02.920
the same type of concepts.

1:19:02.920,1:19:05.620
It's good practice[br]for the final as well.

1:19:05.620,1:19:06.520
All right, good luck.

1:19:06.520,1:19:08.620
I'll see you Tuesday.

1:19:08.620,1:19:12.870
Let me know by email how[br]it goes with the problems.

1:19:12.870,1:19:18.789