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00:00:00,000 --> 00:00:01,708
MAGDALENA TODA: I'm
starting early, am I?

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00:00:01,708 --> 00:00:04,000
It's exactly 12:30.

3
00:00:04,000 --> 00:00:07,000
The weather is getting
better, hopefully,

4
00:00:07,000 --> 00:00:14,000
and not too many people
should miss class today.

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Can you start an attendance
sheet for me [INAUDIBLE]?

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I know I can count on you.

7
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OK.

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I have good markers today.

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I'm going to go ahead
and talk about 12.3,

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00:00:31,304 --> 00:00:34,280
double integrals in
polar coordinates.

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00:00:34,280 --> 00:00:36,264
These are all friends of yours.

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00:00:36,264 --> 00:00:55,112


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You've seen until now
only double integrals that

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involve the rectangles, either
a rectangle, we saw [INAUDIBLE],

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00:01:05,580 --> 00:01:10,610
and we saw some type
of double integrals,

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00:01:10,610 --> 00:01:17,600
of course that involved
x and y, so-called type

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00:01:17,600 --> 00:01:21,210
1 and type 2
regions, which were--

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so we saw the rectangular case.

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You have ab plus
cd, a rectangle.

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You have what other kind
of a velocity [INAUDIBLE]

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00:01:33,253 --> 00:01:43,400
over the the main of the shape
x between a and be and y.

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You write wild and happy
from bottom to top.

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00:01:49,070 --> 00:01:54,480
That's called the wild--
not wild, the vertical strip

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00:01:54,480 --> 00:01:58,994
method, where y will be
between the bottom function

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00:01:58,994 --> 00:02:02,906
f of x and the top
function f of x.

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And last time I
took examples where

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f and g were both positive, but
remember, you don't have to.

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All you have to have is that
g is always greater than f,

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or equal at some point.

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And then what else do
we have for these cases?

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These are all
continuous functions.

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What else did we have?

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We had two domains.

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Had one and had two.

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Where what was going on,
we have played a little bit

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around with y between c
and d limits with points.

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These are horizontal,
so we take the domain

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as being defined by these
horizontal strips between let's

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say a function.

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Again, I need to rotate my head,
but I didn't do my yoga today,

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so it's a little bit sticky.

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I'll try.

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x equals F of y, and x equals
G of y, assuming, of course,

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that f of y is always greater
than or equal to g of y,

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and the rest of the
apparatus is in place.

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Those are not so
hard to understand.

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00:03:31,680 --> 00:03:33,080
We played around.

51
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We switched the integrals.

52
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We changed the order of
integration from dy dx

53
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to dx dy, so we have
to change the domain.

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We went from
vertical strip method

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to horizontal strip method
or the other way around.

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And for what kind of
example, something

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like that-- I think it
was a leaf like that,

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we said, let's compute
the area or compute

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another kind of double
integral over this leaf in two

60
00:04:09,750 --> 00:04:10,840
different ways.

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And we did it with
vertical strips,

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and we did the same
with horizontal strips.

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So we reversed the
order of integration,

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and we said, I'm having the
double integral over domain

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00:04:27,100 --> 00:04:31,555
of God knows what, f of
xy, continuous function,

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00:04:31,555 --> 00:04:37,440
positive, continuous whenever
you want, and we said da.

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00:04:37,440 --> 00:04:40,160
We didn't quite specify
the meaning of da.

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00:04:40,160 --> 00:04:43,025
We said that da is
the area element,

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00:04:43,025 --> 00:04:47,270
but that sounds a little
bit weird, because it makes

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00:04:47,270 --> 00:04:51,500
you think of surfaces,
and an area element

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00:04:51,500 --> 00:04:53,885
doesn't have to be a
little square in general.

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00:04:53,885 --> 00:04:59,290
It could be something like a
patch on a surface, bounded

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00:04:59,290 --> 00:05:04,100
by two curves within your
segments in each direction.

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00:05:04,100 --> 00:05:06,466
So you think, well, I
don't know what that is.

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00:05:06,466 --> 00:05:07,840
I'll tell you
today what that is.

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It's a mysterious thing,
it's really beautiful,

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and we'll talk about it.

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Now, what did we do last time?

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00:05:15,440 --> 00:05:19,360
We applied the two
theorems that allowed

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00:05:19,360 --> 00:05:23,780
us to do this both ways.

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00:05:23,780 --> 00:05:29,360
Integral from a to b, what was
my usual [? wrist ?] is down,

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00:05:29,360 --> 00:05:32,426
f of x is in g of x, right?

84
00:05:32,426 --> 00:05:36,130


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00:05:36,130 --> 00:05:37,340
dy dx.

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00:05:37,340 --> 00:05:39,750
So if you do it in
this order, it's

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00:05:39,750 --> 00:05:44,780
going to be the same as if
you do it in the other order.

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ab are these guys, and then
this was cd on the y-axis.

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00:05:53,464 --> 00:05:56,810
This is the range between
c and d in altitudes.

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So we have integral from
c to d, integral from,

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00:06:00,740 --> 00:06:02,910
I don't know what they will be.

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00:06:02,910 --> 00:06:07,140
This big guy I'm talking--
which one is the one?

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00:06:07,140 --> 00:06:11,335
This one, that's going to
be called x equals f of y,

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00:06:11,335 --> 00:06:17,596
or g of y, and let's put the
big one G and the smaller one,

95
00:06:17,596 --> 00:06:19,560
x equals F of y.

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00:06:19,560 --> 00:06:24,038
So you have to [? re-denote ?]
these functions,

97
00:06:24,038 --> 00:06:31,430
these inverse functions, and
use them as functions of y.

98
00:06:31,430 --> 00:06:34,640
So it makes sense to
say-- what did we do?

99
00:06:34,640 --> 00:06:39,620
We first integrated respect to
x between two functions of y.

100
00:06:39,620 --> 00:06:44,170
That was the so-called
horizontal strip method, dy.

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00:06:44,170 --> 00:06:48,320
So I have summarized
the ideas from last time

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00:06:48,320 --> 00:06:53,350
that we worked with, generally
with corners x and y.

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We were very happy about them.

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00:06:55,953 --> 00:07:00,050
We had the rectangular
domain, where x was between ab

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00:07:00,050 --> 00:07:01,770
and y was between cd.

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Then we went to type 1, not
diabetes, just type 1 region,

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00:07:05,640 --> 00:07:09,070
type 2, and those
guys are related.

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So if you understood 1 and
understood the other one,

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and if you have a
nice domain like that,

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you can compute the
area or something.

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The area will correspond
to x equals 1.

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So if f is 1, then
that's the area.

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That will also be a
volume of a cylinder based

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on that region with height 1.

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Imagine a can of Coke
that has height 1,

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and-- maybe better,
chocolate cake,

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that has the shape of
this leaf on the bottom,

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and then its height
is 1 everywhere.

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So if you put 1 here, and
you get the area element,

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and then everything
else can be done

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by reversing the order of
integration if f is continuous.

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00:07:59,960 --> 00:08:02,860
But for polar
coordinates, the situation

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has to be reconsidered almost
entirely, because the area

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element, da is called
the area element for us,

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was equal to dx dy for the
cartesian coordinate case.

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And here I'm making a
weird face, I'm weird, no?

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Saying, what am I going
to do, what is this

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going to become for
polar coordinates?

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And now you go, oh my God,
not polar coordinates.

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Those were my
enemies in Calc II.

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Many people told me that.

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And I tried to go
into my time machine

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and go back something
like 25 years ago

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and see how I felt about
them, and I remember that.

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I didn't get them from
the first 48 hours

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after I was exposed to them.

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Therefore, let's
do some preview.

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What were those
polar coordinates?

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Polar coordinates were
a beautiful thing,

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these guys from trig.

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Trig was your friend hopefully.

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And what did we have
in trigonometry?

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In trigonometry, we had
a point on a circle.

146
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This is not the unit
trigonometric circle,

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it's a circle of--
bless you-- radius r.

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00:09:45,205 --> 00:09:49,760
I'm a little bit shifted
by a phase of phi 0.

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So you have a radius r.

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And let's call that little r.

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00:10:02,710 --> 00:10:06,372
And then, we say, OK,
how about the angle?

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00:10:06,372 --> 00:10:08,800
That's the second
polar coordinate.

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00:10:08,800 --> 00:10:16,395
The angle by measuring
from the, what

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is this called, the x-axis.

156
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Origin, x-axis, o, x,
going counterclockwise,

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because we are mathemeticians.

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Every normal person, when
they mix into a bowl,

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they mix like that.

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Well, I've seen that
most of my colleagues--

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this is just a
psychological test, OK?

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I wanted to see
how they mix when

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they cook, or mix
up-- most of them

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mix in a trigonometric sense.

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I don't know if this has
anything to do with the brain

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00:10:47,610 --> 00:10:51,366
connections, but I think
that's [? kind of weird. ?]

168
00:10:51,366 --> 00:10:54,550
I don't have a statistical
result, but most of the people

169
00:10:54,550 --> 00:10:58,590
I've seen that, and do
mathematics, mix like that.

170
00:10:58,590 --> 00:11:02,530
So trigonometric sense.

171
00:11:02,530 --> 00:11:09,010
What is the connection with the
actual Cartesian coordinates?

172
00:11:09,010 --> 00:11:13,650
D you know what Cartesian
comes from as a word?

173
00:11:13,650 --> 00:11:15,766
Cartesian, that sounds weird.

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00:11:15,766 --> 00:11:17,242
STUDENT: From Descartes.

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MAGDALENA TODA: Exactly.

176
00:11:18,242 --> 00:11:19,702
Who said that?

177
00:11:19,702 --> 00:11:21,178
Roberto, thank you so much.

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I'm impressed.

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Descartes was--

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00:11:22,990 --> 00:11:23,656
STUDENT: French.

181
00:11:23,656 --> 00:11:26,010
MAGDALENA TODA: --a
French mathematician.

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00:11:26,010 --> 00:11:28,980
But actually, I mean,
he was everything.

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00:11:28,980 --> 00:11:30,912
He was a crazy lunatic.

184
00:11:30,912 --> 00:11:34,780
He was a philosopher,
a mathematician,

185
00:11:34,780 --> 00:11:37,360
a scientist in general.

186
00:11:37,360 --> 00:11:40,850
He also knew a lot
about life science.

187
00:11:40,850 --> 00:11:43,790
But at the time, I don't
know if this is true.

188
00:11:43,790 --> 00:11:45,970
I should check with wiki,
or whoever can tell me.

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00:11:45,970 --> 00:11:50,440
One of my professors in college
told me that at that time,

190
00:11:50,440 --> 00:11:53,240
there was a fashion
that people would

191
00:11:53,240 --> 00:11:57,060
change their names like they
do on Facebook nowadays.

192
00:11:57,060 --> 00:11:59,980
So they and change their
name from Francesca

193
00:11:59,980 --> 00:12:04,780
to Frenchy, from Roberto
to Robby, from-- so

194
00:12:04,780 --> 00:12:08,510
if they would have to
clean up Facebook and see

195
00:12:08,510 --> 00:12:14,800
how many names correspond to
the ID, I think less than 20%.

196
00:12:14,800 --> 00:12:16,860
At that time it was the same.

197
00:12:16,860 --> 00:12:22,780
All of the scientists loved
to romanize their names.

198
00:12:22,780 --> 00:12:26,150
And of course he was
of a romance language,

199
00:12:26,150 --> 00:12:30,370
but he said, what if I
made my name a Latin name,

200
00:12:30,370 --> 00:12:32,250
I changed my name
into a Latin name.

201
00:12:32,250 --> 00:12:36,640
So he himself, this is what
my professor told me, he

202
00:12:36,640 --> 00:12:39,785
himself changed his
name to Cartesius.

203
00:12:39,785 --> 00:12:45,513
"Car-teh-see-yus" actually, in
Latin, the way it should be.

204
00:12:45,513 --> 00:12:49,880


205
00:12:49,880 --> 00:12:52,200
OK, very smart guy.

206
00:12:52,200 --> 00:12:56,800
Now, when we look
a x and y, there

207
00:12:56,800 --> 00:13:04,400
has to be a connection between
x, y as the couple, and r theta

208
00:13:04,400 --> 00:13:09,230
as the same-- I mean a
couple, not the couple,

209
00:13:09,230 --> 00:13:10,691
for the same point.

210
00:13:10,691 --> 00:13:11,190
Yes, sir?

211
00:13:11,190 --> 00:13:11,981
STUDENT: Cartesius.

212
00:13:11,981 --> 00:13:14,751
Like meaning flat?

213
00:13:14,751 --> 00:13:15,250
The name?

214
00:13:15,250 --> 00:13:17,416
MAGDALENA TODA: These are
the Cartesian coordinates,

215
00:13:17,416 --> 00:13:20,130
and it sounds like the word map.

216
00:13:20,130 --> 00:13:22,016
I think he had meant

217
00:13:22,016 --> 00:13:23,640
STUDENT: Because the
meaning of carte--

218
00:13:23,640 --> 00:13:24,760
STUDENT: But look, look.

219
00:13:24,760 --> 00:13:27,950
Descartes means from the map.

220
00:13:27,950 --> 00:13:30,250
From the books, or from the map.

221
00:13:30,250 --> 00:13:33,350
So he thought what his
name would really mean,

222
00:13:33,350 --> 00:13:36,260
and so he recalled himself.

223
00:13:36,260 --> 00:13:39,000
There was no fun, no
Twitter, no Facebook.

224
00:13:39,000 --> 00:13:43,550
So they had to do something
to enjoy themselves.

225
00:13:43,550 --> 00:13:46,175
Now, when it comes
to these triangles,

226
00:13:46,175 --> 00:13:49,780
we have to think of the
relationship between x, y

227
00:13:49,780 --> 00:13:52,510
and r, theta.

228
00:13:52,510 --> 00:13:56,160
And could somebody tell me what
the relationship between x, y

229
00:13:56,160 --> 00:13:59,240
and r, theta is?

230
00:13:59,240 --> 00:14:01,261
x represents

231
00:14:01,261 --> 00:14:02,610
STUDENT: R cosine theta.

232
00:14:02,610 --> 00:14:05,260
STUDENT: r cosine
theta, who says that?

233
00:14:05,260 --> 00:14:07,900
Trigonometry taught us
that, because that's

234
00:14:07,900 --> 00:14:14,390
the adjacent side over
the hypotenuse for cosine.

235
00:14:14,390 --> 00:14:18,250
In terms of sine, you
know what you have,

236
00:14:18,250 --> 00:14:22,586
so you're going to have
y equals r sine theta,

237
00:14:22,586 --> 00:14:26,740
and we have to decide
if x and y are allowed

238
00:14:26,740 --> 00:14:28,200
to be anywhere in plane.

239
00:14:28,200 --> 00:14:31,160


240
00:14:31,160 --> 00:14:35,470
For the plane,
I'll also write r2.

241
00:14:35,470 --> 00:14:40,830
R2, not R2 from the movie,
just r2 is the plane,

242
00:14:40,830 --> 00:14:44,100
and r3 is the space,
the [? intriguing ?]

243
00:14:44,100 --> 00:14:46,850
space, three-dimensional one.

244
00:14:46,850 --> 00:14:50,870
r theta, is a couple where?

245
00:14:50,870 --> 00:14:52,440
That's a little bit tricky.

246
00:14:52,440 --> 00:14:54,120
We have to make a restriction.

247
00:14:54,120 --> 00:14:59,340
We allow r to be anywhere
between 0 and infinity.

248
00:14:59,340 --> 00:15:03,950
So it has to be a
positive number.

249
00:15:03,950 --> 00:15:13,050
And theta [INTERPOSING VOICES]
between 0 and 2 pi.

250
00:15:13,050 --> 00:15:14,900
STUDENT: I've been
sick since Tuesday.

251
00:15:14,900 --> 00:15:16,400
MAGDALENA TODA: I
believe you, Ryan.

252
00:15:16,400 --> 00:15:17,640
You sound sick to me.

253
00:15:17,640 --> 00:15:20,780
Take your viruses away from me.

254
00:15:20,780 --> 00:15:21,650
Take the germs away.

255
00:15:21,650 --> 00:15:25,315
I don't even have
the-- I'm kidding,

256
00:15:25,315 --> 00:15:27,746
Alex, I hope you
don't get offended.

257
00:15:27,746 --> 00:15:31,529
So, I hope this works this time.

258
00:15:31,529 --> 00:15:33,070
I'm making a
sarcastic-- it's really,

259
00:15:33,070 --> 00:15:34,522
I hope you're feeling better.

260
00:15:34,522 --> 00:15:35,974
I'm sorry about that.

261
00:15:35,974 --> 00:15:38,900


262
00:15:38,900 --> 00:15:41,760
So you haven't missed much.

263
00:15:41,760 --> 00:15:42,630
Only the jokes.

264
00:15:42,630 --> 00:15:46,730
So x equals r cosine theta,
y equals r sine theta.

265
00:15:46,730 --> 00:15:49,640
Is that your
favorite change that

266
00:15:49,640 --> 00:15:55,995
was a differential
mapping from the set x,

267
00:15:55,995 --> 00:15:58,806
y to the set r,
theta back and forth.

268
00:15:58,806 --> 00:16:02,300


269
00:16:02,300 --> 00:16:05,380
And you are going
to probably say, OK

270
00:16:05,380 --> 00:16:08,330
how do you denote such a map?

271
00:16:08,330 --> 00:16:11,460
I mean, going from x,
y to r, theta and back,

272
00:16:11,460 --> 00:16:14,690
let's suppose that we go
from r, theta to x, y,

273
00:16:14,690 --> 00:16:17,280
and that's going to be a big if.

274
00:16:17,280 --> 00:16:20,430
And going backwards is going
to be the inverse mapping.

275
00:16:20,430 --> 00:16:23,710
So I'm going to
call it f inverse.

276
00:16:23,710 --> 00:16:30,770
So that's a map from a couple
to another couple of number.

277
00:16:30,770 --> 00:16:35,960
And you say, OK, but
why is that a map?

278
00:16:35,960 --> 00:16:38,260
All right, guys,
now let me tell you.

279
00:16:38,260 --> 00:16:43,326
So x, you can do x as a
function of r, theta, right?

280
00:16:43,326 --> 00:16:45,750
It is a function of r and theta.

281
00:16:45,750 --> 00:16:48,420
It's a function
of two variables.

282
00:16:48,420 --> 00:16:51,960
And y is a function
of r and theta.

283
00:16:51,960 --> 00:16:53,735
It's another function
of two variables.

284
00:16:53,735 --> 00:16:58,170
They are both nice
and differentiable.

285
00:16:58,170 --> 00:17:02,710
We assume not only that
they are differentiable,

286
00:17:02,710 --> 00:17:07,356
but the partial derivatives
will be continuous.

287
00:17:07,356 --> 00:17:10,640
So it's really
nice as a mapping.

288
00:17:10,640 --> 00:17:14,660
And you think, could I
write the chain rule?

289
00:17:14,660 --> 00:17:16,232
That is the whole idea.

290
00:17:16,232 --> 00:17:18,040
What is the meaning
of differential?

291
00:17:18,040 --> 00:17:20,050
dx differential dy.

292
00:17:20,050 --> 00:17:23,390
Since I was chatting with
you, once, [? Yuniel ?],

293
00:17:23,390 --> 00:17:28,600
and you asked me to
help you with homework,

294
00:17:28,600 --> 00:17:31,480
I had to go over
differential again.

295
00:17:31,480 --> 00:17:36,350
If you were to define,
like Mr. Leibniz did,

296
00:17:36,350 --> 00:17:39,930
the differential of the
function x with respect

297
00:17:39,930 --> 00:17:44,060
to both variables, that
was the sum, right?

298
00:17:44,060 --> 00:17:45,450
You've done that
in the homework,

299
00:17:45,450 --> 00:17:46,720
it's fresh in your mind.

300
00:17:46,720 --> 00:17:53,505
So you get x sub r,
dr, plus f x sub what?

301
00:17:53,505 --> 00:17:54,130
STUDENT: Theta.

302
00:17:54,130 --> 00:17:56,940
MAGDALENA TODA:
Sub theta d-theta.

303
00:17:56,940 --> 00:18:01,740
And somebody asked me,
what if I see skip the dr?

304
00:18:01,740 --> 00:18:02,490
No, don't do that.

305
00:18:02,490 --> 00:18:05,205
First of all, WeBWorK is not
going to take the answer.

306
00:18:05,205 --> 00:18:09,310
But second of all, the
most important stuff

307
00:18:09,310 --> 00:18:13,190
here to remember is that these
are small, infinitesimally

308
00:18:13,190 --> 00:18:15,190
small, displacements.

309
00:18:15,190 --> 00:18:32,130
Infinitesimally small
displacements in the directions

310
00:18:32,130 --> 00:18:33,990
x and y, respectively.

311
00:18:33,990 --> 00:18:37,435
So you would say, what does
that mean, infinitesimally?

312
00:18:37,435 --> 00:18:39,790
It doesn't mean delta-x small.

313
00:18:39,790 --> 00:18:43,950
Delta-x small would be like
me driving 7 feet, when

314
00:18:43,950 --> 00:18:48,920
I know I have to drive fast to
Amarillo to be there in 1 hour.

315
00:18:48,920 --> 00:18:49,655
Well, OK.

316
00:18:49,655 --> 00:18:51,600
Don't tell anybody.

317
00:18:51,600 --> 00:18:55,160
But, it's about 2 hours, right?

318
00:18:55,160 --> 00:18:58,230
So I cannot be there in an hour.

319
00:18:58,230 --> 00:19:01,940
But driving those seven
feet is like a delta x.

320
00:19:01,940 --> 00:19:07,100
Imagine, however, me
measuring that speed of mine

321
00:19:07,100 --> 00:19:10,130
in a much smaller
fraction of a second.

322
00:19:10,130 --> 00:19:15,999
So shrink that time to
something infinitesimally small,

323
00:19:15,999 --> 00:19:17,450
which is what you have here.

324
00:19:17,450 --> 00:19:19,206
That kind of quantity.

325
00:19:19,206 --> 00:19:25,440
And dy will be y sub r dr
plus y sub theta d-theta.

326
00:19:25,440 --> 00:19:28,750


327
00:19:28,750 --> 00:19:32,718
And now, I'm not going
to go by the book.

328
00:19:32,718 --> 00:19:34,560
I'm going to go
a little bit more

329
00:19:34,560 --> 00:19:39,630
in depth, because in the book--
First of all, let me tell you,

330
00:19:39,630 --> 00:19:43,870
if I went by the book,
what I would come with.

331
00:19:43,870 --> 00:19:48,790
And of course the way
we teach mathematics

332
00:19:48,790 --> 00:19:52,430
all through K-12 and through
college is swallow this theorem

333
00:19:52,430 --> 00:19:53,810
and believe it.

334
00:19:53,810 --> 00:19:58,910
So practically you accept
whatever we give you

335
00:19:58,910 --> 00:20:02,120
without controlling it, without
checking if we're right,

336
00:20:02,120 --> 00:20:05,151
without trying to prove it.

337
00:20:05,151 --> 00:20:06,650
Practically, the
theorem in the book

338
00:20:06,650 --> 00:20:09,300
says that if you
have a bunch of x,

339
00:20:09,300 --> 00:20:14,400
y that is continuous
over a domain, D,

340
00:20:14,400 --> 00:20:21,322
and you do change
the variables over--

341
00:20:21,322 --> 00:20:22,714
STUDENT: I forgot my glasses.

342
00:20:22,714 --> 00:20:25,040
So I'm going to sit very close.

343
00:20:25,040 --> 00:20:28,570
MAGDALENA TODA:
What do you wear?

344
00:20:28,570 --> 00:20:30,632
What [INAUDIBLE]?

345
00:20:30,632 --> 00:20:31,840
STUDENT: I couldn't tell you.

346
00:20:31,840 --> 00:20:33,104
I can see from here.

347
00:20:33,104 --> 00:20:33,568
MAGDALENA TODA: You can?

348
00:20:33,568 --> 00:20:34,032
STUDENT: Yeah.

349
00:20:34,032 --> 00:20:35,073
My vision's not terrible.

350
00:20:35,073 --> 00:20:41,602
MAGDALENA TODA: All
right. f of x, y da.

351
00:20:41,602 --> 00:20:47,120
If I change this da
as dx dy, let's say,

352
00:20:47,120 --> 00:20:49,820
to a perspective
of something else

353
00:20:49,820 --> 00:20:52,470
in terms of polar
coordinates, then

354
00:20:52,470 --> 00:20:57,380
the integral I'm going to get is
over the corresponding domain D

355
00:20:57,380 --> 00:21:00,530
star, whatever that would be.

356
00:21:00,530 --> 00:21:06,480
Then I'm going to have f of
x of r theta, y of r theta,

357
00:21:06,480 --> 00:21:09,870
everything expressed
in terms of r theta.

358
00:21:09,870 --> 00:21:13,990
And instead of
the a-- so we just

359
00:21:13,990 --> 00:21:18,496
feed you this piece of
cake and say, believe it,

360
00:21:18,496 --> 00:21:21,932
believe it and leave us alone.

361
00:21:21,932 --> 00:21:22,432
OK?

362
00:21:22,432 --> 00:21:26,780
That's what it does in
the book in section 11.3.

363
00:21:26,780 --> 00:21:33,450
So without understanding why
you have to-- instead of the r

364
00:21:33,450 --> 00:21:35,790
d theta and multiply it by an r.

365
00:21:35,790 --> 00:21:36,480
What is that?

366
00:21:36,480 --> 00:21:38,106
You don't know why you do that.

367
00:21:38,106 --> 00:21:40,520
And I thought, that's
the way we thought it

368
00:21:40,520 --> 00:21:42,580
for way too many years.

369
00:21:42,580 --> 00:21:45,920
I'm sick and tired
of not explaining why

370
00:21:45,920 --> 00:21:50,740
you multiply that with an r.

371
00:21:50,740 --> 00:21:55,100
So I will tell you something
that's quite interesting,

372
00:21:55,100 --> 00:21:58,290
and something that I learned
late in graduate school.

373
00:21:58,290 --> 00:22:00,630
I was late already.

374
00:22:00,630 --> 00:22:05,790
I was in my 20s when I
studied differential forms

375
00:22:05,790 --> 00:22:07,586
for the first time.

376
00:22:07,586 --> 00:22:12,880
And differential
forms have some sort

377
00:22:12,880 --> 00:22:25,060
of special wedge product, which
is very physical in nature.

378
00:22:25,060 --> 00:22:30,264
So if you love physics, you
will understand more or less

379
00:22:30,264 --> 00:22:34,010
what I'm talking about.

380
00:22:34,010 --> 00:22:40,660
Imagine that you have two
vectors, vector a and vector b.

381
00:22:40,660 --> 00:22:44,170


382
00:22:44,170 --> 00:22:48,250
For these vectors,
you go, oh my God.

383
00:22:48,250 --> 00:22:54,086
If these would be vectors in
a tangent plane to a surface,

384
00:22:54,086 --> 00:22:56,395
you think, some
of these would be

385
00:22:56,395 --> 00:22:59,980
tangent vectors to a surface.

386
00:22:59,980 --> 00:23:02,380
This is the tangent
plane and everything.

387
00:23:02,380 --> 00:23:07,030
You go, OK, if these
were infinitesimally

388
00:23:07,030 --> 00:23:11,630
small displacements-- which they
are not, but assume they would

389
00:23:11,630 --> 00:23:19,490
be-- how would you do the area
of the infinitesimally small

390
00:23:19,490 --> 00:23:22,370
parallelogram that
they have between them.

391
00:23:22,370 --> 00:23:31,166
This is actually the area
element right here, ea.

392
00:23:31,166 --> 00:23:35,150
So instead of dx dy, you're
not going to have dx dy,

393
00:23:35,150 --> 00:23:39,998
you're going to have some
sort of, I don't know,

394
00:23:39,998 --> 00:23:48,170
this is like a
d-something, d u, and this

395
00:23:48,170 --> 00:23:55,180
is a d v. And when I compute
the area of the parallelogram,

396
00:23:55,180 --> 00:23:58,120
I consider these to
be vectors, and I

397
00:23:58,120 --> 00:24:02,180
say, how did we get
it from the vectors

398
00:24:02,180 --> 00:24:06,259
to the area of
the parallelogram?

399
00:24:06,259 --> 00:24:10,060
We took the vectors,
we shook them off.

400
00:24:10,060 --> 00:24:19,320
We made a cross product
of them, and then we

401
00:24:19,320 --> 00:24:23,370
took the norm, the
magnitude of that.

402
00:24:23,370 --> 00:24:26,812
Does this makes sense,
compared to this parallelogram?

403
00:24:26,812 --> 00:24:27,311
Yeah.

404
00:24:27,311 --> 00:24:30,550
Remember, guys, this
was like, how big

405
00:24:30,550 --> 00:24:33,170
is du, a small
infinitesimal displacement,

406
00:24:33,170 --> 00:24:36,340
but that would be like the
width, one of the dimensions.

407
00:24:36,340 --> 00:24:39,994
There's the other of the
dimension of the area element

408
00:24:39,994 --> 00:24:44,050
times-- this area element
is that tiny pixel that

409
00:24:44,050 --> 00:24:49,010
is sitting on the surface
in the tangent plane, yeah?

410
00:24:49,010 --> 00:24:54,220
Sine of the angle
between the guys.

411
00:24:54,220 --> 00:24:54,760
Oh, OK.

412
00:24:54,760 --> 00:25:00,760
So if the guys are not
perpendicular to one another,

413
00:25:00,760 --> 00:25:03,730
if the two displacements are not
perpendicular to one another,

414
00:25:03,730 --> 00:25:07,340
you still have to multiply
the sine of theta.

415
00:25:07,340 --> 00:25:09,187
Otherwise you don't
get the element

416
00:25:09,187 --> 00:25:12,320
of the area of
this parallelogram.

417
00:25:12,320 --> 00:25:17,530
So why did the Cartesian
coordinates not pose a problem?

418
00:25:17,530 --> 00:25:19,560
For Cartesian
coordinates, it's easy.

419
00:25:19,560 --> 00:25:22,615


420
00:25:22,615 --> 00:25:23,490
It's a piece of cake.

421
00:25:23,490 --> 00:25:24,365
Why?

422
00:25:24,365 --> 00:25:32,130
Because this is the x, this is
the y, as little tiny measures

423
00:25:32,130 --> 00:25:33,330
multiplied.

424
00:25:33,330 --> 00:25:37,380
How much is sine of theta
between Cartesian coordinates?

425
00:25:37,380 --> 00:25:37,880
STUDENT: 1.

426
00:25:37,880 --> 00:25:40,970
MAGDALENA TODA: It's 1,
because its 90 degrees.

427
00:25:40,970 --> 00:25:43,160
When they are
orthogonal coordinates,

428
00:25:43,160 --> 00:25:46,884
it's a piece of cake,
because you have 1 there,

429
00:25:46,884 --> 00:25:48,300
and then your life
becomes easier.

430
00:25:48,300 --> 00:25:50,940


431
00:25:50,940 --> 00:25:57,030
So in general, what
is the area limit?

432
00:25:57,030 --> 00:26:02,026
The area limit for
arbitrary coordinates--

433
00:26:02,026 --> 00:26:17,020
So area limit for some
arbitrary coordinates

434
00:26:17,020 --> 00:26:20,310
should be defined
as the sined area.

435
00:26:20,310 --> 00:26:29,320


436
00:26:29,320 --> 00:26:32,190
And you say, what do you
mean that's a sined area,

437
00:26:32,190 --> 00:26:34,580
and why would you do that.?

438
00:26:34,580 --> 00:26:38,280
Well, it's not so
hard to understand.

439
00:26:38,280 --> 00:26:41,740
Imagine that you have a
convention, and you say,

440
00:26:41,740 --> 00:26:54,810
OK, dx times dy equals
negative dy times dx.

441
00:26:54,810 --> 00:26:56,920
And you say, what, what?

442
00:26:56,920 --> 00:27:00,520
If you change the
order of dx dy,

443
00:27:00,520 --> 00:27:06,597
this wedge stuff works exactly
like the-- what is that called?

444
00:27:06,597 --> 00:27:07,790
Cross product.

445
00:27:07,790 --> 00:27:13,150
So the wedge works just
like the cross product.

446
00:27:13,150 --> 00:27:17,509
Just like the cross product.

447
00:27:17,509 --> 00:27:23,320
In some other ways, suppose
that I am here, right?

448
00:27:23,320 --> 00:27:27,720
And this is a vector, like an
infinitesimal displacement,

449
00:27:27,720 --> 00:27:29,370
and that's the other one.

450
00:27:29,370 --> 00:27:33,800
If I multiply them
one after the other,

451
00:27:33,800 --> 00:27:38,060
and I use this strange wedge
[INTERPOSING VOICES] the area,

452
00:27:38,060 --> 00:27:40,970
I'm going to have an orientation
for that tangent line,

453
00:27:40,970 --> 00:27:46,390
and it's going to go
up, the orientation.

454
00:27:46,390 --> 00:27:48,330
The orientation is important.

455
00:27:48,330 --> 00:27:50,990
But if dx dy and
I switched them,

456
00:27:50,990 --> 00:27:56,050
I said, dy, swap with dx,
what's going to happen?

457
00:27:56,050 --> 00:28:01,530
I have to change to
change to clockwise.

458
00:28:01,530 --> 00:28:03,610
And then the
orientation goes down.

459
00:28:03,610 --> 00:28:06,720
And that's what they use
in mechanics when it comes

460
00:28:06,720 --> 00:28:09,130
to the normal to the surface.

461
00:28:09,130 --> 00:28:12,773
So again, you guys remember,
we had 2 vector products,

462
00:28:12,773 --> 00:28:16,370
and we did the cross product,
and we got the normal.

463
00:28:16,370 --> 00:28:18,795
If it's from this
one to this one,

464
00:28:18,795 --> 00:28:20,740
it's counterclockwise
and goes up,

465
00:28:20,740 --> 00:28:23,855
but if it's from this
vector to this other vector,

466
00:28:23,855 --> 00:28:26,990
it's clockwise and goes down.

467
00:28:26,990 --> 00:28:29,820
This is how a
mechanical engineer

468
00:28:29,820 --> 00:28:32,820
will know how the
surface is oriented

469
00:28:32,820 --> 00:28:35,730
based on the partial
velocities, for example

470
00:28:35,730 --> 00:28:39,370
He has the partial
velocities along a surface,

471
00:28:39,370 --> 00:28:42,750
and somebody says, take the
normal, take the unit normal.

472
00:28:42,750 --> 00:28:44,760
He goes, like, are
you a physicist?

473
00:28:44,760 --> 00:28:46,440
No, I'm an engineer.

474
00:28:46,440 --> 00:28:48,790
You don't know how
to take the normal.

475
00:28:48,790 --> 00:28:50,110
And of course, he knows.

476
00:28:50,110 --> 00:28:53,500
He knows the convention
by this right-hand rule,

477
00:28:53,500 --> 00:28:55,190
whatever you guys call it.

478
00:28:55,190 --> 00:28:57,260
I call it the faucet rule.

479
00:28:57,260 --> 00:29:01,400
It goes like this,
or it goes like that.

480
00:29:01,400 --> 00:29:04,272
It's the same for a faucet,
for any type of screw,

481
00:29:04,272 --> 00:29:08,350
for the right-hand
rule, whatever.

482
00:29:08,350 --> 00:29:11,360
What else do you have
to believe me are true?

483
00:29:11,360 --> 00:29:14,560
dx wedge dx is 0.

484
00:29:14,560 --> 00:29:17,740
Can somebody tell me why
that is natural to introduce

485
00:29:17,740 --> 00:29:19,490
such a wedge product?

486
00:29:19,490 --> 00:29:22,364
STUDENT: Because the sine of
the angle between those is 0.

487
00:29:22,364 --> 00:29:23,280
MAGDALENA TODA: Right.

488
00:29:23,280 --> 00:29:28,660
Once you flatten this, once
you flatten the parallelogram,

489
00:29:28,660 --> 00:29:29,830
there is no area.

490
00:29:29,830 --> 00:29:31,470
So the area is 0.

491
00:29:31,470 --> 00:29:34,960
How about dy dy sined area?

492
00:29:34,960 --> 00:29:35,930
0.

493
00:29:35,930 --> 00:29:37,810
So these are all
the properties you

494
00:29:37,810 --> 00:29:41,441
need to know of the
sine area, sined areas.

495
00:29:41,441 --> 00:29:44,270


496
00:29:44,270 --> 00:29:46,530
OK, so now let's
see what happens

497
00:29:46,530 --> 00:29:51,150
if we take this element,
which is a differential,

498
00:29:51,150 --> 00:29:55,350
and wedge it with this element,
which is also a differential.

499
00:29:55,350 --> 00:29:56,370
OK.

500
00:29:56,370 --> 00:29:59,920
Oh my God, I'm shaking
only thinking about it.

501
00:29:59,920 --> 00:30:01,860
I'm going to get
something weird.

502
00:30:01,860 --> 00:30:04,070
But I mean, mad weird.

503
00:30:04,070 --> 00:30:06,663
Let's see what happens.

504
00:30:06,663 --> 00:30:13,960
dx wedge dy equals-- do
you guys have questions?

505
00:30:13,960 --> 00:30:18,334
Let's see what the mechanics are
for this type of computation.

506
00:30:18,334 --> 00:30:21,250


507
00:30:21,250 --> 00:30:27,690
I go-- this is like
a-- displacement wedge

508
00:30:27,690 --> 00:30:29,590
this other displacement.

509
00:30:29,590 --> 00:30:32,736


510
00:30:32,736 --> 00:30:36,110
Think of them as true
vector displacements,

511
00:30:36,110 --> 00:30:41,150
and as if you had a cross
product, or something.

512
00:30:41,150 --> 00:30:42,080
OK.

513
00:30:42,080 --> 00:30:43,657
How does this go?

514
00:30:43,657 --> 00:30:44,820
It's distributed.

515
00:30:44,820 --> 00:30:47,770
It's linear functions,
because we've

516
00:30:47,770 --> 00:30:51,140
studied the
properties of vectors,

517
00:30:51,140 --> 00:30:52,740
this acts by linearity.

518
00:30:52,740 --> 00:30:58,182
So you go and say, first
first, times plus first times

519
00:30:58,182 --> 00:31:02,640
second-- and times is
this guy, this weirdo--

520
00:31:02,640 --> 00:31:06,940
plus second times first,
plus second times second,

521
00:31:06,940 --> 00:31:09,200
where the wedge is
the operator that

522
00:31:09,200 --> 00:31:11,280
has to satisfy these functions.

523
00:31:11,280 --> 00:31:14,060
It's similar to
the cross product.

524
00:31:14,060 --> 00:31:15,190
OK.

525
00:31:15,190 --> 00:31:21,370
Then let's go x
sub r, y sub r, dr

526
00:31:21,370 --> 00:31:26,880
wedge dr. Oh, let's 0 go away.

527
00:31:26,880 --> 00:31:30,340
I say, leave me alone,
you're making my life hard.

528
00:31:30,340 --> 00:31:37,690
Then I go plus x sub r--
this is a small function.

529
00:31:37,690 --> 00:31:40,520
y sub theta, another
small function.

530
00:31:40,520 --> 00:31:44,050
What of this
displacement, dr d theta.

531
00:31:44,050 --> 00:31:46,650
I'm like those d
something, d something,

532
00:31:46,650 --> 00:31:49,350
two small displacements
in the cross product.

533
00:31:49,350 --> 00:31:52,620
OK, plus.

534
00:31:52,620 --> 00:31:55,271
Who is telling me what next?

535
00:31:55,271 --> 00:31:56,020
STUDENT: x theta--

536
00:31:56,020 --> 00:32:05,550
MAGDALENA TODA: x theta
yr, d theta dr. Is it fair?

537
00:32:05,550 --> 00:32:10,200
I did the second guy from the
first one with the first guy

538
00:32:10,200 --> 00:32:11,790
from the second one.

539
00:32:11,790 --> 00:32:14,720
And finally, I'm too
lazy to write it down.

540
00:32:14,720 --> 00:32:15,964
What do I get?

541
00:32:15,964 --> 00:32:16,860
STUDENT: 0.

542
00:32:16,860 --> 00:32:17,020
MAGDALENA TODA: 0.

543
00:32:17,020 --> 00:32:17,680
Why is that?

544
00:32:17,680 --> 00:32:20,070
Because d theta,
always d theta is 0.

545
00:32:20,070 --> 00:32:27,160
It's like you are flattening--
there is no more parallelogram.

546
00:32:27,160 --> 00:32:27,940
OK?

547
00:32:27,940 --> 00:32:32,330
So the two dimensions of
the parallelogram become 0.

548
00:32:32,330 --> 00:32:37,070
The parallelogram would
become [? a secant. ?]

549
00:32:37,070 --> 00:32:39,931
What you get is
something really weak.

550
00:32:39,931 --> 00:32:42,210
And we don't talk
about it in the book,

551
00:32:42,210 --> 00:32:45,022
but that's called the Jacobian.

552
00:32:45,022 --> 00:32:51,150
dr d theta and d theta dr, once
you introduce the sine area,

553
00:32:51,150 --> 00:32:55,920
you finally understand
why you get this r here,

554
00:32:55,920 --> 00:32:57,740
what the Jacobian is.

555
00:32:57,740 --> 00:32:59,370
If you don't introduce
the sine area,

556
00:32:59,370 --> 00:33:02,340
you will never understand,
and you cannot explain it

557
00:33:02,340 --> 00:33:06,140
to anybody, any student have.

558
00:33:06,140 --> 00:33:11,530
OK, so this guy, d theta,
which the r is just

559
00:33:11,530 --> 00:33:13,686
swapping the two displacements.

560
00:33:13,686 --> 00:33:16,990
So it's going to be
minus dr d theta.

561
00:33:16,990 --> 00:33:18,670
Why is that, guys?

562
00:33:18,670 --> 00:33:23,030
Because that's how I said, every
time I swap two displacements,

563
00:33:23,030 --> 00:33:25,440
I'm changing the orientation.

564
00:33:25,440 --> 00:33:28,085
It's like the cross
product between a and b,

565
00:33:28,085 --> 00:33:30,080
and the cross product
between b and a.

566
00:33:30,080 --> 00:33:34,520
So I'm going up or I'm going
down, I'm changing orientation.

567
00:33:34,520 --> 00:33:35,900
What's left in the end?

568
00:33:35,900 --> 00:33:39,000
It's really just this
guy that's really weird.

569
00:33:39,000 --> 00:33:41,286
I'm going to collect the terms.

570
00:33:41,286 --> 00:33:44,930
One from here, one
from here, and a minus.

571
00:33:44,930 --> 00:33:45,430
Go ahead.

572
00:33:45,430 --> 00:33:49,030
STUDENT: Do the wedges
just cancel out?

573
00:33:49,030 --> 00:33:50,350
MAGDALENA TODA: This was 0.

574
00:33:50,350 --> 00:33:52,350
This was 0.

575
00:33:52,350 --> 00:33:57,580
And this dr d theta is nonzero,
but is the common factor.

576
00:33:57,580 --> 00:34:00,015
So I pull him out from here.

577
00:34:00,015 --> 00:34:01,890
I pull him out from here.

578
00:34:01,890 --> 00:34:02,390
Out.

579
00:34:02,390 --> 00:34:08,469
Factor out, and what's
left is this guy over here

580
00:34:08,469 --> 00:34:10,561
who is this guy over here.

581
00:34:10,561 --> 00:34:14,576
And this guy over
here with a minus

582
00:34:14,576 --> 00:34:20,320
who gives me minus d theta yr.

583
00:34:20,320 --> 00:34:21,000
That's all.

584
00:34:21,000 --> 00:34:25,270
So now you will understand
why I am going to get r.

585
00:34:25,270 --> 00:34:30,440
So the general rule will
be that the area element dx

586
00:34:30,440 --> 00:34:35,860
dy, the wedge sined
area, will be--

587
00:34:35,860 --> 00:34:39,210
you have to help me
with this individual,

588
00:34:39,210 --> 00:34:42,989
because he really looks weird.

589
00:34:42,989 --> 00:34:46,480
Do you know of a name for it?

590
00:34:46,480 --> 00:34:49,909
Do you know what
this is going to be?

591
00:34:49,909 --> 00:34:52,400
Linear algebra people,
only two of you.

592
00:34:52,400 --> 00:34:56,650
Maybe you have an idea.

593
00:34:56,650 --> 00:34:59,950
So it's like, I
take this fellow,

594
00:34:59,950 --> 00:35:01,820
and I multiply by that fellow.

595
00:35:01,820 --> 00:35:04,496


596
00:35:04,496 --> 00:35:06,550
Multiply these two.

597
00:35:06,550 --> 00:35:12,970
And I go minus this
fellow times that fellow.

598
00:35:12,970 --> 00:35:14,820
STUDENT: [INAUDIBLE]

599
00:35:14,820 --> 00:35:17,590
MAGDALENA TODA: It's like
a determinant of something.

600
00:35:17,590 --> 00:35:23,380
So when people write
the differential system,

601
00:35:23,380 --> 00:35:26,460
[INTERPOSING VOICES]
51, you will understand

602
00:35:26,460 --> 00:35:27,940
that this is a system.

603
00:35:27,940 --> 00:35:28,440
OK?

604
00:35:28,440 --> 00:35:29,935
It's a system of two equations.

605
00:35:29,935 --> 00:35:32,155


606
00:35:32,155 --> 00:35:34,030
The other little, like,
vector displacements,

607
00:35:34,030 --> 00:35:36,370
you are going to
write it like that.

608
00:35:36,370 --> 00:35:45,950
dx dy will be matrix
multiplication dr d theta.

609
00:35:45,950 --> 00:35:50,210
And how do you multiply
x sub r x sub theta?

610
00:35:50,210 --> 00:35:55,190
So you go first row times
first column give you that.

611
00:35:55,190 --> 00:35:59,510
And second row times the
column gives you this.

612
00:35:59,510 --> 00:36:02,150
y sub r, y sub theta.

613
00:36:02,150 --> 00:36:06,340
This is a magic guy
called Jacobian.

614
00:36:06,340 --> 00:36:09,880
We keep this a secret, and
most Professors don't even

615
00:36:09,880 --> 00:36:13,050
cover 12.8, because
they don't want to tell

616
00:36:13,050 --> 00:36:15,060
people what a Jacobian is.

617
00:36:15,060 --> 00:36:16,890
This is little r.

618
00:36:16,890 --> 00:36:20,855
I know you don't believe me, but
the determinant of this matrix

619
00:36:20,855 --> 00:36:22,520
must be little r.

620
00:36:22,520 --> 00:36:24,910
You have to help me prove that.

621
00:36:24,910 --> 00:36:27,340
And this is the Jacobian.

622
00:36:27,340 --> 00:36:30,385
Do you guys know why
it's called Jacobian?

623
00:36:30,385 --> 00:36:33,355
It's the determinant
of this matrix.

624
00:36:33,355 --> 00:36:43,255
Let's call this
matrix J. And this

625
00:36:43,255 --> 00:36:49,210
is J, determinant
of [? scripture. ?]

626
00:36:49,210 --> 00:36:50,480
This is called Jacobian.

627
00:36:50,480 --> 00:36:54,160


628
00:36:54,160 --> 00:36:55,080
Why is it r?

629
00:36:55,080 --> 00:36:57,850
Let's-- I don't know.

630
00:36:57,850 --> 00:36:59,710
Let's see how we do it.

631
00:36:59,710 --> 00:37:03,540


632
00:37:03,540 --> 00:37:06,900
This is r cosine theta, right?

633
00:37:06,900 --> 00:37:09,790
This is r sine theta.

634
00:37:09,790 --> 00:37:14,790
So dx must be what x sub r?

635
00:37:14,790 --> 00:37:19,730
X sub r, x sub r, cosine theta.

636
00:37:19,730 --> 00:37:21,750
d plus.

637
00:37:21,750 --> 00:37:23,570
What is x sub t?

638
00:37:23,570 --> 00:37:26,385


639
00:37:26,385 --> 00:37:28,550
x sub theta.

640
00:37:28,550 --> 00:37:31,601
I need to differentiate
this with respect to theta.

641
00:37:31,601 --> 00:37:33,600
STUDENT: It's going to
be negative r sine theta.

642
00:37:33,600 --> 00:37:36,390
MAGDALENA TODA: Minus r
sine theta, very good.

643
00:37:36,390 --> 00:37:38,090
And d theta.

644
00:37:38,090 --> 00:37:44,200
Then I go dy was
sine theta-- dr,

645
00:37:44,200 --> 00:37:46,450
I'm looking at these
equations, and I'm

646
00:37:46,450 --> 00:37:49,020
repeating them for my case.

647
00:37:49,020 --> 00:37:52,890
This is true in general for
any kind of coordinates.

648
00:37:52,890 --> 00:37:56,640
So it's a general equation
for any kind of coordinate,

649
00:37:56,640 --> 00:37:58,830
two coordinates,
two coordinates,

650
00:37:58,830 --> 00:38:00,630
any kind of
coordinates in plane,

651
00:38:00,630 --> 00:38:04,940
you can choose any
functions, f of uv, g of uv,

652
00:38:04,940 --> 00:38:06,600
whatever you want.

653
00:38:06,600 --> 00:38:09,460
But for this particular
case of polar coordinates

654
00:38:09,460 --> 00:38:12,270
is going to look really
pretty in the end.

655
00:38:12,270 --> 00:38:15,610
What do I get when I do y theta?

656
00:38:15,610 --> 00:38:17,485
r cosine theta.

657
00:38:17,485 --> 00:38:18,830
Am I right, guys?

658
00:38:18,830 --> 00:38:20,730
Keen an eye on it.

659
00:38:20,730 --> 00:38:27,280
So this will become-- the
area element will become what?

660
00:38:27,280 --> 00:38:31,310
The determinant of this matrix.

661
00:38:31,310 --> 00:38:34,570
Red, red, red, red.

662
00:38:34,570 --> 00:38:35,886
How do I compute a term?

663
00:38:35,886 --> 00:38:39,410
Not everybody knows,
and it's this times

664
00:38:39,410 --> 00:38:45,150
that minus this times that.

665
00:38:45,150 --> 00:38:46,370
OK, let's do that.

666
00:38:46,370 --> 00:38:53,440
So I get r cosine squared
theta minus minus plus r sine

667
00:38:53,440 --> 00:38:56,490
squared theta.

668
00:38:56,490 --> 00:38:58,870
dr, d theta, and our wedge.

669
00:38:58,870 --> 00:39:00,000
What is this?

670
00:39:00,000 --> 00:39:00,600
STUDENT: 1.

671
00:39:00,600 --> 00:39:04,320
MAGDALENA TODA:
Jacobian is r times 1,

672
00:39:04,320 --> 00:39:07,430
because that's the
Pythagorean theorem, right?

673
00:39:07,430 --> 00:39:12,330
So we have r, and this is
the meaning of r, here.

674
00:39:12,330 --> 00:39:16,830
So when I moved from dx dy,
I originally had the wedge

675
00:39:16,830 --> 00:39:19,470
that I didn't tell you about.

676
00:39:19,470 --> 00:39:23,090
And this wedge
becomes r dr d theta,

677
00:39:23,090 --> 00:39:27,290
and that's the
correct way to explain

678
00:39:27,290 --> 00:39:29,890
why you get the Jacobian there.

679
00:39:29,890 --> 00:39:31,505
We don't do that in the book.

680
00:39:31,505 --> 00:39:34,855
We do it later, and we
sort of smuggle through.

681
00:39:34,855 --> 00:39:37,100
We don't do a very thorough job.

682
00:39:37,100 --> 00:39:39,980
When you go into
advanced calculus,

683
00:39:39,980 --> 00:39:43,237
you would see that again the
way I explained it to you.

684
00:39:43,237 --> 00:39:47,370
If you ever want to
go to graduate school,

685
00:39:47,370 --> 00:39:52,440
then you need to take the
Advanced Calculus I, 4350

686
00:39:52,440 --> 00:39:57,530
and 4351 where you are
going to learn about this.

687
00:39:57,530 --> 00:40:01,210
If you take those as a math
major or engineering major,

688
00:40:01,210 --> 00:40:01,960
it doesn't matter.

689
00:40:01,960 --> 00:40:03,920
When you go to
graduate school, they

690
00:40:03,920 --> 00:40:07,470
don't make you take
advanced calculus again

691
00:40:07,470 --> 00:40:09,380
at graduate school.

692
00:40:09,380 --> 00:40:12,740
So it's somewhere borderline
between senior year

693
00:40:12,740 --> 00:40:19,010
and graduate school, it's like
the first course you would take

694
00:40:19,010 --> 00:40:22,020
in graduate school, for many.

695
00:40:22,020 --> 00:40:22,670
OK.

696
00:40:22,670 --> 00:40:29,890
So an example of
this transformation

697
00:40:29,890 --> 00:40:33,270
where we know what
we are talking about.

698
00:40:33,270 --> 00:40:39,130
Let's say I have
a picture, and I

699
00:40:39,130 --> 00:40:42,730
have a domain D, which
is-- this is x squared

700
00:40:42,730 --> 00:40:44,946
plus y squared equals 1.

701
00:40:44,946 --> 00:40:48,369
I have the domain as being
[INTERPOSING VOICES].

702
00:40:48,369 --> 00:40:51,800


703
00:40:51,800 --> 00:40:58,120
And then I say, I would
like-- what would I like?

704
00:40:58,120 --> 00:41:04,290
I would like the
volume of the-- this

705
00:41:04,290 --> 00:41:10,220
is a paraboloid, z equals
x squared plus y squared.

706
00:41:10,220 --> 00:41:12,616
I would like the
volume of this object.

707
00:41:12,616 --> 00:41:13,820
This is my obsession.

708
00:41:13,820 --> 00:41:17,580
I'm going to create a
vase some day like that.

709
00:41:17,580 --> 00:41:22,560
So you want this
piece to be a solid.

710
00:41:22,560 --> 00:41:25,420
In cross section,
it will just this.

711
00:41:25,420 --> 00:41:26,250
In cross section.

712
00:41:26,250 --> 00:41:27,830
And it's a solid of revolution.

713
00:41:27,830 --> 00:41:30,300
In this cross section,
you have to imagine

714
00:41:30,300 --> 00:41:36,100
revolving it around the z-axis,
then creating a heavy object.

715
00:41:36,100 --> 00:41:38,440
From the outside, don't
see what's inside.

716
00:41:38,440 --> 00:41:39,530
It looks like a cylinder.

717
00:41:39,530 --> 00:41:42,460
But you go inside and
you see the valley.

718
00:41:42,460 --> 00:41:46,260
So it's between a
paraboloid and a disc,

719
00:41:46,260 --> 00:41:48,460
a unit disc on the floor.

720
00:41:48,460 --> 00:41:51,400
How are we going
to try and do that?

721
00:41:51,400 --> 00:41:53,960
And what did I
teach you last time?

722
00:41:53,960 --> 00:42:02,020
Last time, I taught you that--
we have to go over a domain D.

723
00:42:02,020 --> 00:42:04,190
But that domain
D, unfortunately,

724
00:42:04,190 --> 00:42:05,780
is hard to express.

725
00:42:05,780 --> 00:42:09,217
How would you express D
in Cartesian coordinates?

726
00:42:09,217 --> 00:42:14,630


727
00:42:14,630 --> 00:42:15,840
You can do it.

728
00:42:15,840 --> 00:42:18,770
It's going to be a headache.

729
00:42:18,770 --> 00:42:22,270
x is between minus 1 and 1.

730
00:42:22,270 --> 00:42:23,770
Am I right, guys?

731
00:42:23,770 --> 00:42:28,270
And y will be between--
now I have two branches.

732
00:42:28,270 --> 00:42:30,230
One, and the other one.

733
00:42:30,230 --> 00:42:33,100
One branch would be square--
I hate square roots.

734
00:42:33,100 --> 00:42:36,250
I absolutely hate them.

735
00:42:36,250 --> 00:42:40,330
y is between 1 minus
square root x squared,

736
00:42:40,330 --> 00:42:43,300
minus square root
1 minus x squared.

737
00:42:43,300 --> 00:42:47,650
So if I were to ask you to do
the integral like last time,

738
00:42:47,650 --> 00:42:50,794
how would you set
up the integral?

739
00:42:50,794 --> 00:42:53,380
You go, OK, I know what this is.

740
00:42:53,380 --> 00:43:01,380
Integral over D of
f of x, y, dx dy.

741
00:43:01,380 --> 00:43:02,900
This is actually a wedge.

742
00:43:02,900 --> 00:43:06,060
In my case, we avoided that.

743
00:43:06,060 --> 00:43:07,540
We said dh.

744
00:43:07,540 --> 00:43:09,910
And we said, what is f of x, y?

745
00:43:09,910 --> 00:43:11,770
x squared plus y
squared, because I

746
00:43:11,770 --> 00:43:16,044
want everything that's under
the graph, not above the graph.

747
00:43:16,044 --> 00:43:18,996
So everything that's
under the graph.

748
00:43:18,996 --> 00:43:26,600
F of x, y is this guy.

749
00:43:26,600 --> 00:43:28,430
And the I have to
start thinking,

750
00:43:28,430 --> 00:43:31,540
because it's a type 1 or type 2?

751
00:43:31,540 --> 00:43:35,700
It's a type 1 the
way I set it up,

752
00:43:35,700 --> 00:43:39,060
but I can make it
type 2 by reversing

753
00:43:39,060 --> 00:43:41,520
the order of integration
like I did last time.

754
00:43:41,520 --> 00:43:44,035
If I treat it like
that, it's going

755
00:43:44,035 --> 00:43:46,420
to be type 1, though, right?

756
00:43:46,420 --> 00:43:50,640
So I have to put
dy first, and then

757
00:43:50,640 --> 00:43:54,570
change the color of the dx.

758
00:43:54,570 --> 00:43:58,280
And since mister y
is the purple guy,

759
00:43:58,280 --> 00:44:03,000
y would be going between
these ugly square roots that

760
00:44:03,000 --> 00:44:04,220
to go on my nerves.

761
00:44:04,220 --> 00:44:10,360


762
00:44:10,360 --> 00:44:17,485
And then x goes
between minus 1 and 1.

763
00:44:17,485 --> 00:44:20,870
It's a little bit of a headache.

764
00:44:20,870 --> 00:44:22,980
Why is it a headache, guys?

765
00:44:22,980 --> 00:44:27,470
Let's anticipate what we need to
do if we do it like last time.

766
00:44:27,470 --> 00:44:32,110
We need to integrate this
ugly fellow in terms of y,

767
00:44:32,110 --> 00:44:35,510
and when we integrate this in
terms of y, what do we get?

768
00:44:35,510 --> 00:44:38,450
Don't write it, because
it's going to be a mess.

769
00:44:38,450 --> 00:44:44,870
We get x squared times
y plus y cubed over 3.

770
00:44:44,870 --> 00:44:47,480
And then, instead of y, I have
to replace those square roots,

771
00:44:47,480 --> 00:44:49,600
and I'll never get rid
of the square roots.

772
00:44:49,600 --> 00:44:52,760
It's going to be a mess, indeed.

773
00:44:52,760 --> 00:44:56,250
And I may even-- in
general, I may not even

774
00:44:56,250 --> 00:44:58,860
be able to solve the
integral, and that's

775
00:44:58,860 --> 00:45:00,780
a bit headache,
because I'll start

776
00:45:00,780 --> 00:45:03,444
crying, I'll get depressed,
I'll take Prozac, whatever

777
00:45:03,444 --> 00:45:04,815
you take for depression.

778
00:45:04,815 --> 00:45:07,560
I don't know, I never took it,
because I'm never depressed.

779
00:45:07,560 --> 00:45:10,960
So what do you do in that case?

780
00:45:10,960 --> 00:45:12,220
STUDENT: Switch to polar.

781
00:45:12,220 --> 00:45:13,720
MAGDALENA TODA: You
switch to polar.

782
00:45:13,720 --> 00:45:18,610
So you use this big polar-switch
theorem, the theorem that

783
00:45:18,610 --> 00:45:23,940
tells you, be smart,
apply this theorem,

784
00:45:23,940 --> 00:45:30,700
and have to understand that
the D, which was this expressed

785
00:45:30,700 --> 00:45:32,970
in [INTERPOSING VOICES]
Cartesian coordinates

786
00:45:32,970 --> 00:45:37,480
is D. If you want express
the same thing as D star,

787
00:45:37,480 --> 00:45:39,600
D star will be in
polar coordinates.

788
00:45:39,600 --> 00:45:44,010
You have to be a little bit
smarter, and say r theta,

789
00:45:44,010 --> 00:45:48,980
where now you have to put
the bounds that limit--

790
00:45:48,980 --> 00:45:49,590
STUDENT: r.

791
00:45:49,590 --> 00:45:50,694
MAGDALENA TODA: r from?

792
00:45:50,694 --> 00:45:51,360
STUDENT: 0 to 1.

793
00:45:51,360 --> 00:45:52,776
MAGDALENA TODA: 0
to 1, excellent.

794
00:45:52,776 --> 00:45:56,899
You cannot let r go to
infinity, because the vase is

795
00:45:56,899 --> 00:45:57,440
increasingly.

796
00:45:57,440 --> 00:46:01,312
You only needs the vase that
has the radius 1 on the bottom.

797
00:46:01,312 --> 00:46:08,723
So r is 0 to 1, and
theta is 0 to 1 pi.

798
00:46:08,723 --> 00:46:10,640
And there you have
your domain this time.

799
00:46:10,640 --> 00:46:15,746
So I need to be smart
and say integral.

800
00:46:15,746 --> 00:46:18,000
Integral, what do
you want to do first?

801
00:46:18,000 --> 00:46:21,850
Well, it doesn't matter, dr,
d theta, whatever you want.

802
00:46:21,850 --> 00:46:26,310
So mister theta will
be the last of the two.

803
00:46:26,310 --> 00:46:32,270
So theta will be between 0
and 2 pi, a complete rotation.

804
00:46:32,270 --> 00:46:35,856
r between 0 and 1.

805
00:46:35,856 --> 00:46:37,970
And inside here I
have to be smart

806
00:46:37,970 --> 00:46:41,710
and see that life
can be fun when

807
00:46:41,710 --> 00:46:44,320
I work with polar coordinates.

808
00:46:44,320 --> 00:46:45,642
Why?

809
00:46:45,642 --> 00:46:47,060
What is the integral?

810
00:46:47,060 --> 00:46:48,110
x squared plus y squared.

811
00:46:48,110 --> 00:46:50,680
I've seen him
somewhere before when

812
00:46:50,680 --> 00:46:54,989
it came to polar coordinates.

813
00:46:54,989 --> 00:46:55,780
STUDENT: R squared.

814
00:46:55,780 --> 00:46:57,113
STUDENT: That will be r squared.

815
00:46:57,113 --> 00:46:59,600
MAGDALENA TODA: That
will be r squared.

816
00:46:59,600 --> 00:47:04,482
r squared times-- never
forget the Jacobian,

817
00:47:04,482 --> 00:47:07,910
and the Jacobian is mister r.

818
00:47:07,910 --> 00:47:13,030
And now I'm going to
take all this integral.

819
00:47:13,030 --> 00:47:16,490
I'll finally compute
the volume of my vase.

820
00:47:16,490 --> 00:47:19,960
Imagine if this vase
would be made of gold.

821
00:47:19,960 --> 00:47:21,690
This is my dream.

822
00:47:21,690 --> 00:47:24,970
So imagine that this
vase would have,

823
00:47:24,970 --> 00:47:26,790
I don't know what dimensions.

824
00:47:26,790 --> 00:47:29,390
I need to find the
volume, and multiply it

825
00:47:29,390 --> 00:47:32,405
by the density of gold
and find out-- yes, sir?

826
00:47:32,405 --> 00:47:35,660
STUDENT: Professor, like in this
question, b time is dt by dr,

827
00:47:35,660 --> 00:47:38,062
but you can't switch it--

828
00:47:38,062 --> 00:47:39,270
MAGDALENA TODA: Yes, you can.

829
00:47:39,270 --> 00:47:41,320
That's exactly my point.

830
00:47:41,320 --> 00:47:42,690
I'll tell you in a second.

831
00:47:42,690 --> 00:47:47,980
When can you replace d theta dr?

832
00:47:47,980 --> 00:47:52,450
You can always do that when
you have something under here,

833
00:47:52,450 --> 00:47:55,690
which is a big
function of theta times

834
00:47:55,690 --> 00:48:01,630
a bit function of r, because
you can treat them differently.

835
00:48:01,630 --> 00:48:05,050
We will work about this later.

836
00:48:05,050 --> 00:48:08,640
Now, this has no theta.

837
00:48:08,640 --> 00:48:13,720
So actually, the
theta is not going

838
00:48:13,720 --> 00:48:18,700
to affect your computation.

839
00:48:18,700 --> 00:48:22,410
Let's not even think about
theta for the time being.

840
00:48:22,410 --> 00:48:29,904
What you have inside is Calculus
I. When you have a product,

841
00:48:29,904 --> 00:48:31,395
you can always switch.

842
00:48:31,395 --> 00:48:33,880
And I'll give you
a theorem later.

843
00:48:33,880 --> 00:48:39,150
0 over 1, r cubed,
thank God, this

844
00:48:39,150 --> 00:48:42,500
is Calc I. Integral
from 0 to 1, r

845
00:48:42,500 --> 00:48:47,000
cubed dr. That's Calc
I. How much is that?

846
00:48:47,000 --> 00:48:47,620
I'm lazy.

847
00:48:47,620 --> 00:48:50,110
I don't want to do it.

848
00:48:50,110 --> 00:48:51,179
STUDENT: 1/4.

849
00:48:51,179 --> 00:48:52,220
MAGDALENA TODA: It's 1/4.

850
00:48:52,220 --> 00:48:52,720
Very good.

851
00:48:52,720 --> 00:48:53,910
Thank you.

852
00:48:53,910 --> 00:48:58,460
And if I get further, and I'm a
little bi lazy, what do I get?

853
00:48:58,460 --> 00:49:01,500
1/4 is the constant,
it pulls out.

854
00:49:01,500 --> 00:49:03,140
STUDENT: So, they don't--

855
00:49:03,140 --> 00:49:09,780
MAGDALENA TODA: So I get 2 pi
over 4, which is pi over 2.

856
00:49:09,780 --> 00:49:10,535
Am I right?

857
00:49:10,535 --> 00:49:11,118
STUDENT: Yeah.

858
00:49:11,118 --> 00:49:12,867
MAGDALENA TODA: So
this constant gets out,

859
00:49:12,867 --> 00:49:14,200
integral comes in through 2 pi.

860
00:49:14,200 --> 00:49:16,225
It will be 2 pi, and
this is my answer.

861
00:49:16,225 --> 00:49:19,520
So pi over 2 is the volume.

862
00:49:19,520 --> 00:49:22,570
If I have a 1-inch
diameter, and I

863
00:49:22,570 --> 00:49:26,536
have this vase made of gold,
which is a piece of jewelry,

864
00:49:26,536 --> 00:49:34,160
really beautiful, then I'm going
to have pi over 2 the volume.

865
00:49:34,160 --> 00:49:36,330
That will be a little
bit hard to see

866
00:49:36,330 --> 00:49:38,930
what we have in square inches.

867
00:49:38,930 --> 00:49:43,920
We have 1.5-something
square inches, and then--

868
00:49:43,920 --> 00:49:45,105
STUDENT: More.

869
00:49:45,105 --> 00:49:46,480
MAGDALENA TODA:
And then multiply

870
00:49:46,480 --> 00:49:50,350
by the density of
gold, and estimate,

871
00:49:50,350 --> 00:49:57,730
based on the mass, how much
money that's going to be.

872
00:49:57,730 --> 00:49:59,880
What did I want to
tell [? Miteish? ?]

873
00:49:59,880 --> 00:50:02,633
I don't want to forget what
he asked me, because that

874
00:50:02,633 --> 00:50:04,240
was a smart question.

875
00:50:04,240 --> 00:50:08,620
When can we reverse the
order of integration?

876
00:50:08,620 --> 00:50:11,995
In general, it's
hard to compute.

877
00:50:11,995 --> 00:50:14,540
But in this case, I'm you
are the luckiest person

878
00:50:14,540 --> 00:50:16,790
in the world, because
just take a look at me.

879
00:50:16,790 --> 00:50:22,180
I have, let's see, my
r between 0 and 2 pi,

880
00:50:22,180 --> 00:50:29,470
and my theta between 0 and 2
pi, and my r between 0 and 1.

881
00:50:29,470 --> 00:50:31,970
Whatever, it doesn't matter,
it could be anything.

882
00:50:31,970 --> 00:50:36,390
And here I have a function of r
and a function g of theta only.

883
00:50:36,390 --> 00:50:38,060
And it's a product.

884
00:50:38,060 --> 00:50:40,790
The variables are separate.

885
00:50:40,790 --> 00:50:45,800
When I do-- what do I
do for dr or d theta?

886
00:50:45,800 --> 00:50:49,240
dr. When I do dr--
with respect to dr,

887
00:50:49,240 --> 00:50:52,702
this fellow goes, I
don't belong in here.

888
00:50:52,702 --> 00:50:55,650
I'm mister theta that
doesn't belong in here.

889
00:50:55,650 --> 00:50:56,930
I'm independent.

890
00:50:56,930 --> 00:50:59,160
I want to go out.

891
00:50:59,160 --> 00:51:01,600
And he wants out.

892
00:51:01,600 --> 00:51:10,480
So you have some integrals
that you got out a g of theta,

893
00:51:10,480 --> 00:51:16,440
and another integral, and you
have f of r dr in a bracket,

894
00:51:16,440 --> 00:51:20,880
and then you go d theta.

895
00:51:20,880 --> 00:51:23,080
What is going to happen next?

896
00:51:23,080 --> 00:51:26,790
You solve this integral, and
it's going to be a number.

897
00:51:26,790 --> 00:51:30,400
This number could be 8,
7, 9.2, God knows what.

898
00:51:30,400 --> 00:51:33,230
Why don't you pull that
constant out right now?

899
00:51:33,230 --> 00:51:35,480
So you say, OK, I can do that.

900
00:51:35,480 --> 00:51:37,130
It's just a number.

901
00:51:37,130 --> 00:51:37,630
Whatever.

902
00:51:37,630 --> 00:51:41,610
That's going to be
integral f dr, times

903
00:51:41,610 --> 00:51:44,320
what do you have left
when you pull that out?

904
00:51:44,320 --> 00:51:44,820
A what?

905
00:51:44,820 --> 00:51:45,623
STUDENT: Integral.

906
00:51:45,623 --> 00:51:49,460
MAGDALENA TODA: Integral of
G, the integral of g of theta,

907
00:51:49,460 --> 00:51:51,000
d theta.

908
00:51:51,000 --> 00:51:54,440
So we just proved a theorem
that is really pretty.

909
00:51:54,440 --> 00:51:59,238
If you have to integrate,
and I will try to do it here.

910
00:51:59,238 --> 00:52:03,201


911
00:52:03,201 --> 00:52:03,700
No--

912
00:52:03,700 --> 00:52:06,241
STUDENT: So essentially, when
you're integrating with respect

913
00:52:06,241 --> 00:52:11,243
to r, you can treat any function
of only theta as a constant?

914
00:52:11,243 --> 00:52:12,230
MAGDALENA TODA: Yeah.

915
00:52:12,230 --> 00:52:15,050
I'll tell you in a second
what it means, because--

916
00:52:15,050 --> 00:52:15,809
STUDENT: Sorry.

917
00:52:15,809 --> 00:52:16,975
MAGDALENA TODA: You're fine.

918
00:52:16,975 --> 00:52:21,601
Integrate for domain,
rectangular domains,

919
00:52:21,601 --> 00:52:25,770
let's say u between alpha,
beta, u between gamma,

920
00:52:25,770 --> 00:52:29,710
delta, then what's
going to happen?

921
00:52:29,710 --> 00:52:35,383
As you said very well,
integral from-- what

922
00:52:35,383 --> 00:52:38,332
do you want first, dv or du?

923
00:52:38,332 --> 00:52:41,132
dv, du, it doesn't matter.

924
00:52:41,132 --> 00:52:44,108
v is between gamma, delta.

925
00:52:44,108 --> 00:52:47,084
v is the first guy inside, OK.

926
00:52:47,084 --> 00:52:48,572
Gamma, delta.

927
00:52:48,572 --> 00:52:50,060
I should have cd.

928
00:52:50,060 --> 00:52:51,080
It's all Greek to me.

929
00:52:51,080 --> 00:52:55,060
Why did I pick
that three people?

930
00:52:55,060 --> 00:52:59,600
If this is going to be a product
of two functions, one is in u

931
00:52:59,600 --> 00:53:06,210
and one is in v. Let's
say A of u and B of v,

932
00:53:06,210 --> 00:53:11,100
I can go ahead and say
product of two constants.

933
00:53:11,100 --> 00:53:14,040
And who are those two
constants I was referring to?

934
00:53:14,040 --> 00:53:16,000
You can do that directly.

935
00:53:16,000 --> 00:53:18,940
If the two variables are
separated through a product,

936
00:53:18,940 --> 00:53:22,730
you have a product of
two separate variables.

937
00:53:22,730 --> 00:53:26,320
A is only in u, it
depends only on u.

938
00:53:26,320 --> 00:53:30,820
And B is only on v. They have
nothing to do with one another.

939
00:53:30,820 --> 00:53:35,152
Then you can go ahead and do
the first integral with respect

940
00:53:35,152 --> 00:53:43,310
to u only of a of u, du,
u between alpha, beta.

941
00:53:43,310 --> 00:53:45,940
That was your first variable.

942
00:53:45,940 --> 00:53:48,615
Times this other constant.

943
00:53:48,615 --> 00:53:54,490
Integral of B of v,
where v is moving,

944
00:53:54,490 --> 00:53:59,070
v is moving between
gamma, delta.

945
00:53:59,070 --> 00:54:00,980
Instead of alpha,
beta, gamma, delta,

946
00:54:00,980 --> 00:54:03,970
put any numbers you want.

947
00:54:03,970 --> 00:54:04,854
OK?

948
00:54:04,854 --> 00:54:06,180
This is the lucky case.

949
00:54:06,180 --> 00:54:09,200
So you're always hoping
that on the final,

950
00:54:09,200 --> 00:54:12,840
you can get something
where you can separate.

951
00:54:12,840 --> 00:54:13,950
Here you have no theta.

952
00:54:13,950 --> 00:54:16,330
This is the luckiest
case in the world.

953
00:54:16,330 --> 00:54:18,550
So it's just r
cubed times theta.

954
00:54:18,550 --> 00:54:21,440
But you can still
have a lucky case

955
00:54:21,440 --> 00:54:24,530
when you have something
like a function of r

956
00:54:24,530 --> 00:54:25,940
times a function of theta.

957
00:54:25,940 --> 00:54:28,550
And then you have
another beautiful polar

958
00:54:28,550 --> 00:54:31,600
coordinate integral
that you're not going

959
00:54:31,600 --> 00:54:35,000
to struggle with for very long.

960
00:54:35,000 --> 00:54:37,450
OK, I'm going to erase here.

961
00:54:37,450 --> 00:54:56,100


962
00:54:56,100 --> 00:55:01,560
For example, let me
give you another one.

963
00:55:01,560 --> 00:55:04,110
Suppose that somebody
was really mean to you,

964
00:55:04,110 --> 00:55:08,399
and wanted to kill
you in the final,

965
00:55:08,399 --> 00:55:10,065
and they gave you the
following problem.

966
00:55:10,065 --> 00:55:12,590


967
00:55:12,590 --> 00:55:17,370
Assume the domain D-- they
don't even tell you what it is.

968
00:55:17,370 --> 00:55:19,350
They just want to
challenge you--

969
00:55:19,350 --> 00:55:25,472
will be x, y with the
property that x squared plus y

970
00:55:25,472 --> 00:55:32,080
squared is between a 1 and a 4.

971
00:55:32,080 --> 00:55:36,370


972
00:55:36,370 --> 00:55:52,922
Compute the integral over D of
r [? pan ?] of y over x and da,

973
00:55:52,922 --> 00:55:57,330
where bi would be ds dy.

974
00:55:57,330 --> 00:56:00,710
So you look at this
cross-eyed and say, gosh,

975
00:56:00,710 --> 00:56:04,220
whoever-- we don't do that.

976
00:56:04,220 --> 00:56:05,310
But I've seen schools.

977
00:56:05,310 --> 00:56:08,600
I've seen this given at a
school, when they covered

978
00:56:08,600 --> 00:56:11,620
this particular
example, they've covered

979
00:56:11,620 --> 00:56:14,710
something like the previous
one that I showed you.

980
00:56:14,710 --> 00:56:16,200
But they never covered this.

981
00:56:16,200 --> 00:56:18,460
And they said,
OK, they're smart,

982
00:56:18,460 --> 00:56:19,990
let them figure this out.

983
00:56:19,990 --> 00:56:23,360
And I think it was Texas A&M.
They gave something like that

984
00:56:23,360 --> 00:56:26,350
without working this in class.

985
00:56:26,350 --> 00:56:28,576
They assumed that
the students should

986
00:56:28,576 --> 00:56:31,120
be good enough to
figure out what

987
00:56:31,120 --> 00:56:35,360
this is in polar coordinates.

988
00:56:35,360 --> 00:56:39,790
So in polar coordinates,
what does the theorem say?

989
00:56:39,790 --> 00:56:44,360
We should switch to a domain
D star that corresponds to D.

990
00:56:44,360 --> 00:56:48,220
Now, D was given like that.

991
00:56:48,220 --> 00:56:50,660
But we have to say
the corresponding D

992
00:56:50,660 --> 00:56:55,090
star, reinterpreted
in polar coordinates,

993
00:56:55,090 --> 00:56:59,710
r theta has to be also
written beautifully out.

994
00:56:59,710 --> 00:57:03,910
Unless you draw the picture,
first of all, you cannot do it.

995
00:57:03,910 --> 00:57:07,790
So the prof at Texas A&M didn't
even say, draw the picture,

996
00:57:07,790 --> 00:57:10,700
and think of the
meaning of that.

997
00:57:10,700 --> 00:57:14,950
What is the meaning of
this set, geometric set,

998
00:57:14,950 --> 00:57:17,072
geometric locus of points.

999
00:57:17,072 --> 00:57:18,880
STUDENT: You've
got a circle sub-

1000
00:57:18,880 --> 00:57:21,580
MAGDALENA TODA: You
have concentric circles,

1001
00:57:21,580 --> 00:57:26,950
sub-radius 1 and 2, and it's
like a ring, it's an annulus.

1002
00:57:26,950 --> 00:57:30,020
And he said, well,
I didn't do it.

1003
00:57:30,020 --> 00:57:33,020
I mean they were smart.

1004
00:57:33,020 --> 00:57:35,450
I gave it to them to do.

1005
00:57:35,450 --> 00:57:40,670
So if the students don't see
at least an example like that,

1006
00:57:40,670 --> 00:57:44,550
they have difficulty,
in my experience.

1007
00:57:44,550 --> 00:57:47,300
OK, for this kind
of annulus, you

1008
00:57:47,300 --> 00:57:50,810
see the radius would start
here, but the dotted part

1009
00:57:50,810 --> 00:57:53,490
is not included in your domain.

1010
00:57:53,490 --> 00:57:57,107
So you have to be smart and
say, wait a minute, my radius

1011
00:57:57,107 --> 00:57:58,550
is not starting at 0.

1012
00:57:58,550 --> 00:58:01,534
It's starting at 1
and it's ending at 2.

1013
00:58:01,534 --> 00:58:05,980
And I put that here.

1014
00:58:05,980 --> 00:58:11,016
And theta is the whole
ring, so from 0 to 2 pi.

1015
00:58:11,016 --> 00:58:14,490


1016
00:58:14,490 --> 00:58:18,250
Whether you do that
over the open set,

1017
00:58:18,250 --> 00:58:21,360
that's called annulus
without the boundaries,

1018
00:58:21,360 --> 00:58:25,265
or you do it about the
one with the boundaries,

1019
00:58:25,265 --> 00:58:28,235
it doesn't matter, the integral
is not going to change.

1020
00:58:28,235 --> 00:58:33,185
And you are going to learn
that in Advanced Calculus, why

1021
00:58:33,185 --> 00:58:36,650
it doesn't matter that if
you remove the boundary,

1022
00:58:36,650 --> 00:58:38,630
you put back the boundary.

1023
00:58:38,630 --> 00:58:42,970
That is a certain set of a
measure 0 for your integration.

1024
00:58:42,970 --> 00:58:46,000
It's not going to
change your results.

1025
00:58:46,000 --> 00:58:48,740
So no matter how you
express it-- maybe

1026
00:58:48,740 --> 00:58:51,590
you want to express
it like an open set.

1027
00:58:51,590 --> 00:58:55,362
You still have
the same integral.

1028
00:58:55,362 --> 00:58:57,870
Double integral
of D star, this is

1029
00:58:57,870 --> 00:59:01,684
going to give me a headache,
unless you help me.

1030
00:59:01,684 --> 00:59:05,661
What is this in
polar coordinates?

1031
00:59:05,661 --> 00:59:06,494
STUDENT: [INAUDIBLE]

1032
00:59:06,494 --> 00:59:09,785


1033
00:59:09,785 --> 00:59:11,410
MAGDALENA TODA: I
know when-- once I've

1034
00:59:11,410 --> 00:59:13,240
figured out the
integrand, I'm going

1035
00:59:13,240 --> 00:59:16,730
to remember to always
multiply by an r,

1036
00:59:16,730 --> 00:59:18,620
because if I don't,
I'm in big trouble.

1037
00:59:18,620 --> 00:59:23,540
And then I go dr d theta.

1038
00:59:23,540 --> 00:59:26,310
But I don't know what this is.

1039
00:59:26,310 --> 00:59:28,276
STUDENT: r.

1040
00:59:28,276 --> 00:59:34,310
MAGDALENA TODA: Nope, but
you're-- so r cosine theta is

1041
00:59:34,310 --> 00:59:37,680
x, r sine theta is y.

1042
00:59:37,680 --> 00:59:41,260
When you do y over
x, what do you get?

1043
00:59:41,260 --> 00:59:43,710
Always tangent of theta.

1044
00:59:43,710 --> 00:59:47,680
And if you do arctangent
of tangent, you get theta.

1045
00:59:47,680 --> 00:59:50,516
So that was not hard,
but the students did

1046
00:59:50,516 --> 00:59:53,130
not-- in that
class, I was talking

1047
00:59:53,130 --> 00:59:56,600
to whoever gave the exam,
70-something percent

1048
00:59:56,600 --> 00:59:58,920
of the students did
not know how to do it,

1049
00:59:58,920 --> 01:00:01,490
because they had never
seen something similar,

1050
01:00:01,490 --> 01:00:07,230
and they didn't think how
to express this theta in r.

1051
01:00:07,230 --> 01:00:08,860
So what do we mean to do?

1052
01:00:08,860 --> 01:00:11,630
We mean, is this a product?

1053
01:00:11,630 --> 01:00:13,170
It's a beautiful product.

1054
01:00:13,170 --> 01:00:17,620
They are separate variables like
[INAUDIBLE] [? shafts. ?] Now,

1055
01:00:17,620 --> 01:00:19,830
you see, you can separate them.

1056
01:00:19,830 --> 01:00:26,730
The r is between 1 and 2,
so I can do-- eventually I

1057
01:00:26,730 --> 01:00:27,980
can do the r first.

1058
01:00:27,980 --> 01:00:33,320
And theta is between 0 and
2 pi, and as I taught you

1059
01:00:33,320 --> 01:00:37,650
by the previous theorem, you
can separate the two integrals,

1060
01:00:37,650 --> 01:00:39,967
because this one gets out.

1061
01:00:39,967 --> 01:00:41,280
It's a constant.

1062
01:00:41,280 --> 01:00:46,580
So you're left with integral
from 0 to 2 pi theta d

1063
01:00:46,580 --> 01:01:04,930
theta, and the integral from 1
to 2 r dr. r dr theta d theta.

1064
01:01:04,930 --> 01:01:06,400
This should be a piece of cake.

1065
01:01:06,400 --> 01:01:13,940
The only thing we have to
do is some easy Calculus I.

1066
01:01:13,940 --> 01:01:18,440
So what is integral
of theta d theta?

1067
01:01:18,440 --> 01:01:20,480
I'm not going to rush anywhere.

1068
01:01:20,480 --> 01:01:27,160
Theta squared over 2
between theta equals 0 down

1069
01:01:27,160 --> 01:01:30,945
and theta equals 2 pi up.

1070
01:01:30,945 --> 01:01:32,352
Right?

1071
01:01:32,352 --> 01:01:33,735
STUDENT: [INAUDIBLE]

1072
01:01:33,735 --> 01:01:34,610
MAGDALENA TODA: Yeah.

1073
01:01:34,610 --> 01:01:35,560
I'll do that later.

1074
01:01:35,560 --> 01:01:36,520
I don't care.

1075
01:01:36,520 --> 01:01:41,140
This is going to be r squared
over 2 between 1 and 2.

1076
01:01:41,140 --> 01:01:44,400
So the numerical
answer, if I know

1077
01:01:44,400 --> 01:01:51,055
how to do any math like
that, is going to be--

1078
01:01:51,055 --> 01:01:52,125
STUDENT: 2 pi squared.

1079
01:01:52,125 --> 01:01:53,750
MAGDALENA TODA: 2 pi
squared, because I

1080
01:01:53,750 --> 01:01:57,770
have 4 pi squared over
2, so the first guy

1081
01:01:57,770 --> 01:02:07,890
is 2 pi squared, times-- I
get a 4 and 4 minus 1-- are

1082
01:02:07,890 --> 01:02:09,250
you guys with me?

1083
01:02:09,250 --> 01:02:12,620
So I get a-- let me
write it like that.

1084
01:02:12,620 --> 01:02:16,530
4 over 2 minus 1 over 2.

1085
01:02:16,530 --> 01:02:18,775
What's going to
happen to the over 2?

1086
01:02:18,775 --> 01:02:20,200
We'll simplify.

1087
01:02:20,200 --> 01:02:23,540
So this is going
to be 3 pi squared.

1088
01:02:23,540 --> 01:02:24,870
Okey Dokey?

1089
01:02:24,870 --> 01:02:25,375
Yes, sir?

1090
01:02:25,375 --> 01:02:28,380
STUDENT: How did you split it
into two integrals, right here?

1091
01:02:28,380 --> 01:02:31,100
MAGDALENA TODA: That's exactly
what I taught you before.

1092
01:02:31,100 --> 01:02:34,040
So if I had not
taught you before,

1093
01:02:34,040 --> 01:02:36,830
how did I prove that theorem?

1094
01:02:36,830 --> 01:02:41,430
The theorem that was
before was like that.

1095
01:02:41,430 --> 01:02:44,380
What was it?

1096
01:02:44,380 --> 01:02:48,700
Suppose I have a function of
theta, and a function of r,

1097
01:02:48,700 --> 01:02:52,543
and I have d theta
dr. And I think

1098
01:02:52,543 --> 01:02:55,780
this weather got to us,
because several people have

1099
01:02:55,780 --> 01:02:57,772
the cold and the flu.

1100
01:02:57,772 --> 01:02:59,266
Wash your hands a lot.

1101
01:02:59,266 --> 01:03:03,620
It's full of--
mathematicians full of germs.

1102
01:03:03,620 --> 01:03:08,559
So theta, you want theta to
be between whatever you want.

1103
01:03:08,559 --> 01:03:11,030
Any two numbers.

1104
01:03:11,030 --> 01:03:12,290
Alpha and beta.

1105
01:03:12,290 --> 01:03:14,840
And r between gamma, delta.

1106
01:03:14,840 --> 01:03:17,500
This is what I
explained last time.

1107
01:03:17,500 --> 01:03:22,450
So when you integrate with
respect to theta first inside,

1108
01:03:22,450 --> 01:03:26,420
g of r says I have nothing
to do with these guys.

1109
01:03:26,420 --> 01:03:28,390
They're not my type,
they're not my gang.

1110
01:03:28,390 --> 01:03:31,360
I'm going out, have
a beer by myself.

1111
01:03:31,360 --> 01:03:39,220
So he goes out and
joins the r group,

1112
01:03:39,220 --> 01:03:41,420
because theta and r
have nothing in common.

1113
01:03:41,420 --> 01:03:44,560
They are separate variables.

1114
01:03:44,560 --> 01:03:46,336
This is a function
of r only, and that's

1115
01:03:46,336 --> 01:03:48,100
a function of theta only.

1116
01:03:48,100 --> 01:03:50,410
This is what I'm talking about.

1117
01:03:50,410 --> 01:03:52,416
OK, so that's a constant.

1118
01:03:52,416 --> 01:03:55,620
That constant pulls out.

1119
01:03:55,620 --> 01:03:59,515
So in the end, what you have is
that constant that pulled out

1120
01:03:59,515 --> 01:04:06,270
is going to be alpha, beta, f of
beta d theta as a number, times

1121
01:04:06,270 --> 01:04:07,660
what's left inside?

1122
01:04:07,660 --> 01:04:11,250
Integral from gamma
to delta g of r

1123
01:04:11,250 --> 01:04:17,780
dr. So when the two functions
F and G are functions of theta,

1124
01:04:17,780 --> 01:04:22,170
respectively, r only, they have
nothing to do with one another,

1125
01:04:22,170 --> 01:04:24,740
and you can write
the original integral

1126
01:04:24,740 --> 01:04:28,570
as the product of integrals,
and it's really a lucky case.

1127
01:04:28,570 --> 01:04:33,260
But you are going to encounter
this lucky case many times

1128
01:04:33,260 --> 01:04:38,900
in your final, in the midterm,
in-- OK, now thinking of what

1129
01:04:38,900 --> 01:04:41,368
I wanted to put on the midterm.

1130
01:04:41,368 --> 01:04:45,310


1131
01:04:45,310 --> 01:04:47,885
Somebody asked me if I'm going
to put-- they looked already

1132
01:04:47,885 --> 01:04:52,180
at the homework and at the
book, and they asked me,

1133
01:04:52,180 --> 01:04:57,570
are we going to have something
like the area of the cardioid?

1134
01:04:57,570 --> 01:05:01,130
Maybe not necessarily
that-- or area

1135
01:05:01,130 --> 01:05:05,430
between a cardioid and a circle
that intersect each other.

1136
01:05:05,430 --> 01:05:10,130
Those were doable
even with Calc II.

1137
01:05:10,130 --> 01:05:12,710
Something like that, that
was doable with Calc II,

1138
01:05:12,710 --> 01:05:16,370
I don't want to do it with a
double integral in Calc III,

1139
01:05:16,370 --> 01:05:22,585
and I want to give some problems
that are relevant to you guys.

1140
01:05:22,585 --> 01:05:26,590


1141
01:05:26,590 --> 01:05:29,220
The question, what's going
to be on the midterm?

1142
01:05:29,220 --> 01:05:32,820
is not-- OK, what's going
to be on the midterm?

1143
01:05:32,820 --> 01:05:36,060
It's going to be something
very similar to the sample

1144
01:05:36,060 --> 01:05:37,816
that I'm going to write.

1145
01:05:37,816 --> 01:05:40,690
And I have already
included in that sample

1146
01:05:40,690 --> 01:05:44,730
the volume of a
sphere of radius r.

1147
01:05:44,730 --> 01:05:50,390
So how do you compute out
the weight-- exercise 3 or 4,

1148
01:05:50,390 --> 01:06:07,410
whatever that is-- we compute
the volume of a sphere using

1149
01:06:07,410 --> 01:06:08,357
double integrals.

1150
01:06:08,357 --> 01:06:16,640


1151
01:06:16,640 --> 01:06:20,210
I don't know if we have time to
do this problem, but if we do,

1152
01:06:20,210 --> 01:06:25,390
that will be the last problem--
when you ask you teacher,

1153
01:06:25,390 --> 01:06:28,996
why is the volume inside the
sphere, volume of a ball,

1154
01:06:28,996 --> 01:06:29,890
actually.

1155
01:06:29,890 --> 01:06:33,210
Well, the size-- the solid ball.

1156
01:06:33,210 --> 01:06:35,830
Why is it 4 pi r cubed over 2?

1157
01:06:35,830 --> 01:06:38,440
Your, did she tell
you, or she told

1158
01:06:38,440 --> 01:06:42,840
you something that you asked,
Mr. [? Jaime ?], for example?

1159
01:06:42,840 --> 01:06:47,512
They were supposed to tell
you that you can prove that

1160
01:06:47,512 --> 01:06:49,020
with Calc II or Calc III.

1161
01:06:49,020 --> 01:06:51,060
It's not easy.

1162
01:06:51,060 --> 01:06:52,885
It's not an elementary formula.

1163
01:06:52,885 --> 01:06:54,260
In the ancient
times, they didn't

1164
01:06:54,260 --> 01:06:57,030
know how to do it, because
they didn't know calculus.

1165
01:06:57,030 --> 01:07:00,496
So what they tried to is
to approximate it and see

1166
01:07:00,496 --> 01:07:02,770
how it goes.

1167
01:07:02,770 --> 01:07:07,300
Assume you have the
sphere of radius r,

1168
01:07:07,300 --> 01:07:09,490
and r is from here
to here, and I'm

1169
01:07:09,490 --> 01:07:12,932
going to go ahead and draw the
equator, the upper hemisphere,

1170
01:07:12,932 --> 01:07:18,510
the lower hemisphere, and
you shouldn't help me,

1171
01:07:18,510 --> 01:07:25,420
because isn't enough to say
it's twice the upper hemisphere

1172
01:07:25,420 --> 01:07:28,640
volume, right?

1173
01:07:28,640 --> 01:07:34,275
So if I knew the--
what is this called?

1174
01:07:34,275 --> 01:07:36,560
If I knew the
expression z equals

1175
01:07:36,560 --> 01:07:41,145
f of x, y of the spherical
cap of the hemisphere,

1176
01:07:41,145 --> 01:07:45,390
of the northern hemisphere,
I would be in business.

1177
01:07:45,390 --> 01:07:49,750
So if somebody even
tries-- one of my students,

1178
01:07:49,750 --> 01:07:53,220
I gave him that, he didn't know
polar coordinates very well,

1179
01:07:53,220 --> 01:07:57,620
so what he tried to do,
he was trying to do,

1180
01:07:57,620 --> 01:08:03,870
let's say z is going
to be square root of r

1181
01:08:03,870 --> 01:08:09,770
squared minus z squared minus
y squared over the domain.

1182
01:08:09,770 --> 01:08:13,300
So D will be what
domain? x squared

1183
01:08:13,300 --> 01:08:21,689
plus y squared between 0 and
r squared, am I right guys?

1184
01:08:21,689 --> 01:08:25,892
So the D is on
the floor, means x

1185
01:08:25,892 --> 01:08:28,620
squared plus y squared
between 0 and r squared.

1186
01:08:28,620 --> 01:08:32,345
This is the D that we have.

1187
01:08:32,345 --> 01:08:35,890
This is D So twice what?

1188
01:08:35,890 --> 01:08:37,109
f of x, y.

1189
01:08:37,109 --> 01:08:40,420


1190
01:08:40,420 --> 01:08:42,010
The volume of the
upper hemisphere

1191
01:08:42,010 --> 01:08:44,965
is the volume of everything
under this graph, which

1192
01:08:44,965 --> 01:08:46,380
is like a half.

1193
01:08:46,380 --> 01:08:49,910
It's the northern hemisphere.

1194
01:08:49,910 --> 01:08:52,819
dx dy, whatever is dx.

1195
01:08:52,819 --> 01:08:55,149
So he tried to do
it, and he came up

1196
01:08:55,149 --> 01:08:58,456
with something very ugly.

1197
01:08:58,456 --> 01:09:02,080
Of course you can imagine
what he came up with.

1198
01:09:02,080 --> 01:09:03,260
What would it be?

1199
01:09:03,260 --> 01:09:04,180
I don't know.

1200
01:09:04,180 --> 01:09:05,859
Oh, God.

1201
01:09:05,859 --> 01:09:10,336
x between minus r to r.

1202
01:09:10,336 --> 01:09:30,685
y would be between 0
and-- you have to draw it.

1203
01:09:30,685 --> 01:09:32,060
STUDENT: It's
going to be 0 or r.

1204
01:09:32,060 --> 01:09:32,319
STUDENT: Yeah.

1205
01:09:32,319 --> 01:09:33,080
STUDENT: Oh, no.

1206
01:09:33,080 --> 01:09:35,337
MAGDALENA TODA: So x
is between minus r--

1207
01:09:35,337 --> 01:09:36,322
STUDENT: It's going to
be as a function of x.

1208
01:09:36,322 --> 01:09:37,798
MAGDALENA TODA: And this is x.

1209
01:09:37,798 --> 01:09:39,375
And it's a function of x.

1210
01:09:39,375 --> 01:09:44,555
And then you go square root
r squared minus x squared.

1211
01:09:44,555 --> 01:09:47,046
It looks awful in
Cartesian coordinates.

1212
01:09:47,046 --> 01:09:53,609
And then for f, he just
plugged in that thingy,

1213
01:09:53,609 --> 01:09:55,570
and he said dy dx.

1214
01:09:55,570 --> 01:09:58,060
And he would be
right, except that I

1215
01:09:58,060 --> 01:09:59,530
would get a headache
just looking

1216
01:09:59,530 --> 01:10:03,590
at it, because it's a mess.

1217
01:10:03,590 --> 01:10:05,930
It's a horrible, horrible mess.

1218
01:10:05,930 --> 01:10:09,100
I don't like it.

1219
01:10:09,100 --> 01:10:13,860
So how am I going to solve
this in polar coordinates?

1220
01:10:13,860 --> 01:10:15,539
I still have the 2.

1221
01:10:15,539 --> 01:10:16,810
I cannot get rid of the 2.

1222
01:10:16,810 --> 01:10:21,350
How do I express--
in polar coordinates,

1223
01:10:21,350 --> 01:10:25,770
the 2 would be one for the upper
part, one for the lower part--

1224
01:10:25,770 --> 01:10:29,337
How do I express in polar
coordinates the disc?

1225
01:10:29,337 --> 01:10:31,213
Rho or r.

1226
01:10:31,213 --> 01:10:37,970
r between 0 to R, and theta,
all the way from 0 to 2 pi.

1227
01:10:37,970 --> 01:10:41,140
So I'm still sort of lucky
that I'm in business.

1228
01:10:41,140 --> 01:10:46,620
I go 0 to 2 pi
integral from 0 to r,

1229
01:10:46,620 --> 01:10:51,030
and for that guy, that
is in the integrand,

1230
01:10:51,030 --> 01:10:54,260
I'm going to say squared of z.

1231
01:10:54,260 --> 01:11:03,600
z is r squared minus-- who
is z squared plus y squared

1232
01:11:03,600 --> 01:11:06,682
in polar coordinates?

1233
01:11:06,682 --> 01:11:10,270
r squared. very good. r squared.

1234
01:11:10,270 --> 01:11:13,640
Don't forget that
instead of dy dx,

1235
01:11:13,640 --> 01:11:19,575
you have to say times r,
the Jacobian, dr d theta.

1236
01:11:19,575 --> 01:11:23,565
Can we solve this, and
find the correct formula?

1237
01:11:23,565 --> 01:11:25,840
That's what I'm talking about.

1238
01:11:25,840 --> 01:11:27,410
We need the u substitution.

1239
01:11:27,410 --> 01:11:30,800
Without the u substitution,
we will be dead meat.

1240
01:11:30,800 --> 01:11:33,060
But I don't know how
to do u substitution,

1241
01:11:33,060 --> 01:11:35,384
so I need your help.

1242
01:11:35,384 --> 01:11:37,769
Of course you can help me.

1243
01:11:37,769 --> 01:11:39,200
Who is the constant?

1244
01:11:39,200 --> 01:11:41,108
R is the constant.

1245
01:11:41,108 --> 01:11:43,030
It's a number.

1246
01:11:43,030 --> 01:11:46,312
Little r is a variable.

1247
01:11:46,312 --> 01:11:48,260
Little r is a variable.

1248
01:11:48,260 --> 01:11:53,617


1249
01:11:53,617 --> 01:11:55,570
STUDENT: r squared
is going to be the u.

1250
01:11:55,570 --> 01:11:56,780
MAGDALENA TODA: u, very good.

1251
01:11:56,780 --> 01:11:58,880
r squared minus r squared.

1252
01:11:58,880 --> 01:12:01,735
How come this is
working so well?

1253
01:12:01,735 --> 01:12:07,345
Look why du will be
constant prime 0 minus 2rdr.

1254
01:12:07,345 --> 01:12:10,010


1255
01:12:10,010 --> 01:12:18,430
So I take this couple
called rdr, this block,

1256
01:12:18,430 --> 01:12:21,778
and I identify the
block over here.

1257
01:12:21,778 --> 01:12:31,110
And rdr represents du
over minus 2, right?

1258
01:12:31,110 --> 01:12:32,820
So I have to be
smart and attentive,

1259
01:12:32,820 --> 01:12:36,622
because if I make a mistake
at the end, it's all over.

1260
01:12:36,622 --> 01:12:41,330
So 2 tiomes integral
from 0 to 2 pi.

1261
01:12:41,330 --> 01:12:44,620
I could get rid of
that and say just 2 pi.

1262
01:12:44,620 --> 01:12:46,280
Are you guys with me?

1263
01:12:46,280 --> 01:12:53,380
I could say 1 is theta-- as
the product, go out-- times--

1264
01:12:53,380 --> 01:12:57,381
and this is my integral that
I'm worried about, the one only

1265
01:12:57,381 --> 01:13:00,327
in r.

1266
01:13:00,327 --> 01:13:01,800
Let me review it.

1267
01:13:01,800 --> 01:13:06,720


1268
01:13:06,720 --> 01:13:09,140
This is the only one
I'm worried about.

1269
01:13:09,140 --> 01:13:10,840
This is a piece of cake.

1270
01:13:10,840 --> 01:13:12,610
This is 2, this is 2 pi.

1271
01:13:12,610 --> 01:13:14,010
This whole thing is 4 pi a.

1272
01:13:14,010 --> 01:13:18,360
At least I got some 4 pi out.

1273
01:13:18,360 --> 01:13:19,970
What have I done in here?

1274
01:13:19,970 --> 01:13:23,300
I've applied the u
substitution, and I

1275
01:13:23,300 --> 01:13:25,100
have to be doing a better job.

1276
01:13:25,100 --> 01:13:30,690
I get 4 pi times what is
it after u substitution.

1277
01:13:30,690 --> 01:13:37,080
This guy was minus
1/2 du, right?

1278
01:13:37,080 --> 01:13:40,295
This fellow is squared
u, [? squared ?]

1279
01:13:40,295 --> 01:13:42,106
squared u as a power.

1280
01:13:42,106 --> 01:13:43,110
STUDENT: u to the 1/2.

1281
01:13:43,110 --> 01:13:44,820
MAGDALENA TODA: u
to the one half.

1282
01:13:44,820 --> 01:13:51,877
And for the integral, what
in the world do I write?

1283
01:13:51,877 --> 01:13:52,710
STUDENT: r squared--

1284
01:13:52,710 --> 01:13:54,460
MAGDALENA TODA: OK.

1285
01:13:54,460 --> 01:14:03,232
So when little r is 0, u
is going to be r squared.

1286
01:14:03,232 --> 01:14:08,790
When little r is
big R, you get 0.

1287
01:14:08,790 --> 01:14:11,020
Now you have to
help me finish this.

1288
01:14:11,020 --> 01:14:12,710
It should be a piece of cake.

1289
01:14:12,710 --> 01:14:15,650
I cannot believe it's hard.

1290
01:14:15,650 --> 01:14:19,442
What is the integral of 4 pi?

1291
01:14:19,442 --> 01:14:20,858
Copy and paste.

1292
01:14:20,858 --> 01:14:25,188
Minus 1/2, integrate
y to the 1/2.

1293
01:14:25,188 --> 01:14:27,172
STUDENT: 2/3u to the 3/2.

1294
01:14:27,172 --> 01:14:34,315
MAGDALENA TODA: 2/3 u to the
3/2, between u equals 0 up,

1295
01:14:34,315 --> 01:14:37,568
and u equals r squared down.

1296
01:14:37,568 --> 01:14:38,610
It still looks bad, but--

1297
01:14:38,610 --> 01:14:40,109
STUDENT: You've got
a negative sign.

1298
01:14:40,109 --> 01:14:41,960
MAGDALENA TODA: I've
got a negative sign.

1299
01:14:41,960 --> 01:14:42,940
STUDENT: Where is it--

1300
01:14:42,940 --> 01:14:46,090
MAGDALENA TODA: So when
I go this minus that,

1301
01:14:46,090 --> 01:14:47,780
it's going to be very nice.

1302
01:14:47,780 --> 01:14:48,466
Why?

1303
01:14:48,466 --> 01:14:56,390
I'm going to say minus 4
pi over 2 times 2 over 3.

1304
01:14:56,390 --> 01:14:59,190
I should have simplified
them from the beginning.

1305
01:14:59,190 --> 01:15:05,220
I have minus 5 pi over
3 times at 0 I have 0.

1306
01:15:05,220 --> 01:15:09,265
At r squared, I have r
squared, and the square root

1307
01:15:09,265 --> 01:15:11,738
is r, r cubed.

1308
01:15:11,738 --> 01:15:12,730
r cubed.

1309
01:15:12,730 --> 01:15:19,690


1310
01:15:19,690 --> 01:15:22,060
Oh my God, look how
beautiful it is.

1311
01:15:22,060 --> 01:15:24,000
Two minuses in a row.

1312
01:15:24,000 --> 01:15:27,154
Multiply, give me a plus.

1313
01:15:27,154 --> 01:15:28,320
STUDENT: This is the answer.

1314
01:15:28,320 --> 01:15:29,770
MAGDALENA TODA: Plus.

1315
01:15:29,770 --> 01:15:37,150
4 pi up over 3 down, r cubed.

1316
01:15:37,150 --> 01:15:40,670
So we proved something
that is essential,

1317
01:15:40,670 --> 01:15:42,900
and we knew it from
when we were in school,

1318
01:15:42,900 --> 01:15:46,140
but they told us that
we cannot prove it,

1319
01:15:46,140 --> 01:15:50,555
because we couldn't prove that
the volume of a ball was 4 pi r

1320
01:15:50,555 --> 01:15:51,700
cubed over 3.

1321
01:15:51,700 --> 01:15:52,760
Yes, sir?

1322
01:15:52,760 --> 01:15:55,719
STUDENT: Why are the limits
of integration reversed?

1323
01:15:55,719 --> 01:15:57,010
Why is r squared on the bottom?

1324
01:15:57,010 --> 01:16:02,350
MAGDALENA TODA: Because
first comes little r, 0,

1325
01:16:02,350 --> 01:16:06,310
and then comes little r to
be big R. When I plug them

1326
01:16:06,310 --> 01:16:09,640
in in this order-- so
let's plug them in first,

1327
01:16:09,640 --> 01:16:11,050
little r equals 0.

1328
01:16:11,050 --> 01:16:15,570
I get, for the bottom part,
I get u equals r squared,

1329
01:16:15,570 --> 01:16:18,930
and when little
r equals big R, I

1330
01:16:18,930 --> 01:16:21,806
get big R squared minus
big R squared equals 0.

1331
01:16:21,806 --> 01:16:24,060
And that's the good
thing, because when

1332
01:16:24,060 --> 01:16:28,750
I do that, I get a minus, and
with the minus I already had,

1333
01:16:28,750 --> 01:16:29,800
I get a plus.

1334
01:16:29,800 --> 01:16:33,470
And the volume is a positive
volume, like every volume.

1335
01:16:33,470 --> 01:16:36,110
4 pi [INAUDIBLE].

1336
01:16:36,110 --> 01:16:39,380
So that's it for today.

1337
01:16:39,380 --> 01:16:42,050
We finished 12-- what is that?

1338
01:16:42,050 --> 01:16:44,074
12.3, polar coordinates.

1339
01:16:44,074 --> 01:16:49,541
And we will next time
do some homework.

1340
01:16:49,541 --> 01:16:52,026
Ah, I opened the
homework for you.

1341
01:16:52,026 --> 01:16:55,008
So go ahead and do at least
the first 10 problems.

1342
01:16:55,008 --> 01:16:57,990
If you have difficulties,
let me know on Tuesday,

1343
01:16:57,990 --> 01:17:02,463
so we can work some in class.

1344
01:17:02,463 --> 01:17:04,948
STUDENT: [? You do ?] so much.

1345
01:17:04,948 --> 01:17:09,582
STUDENT: So, I went to the
[INAUDIBLE], and I asked them,

1346
01:17:09,582 --> 01:17:10,415
[INTERPOSING VOICES]

1347
01:17:10,415 --> 01:17:13,894


1348
01:17:13,894 --> 01:17:16,379
[SIDE CONVERSATION]

1349
01:17:16,379 --> 01:18:34,302


1350
01:18:34,302 --> 01:18:35,799
STUDENT: Can you
imagine two years

1351
01:18:35,799 --> 01:18:38,294
of a calculus that's the
equivalent to [? American ?]

1352
01:18:38,294 --> 01:18:39,664
and only two credits?

1353
01:18:39,664 --> 01:18:41,288
MAGDALENA TODA:
Because in your system,

1354
01:18:41,288 --> 01:18:44,282
everything was pretty
much accelerated.

1355
01:18:44,282 --> 01:18:46,777
STUDENT: Yeah, and
they say, no, no, no--

1356
01:18:46,777 --> 01:18:48,274
I had to ask him again.

1357
01:18:48,274 --> 01:18:52,765
[INAUDIBLE] calculus,
in two years,

1358
01:18:52,765 --> 01:18:56,258
that is only equivalent
to two credits.

1359
01:18:56,258 --> 01:18:58,251
I was like--

1360
01:18:58,251 --> 01:18:59,751
MAGDALENA TODA:
Anyway, what happens

1361
01:18:59,751 --> 01:19:03,244
is that we used to have
very good evaluators

1362
01:19:03,244 --> 01:19:06,238
in the registrar's office, and
most of those people retired

1363
01:19:06,238 --> 01:19:09,232
or they got promoted in other
administrative positions.

1364
01:19:09,232 --> 01:19:11,727
So they have three new hires.

1365
01:19:11,727 --> 01:19:14,599
Those guys, they don't
know what they are doing.

1366
01:19:14,599 --> 01:19:17,064
Imagine, you would
finish, graduate, today,

1367
01:19:17,064 --> 01:19:19,529
next week, you go
for the registrar.

1368
01:19:19,529 --> 01:19:21,501
You don't know
what you're doing.

1369
01:19:21,501 --> 01:19:22,480
You need time.

1370
01:19:22,480 --> 01:19:22,980
Yes?

1371
01:19:22,980 --> 01:19:25,445
STUDENT: I had a question
about the homework.

1372
01:19:25,445 --> 01:19:27,403
I'll wait for [INAUDIBLE].

1373
01:19:27,403 --> 01:19:28,403
MAGDALENA TODA: It's OK.

1374
01:19:28,403 --> 01:19:30,375
Do you have secrets?

1375
01:19:30,375 --> 01:19:31,854
STUDENT: No, I don't.

1376
01:19:31,854 --> 01:19:33,826
MAGDALENA TODA: Homework
is due the 32st.

1377
01:19:33,826 --> 01:19:34,812
STUDENT: No, I had a
question from the homework.

1378
01:19:34,812 --> 01:19:35,305
Like I had a problem that I
was working on, and I was like

1379
01:19:35,305 --> 01:19:36,721
MAGDALENA TODA:
From the homework.

1380
01:19:36,721 --> 01:19:39,249
OK You can wait.

1381
01:19:39,249 --> 01:19:42,207
You guys have other,
more basic questions?

1382
01:19:42,207 --> 01:19:43,040
[INTERPOSING VOICES]

1383
01:19:43,040 --> 01:19:49,211


1384
01:19:49,211 --> 01:19:50,960
MAGDALENA TODA: There
is only one meeting.

1385
01:19:50,960 --> 01:19:53,930
Oh, you mean-- Ah.

1386
01:19:53,930 --> 01:19:55,415
Yes, I do.

1387
01:19:55,415 --> 01:19:59,870
I have the following
three-- Tuesday,

1388
01:19:59,870 --> 01:20:04,992
Wednesday, and Friday-
no, Tuesday, Wednesday,

1389
01:20:04,992 --> 01:20:05,620
and Thursday.

1390
01:20:05,620 --> 01:20:09,870
On Friday we can have something,
some special arrangement.

1391
01:20:09,870 --> 01:20:12,410
This Friday?

1392
01:20:12,410 --> 01:20:16,750
OK, how about like 11:15.

1393
01:20:16,750 --> 01:20:22,574
Today, I have--
I have right now.

1394
01:20:22,574 --> 01:20:23,516
2:00.

1395
01:20:23,516 --> 01:20:26,784
And I think the grad
students will come later.

1396
01:20:26,784 --> 01:20:28,768
So you can just right now.

1397
01:20:28,768 --> 01:20:32,240
And tomorrow around 11:15,
because I have meetings

1398
01:20:32,240 --> 01:20:34,730
before 11 at the college.

1399
01:20:34,730 --> 01:20:37,260
STUDENT: Do you mind if I go
get something to eat first?

1400
01:20:37,260 --> 01:20:39,134
Or how long do you think
they'll be in your office?

1401
01:20:39,134 --> 01:20:40,122
MAGDALENA TODA:
Even if they come,

1402
01:20:40,122 --> 01:20:42,098
I'm going to stop
them and talk to you,

1403
01:20:42,098 --> 01:20:43,580
so don't worry about it.

1404
01:20:43,580 --> 01:20:44,074
STUDENT: Thank you very much.

1405
01:20:44,074 --> 01:20:44,568
I'll see you later.

1406
01:20:44,568 --> 01:20:46,050
STUDENT: I just wanted to say
I'm sorry for coming in late.

1407
01:20:46,050 --> 01:20:47,038
I slept in a little
bit this morning--

1408
01:20:47,038 --> 01:20:49,461
MAGDALENA TODA: Did you
get the chance to sign?

1409
01:20:49,461 --> 01:20:50,002
STUDENT: Yes.

1410
01:20:50,002 --> 01:20:50,990
MAGDALENA TODA:
There is no problem.

1411
01:20:50,990 --> 01:20:51,490
I'm--

1412
01:20:51,490 --> 01:20:55,930
STUDENT: I woke up at like
12:30-- I woke up at like 11:30

1413
01:20:55,930 --> 01:20:59,614
and I just fell right back
asleep, and then I got up

1414
01:20:59,614 --> 01:21:01,364
and I looked at my
phone and it was 12:30,

1415
01:21:01,364 --> 01:21:03,340
and I was like, I
have class right now.

1416
01:21:03,340 --> 01:21:04,822
And so what happened was like--

1417
01:21:04,822 --> 01:21:05,810
MAGDALENA TODA: You were tired.

1418
01:21:05,810 --> 01:21:06,830
You were doing
homework until late.

1419
01:21:06,830 --> 01:21:08,610
STUDENT: --homework
and like, I usually

1420
01:21:08,610 --> 01:21:10,972
am on for an
earlier class, and I

1421
01:21:10,972 --> 01:21:12,930
didn't go to bed earlier
than I did last night,

1422
01:21:12,930 --> 01:21:14,774
and so I just overslept.

1423
01:21:14,774 --> 01:21:17,134
MAGDALENA TODA: I
did the same, anyway.

1424
01:21:17,134 --> 01:21:18,674
I have similar experience.

1425
01:21:18,674 --> 01:21:20,090
STUDENT: You have
a very nice day.

1426
01:21:20,090 --> 01:21:21,173
MAGDALENA TODA: Thank you.

1427
01:21:21,173 --> 01:21:21,890
You too.

1428
01:21:21,890 --> 01:21:24,338
So, show me what
you want to ask.

1429
01:21:24,338 --> 01:21:25,534
STUDENT: There it was.

1430
01:21:25,534 --> 01:21:27,200
I looked at that
problem, and I thought,

1431
01:21:27,200 --> 01:21:29,871
that's extremely
simple, acceleration--

1432
01:21:29,871 --> 01:21:31,746
MAGDALENA TODA: Are they
independent, really?

1433
01:21:31,746 --> 01:21:32,287
STUDENT: Huh?

1434
01:21:32,287 --> 01:21:34,930
MAGDALENA TODA: Are they--
b and t are independent?

1435
01:21:34,930 --> 01:21:36,660
I need to stop.

1436
01:21:36,660 --> 01:21:39,110
STUDENT: But I
didn't even bother.