Order of operations: PEMDAS

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:03.360
Now that we've got the basics
of order of operations out of

00:00:03.360 --> 00:00:06.100
the way, let's try to tackle a
really hairy and

00:00:06.100 --> 00:00:07.240
beastly problem.

00:00:07.240 --> 00:00:08.990
So here, we have all
sorts of parentheses and

00:00:08.990 --> 00:00:10.340
numbers flying around.

00:00:10.340 --> 00:00:13.190
But in any of these order of
operations problems, you really

00:00:13.190 --> 00:00:15.650
just have to take a deep breath
and remember, we're going

00:00:15.650 --> 00:00:17.870
to do parentheses first.

00:00:17.870 --> 00:00:18.810
Parentheses.

00:00:18.810 --> 00:00:19.820
P for parentheses.

00:00:19.820 --> 00:00:20.760
Then exponents.

00:00:20.760 --> 00:00:22.330
Don't worry if you don't know
what exponents are, because

00:00:22.330 --> 00:00:23.910
this has no exponents in them.

00:00:23.910 --> 00:00:25.930
Then you're going to do
multiplication and division.

00:00:25.930 --> 00:00:27.220
They're at the same level.

00:00:27.220 --> 00:00:28.980
Then you do addition
and subtraction.

00:00:28.980 --> 00:00:31.330
So some people remember PEMDAS.

00:00:31.330 --> 00:00:33.640
But if you remember PEMDAS,
remember multiplication,

00:00:33.640 --> 00:00:34.900
division, same level.

00:00:34.900 --> 00:00:38.500
Addition and subtraction,
also at the same level.

00:00:38.500 --> 00:00:42.360
So let's figure what the
order of operations say that

00:00:42.360 --> 00:00:44.290
this should evaluate to.

00:00:44.290 --> 00:00:46.100
So the first thing we're going
to do is our parentheses.

00:00:46.100 --> 00:00:47.590
And we have a lot of
parentheses here.

00:00:47.590 --> 00:00:50.800
We have this expression in
parentheses right there, and

00:00:50.800 --> 00:00:54.930
then even within that we
have these parentheses.

00:00:54.930 --> 00:00:58.370
So our order of operations say,
look, do your parentheses

00:00:58.370 --> 00:01:01.120
first, but in order to evaluate
this outer parentheses-- this

00:01:01.120 --> 00:01:02.910
orange thing-- we're going to
have to evaluate this thing

00:01:02.910 --> 00:01:04.750
in yellow right there.

00:01:04.750 --> 00:01:07.000
So let's evaluate
this whole thing.

00:01:07.000 --> 00:01:08.530
So how can we simplify it?

00:01:08.530 --> 00:01:10.910
Well, if we look at just inside
of it, the first thing we want

00:01:10.910 --> 00:01:12.690
to do is simplify the
parentheses inside

00:01:12.690 --> 00:01:13.940
the parentheses.

00:01:13.940 --> 00:01:16.590
So you see this 5
minus 2 right there?

00:01:16.590 --> 00:01:18.560
We're going to do that
first no matter what.

00:01:18.560 --> 00:01:20.000
And that's easy to evaluate.

00:01:20.000 --> 00:01:23.270
5 minus 2 is 3.

00:01:23.270 --> 00:01:26.690
And so this simplifies to--
I'll do it step by step.

00:01:26.690 --> 00:01:28.480
Once you get the hang of
it, you can do multiple

00:01:28.480 --> 00:01:29.570
steps at once.

00:01:29.570 --> 00:01:34.770
So this is going to be
7 plus 3 times the 5

00:01:34.770 --> 00:01:36.710
minus 2, which is 3.

00:01:36.710 --> 00:01:39.620
And all of those have
parentheses around it.

00:01:39.620 --> 00:01:41.780
And of course, you have
all the stuff on either

00:01:41.780 --> 00:01:44.960
side-- the divide 4-- no.

00:01:44.960 --> 00:01:45.820
Oops.

00:01:45.820 --> 00:01:46.650
That's not what I want.

00:01:46.650 --> 00:01:48.760
I wanted to copy and paste.

00:01:48.760 --> 00:01:50.840
I want to copy and paste
that right there.

00:01:50.840 --> 00:01:55.246
So copy, then-- no, that's
giving me the wrong thing.

00:01:55.246 --> 00:01:57.290
It would've been easier--
let me just rewrite it.

00:01:57.290 --> 00:01:58.230
That's the easiest thing.

00:01:58.230 --> 00:02:00.590
I'm having technical
difficulties.

00:02:00.590 --> 00:02:04.660
So divided by 4 times 2.

00:02:04.660 --> 00:02:10.200
And on this side, you had that
7 times 2 plus this thing in

00:02:10.200 --> 00:02:11.760
orange parentheses there.

00:02:11.760 --> 00:02:14.060
Now, at any step you
just look again.

00:02:14.060 --> 00:02:15.690
We always want to do
parentheses first.

00:02:15.690 --> 00:02:17.140
Well, you keep wanting to
do and is there really

00:02:17.140 --> 00:02:18.750
no parentheses left?

00:02:18.750 --> 00:02:22.210
So we have to evaluate this
parentheses in orange here.

00:02:22.210 --> 00:02:25.160
So we have to evaluate
this thing first.

00:02:25.160 --> 00:02:26.550
But in order to evaluate
this thing, we have

00:02:26.550 --> 00:02:27.810
to look inside of it.

00:02:27.810 --> 00:02:31.290
And when you look inside of it,
you have 7 plus 3 times 3.

00:02:31.290 --> 00:02:33.970
So if you just had 7 plus
3 times 3, how would

00:02:33.970 --> 00:02:35.150
you evaluate it?

00:02:35.150 --> 00:02:37.260
Well, look back to your
order of operations.

00:02:37.260 --> 00:02:40.550
We're inside the parentheses
here, so inside of it there are

00:02:40.550 --> 00:02:42.640
no longer any parentheses.

00:02:42.640 --> 00:02:45.000
So the next thing we should do
is-- there are no exponents.

00:02:45.000 --> 00:02:46.200
There is multiplication.

00:02:46.200 --> 00:02:49.970
So we do that before we do
any addition or subtraction.

00:02:49.970 --> 00:02:54.250
So we want to do the 3 times
3 before we add the 7.

00:02:54.250 --> 00:02:59.090
So this is going to be 7
plus-- and the 3 times

00:02:59.090 --> 00:03:00.270
3 we want to do first.

00:03:00.270 --> 00:03:01.850
We want to do the
multiplication first.

00:03:01.850 --> 00:03:04.380
7 plus 9.

00:03:04.380 --> 00:03:06.360
That's going to be in
the orange parentheses.

00:03:06.360 --> 00:03:10.020
And then you have the 7
times 2 plus that, on

00:03:10.020 --> 00:03:10.870
the left hand side.

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You have the divided by 4 times
2 on the right hand side.

00:03:15.060 --> 00:03:17.370
And now this-- the thing in
parentheses-- because we

00:03:17.370 --> 00:03:19.200
still want to do the
parentheses first.

00:03:19.200 --> 00:03:20.650
Pretty easy to evaluate.

00:03:20.650 --> 00:03:23.770
What's 7 plus 9?

00:03:23.770 --> 00:03:27.040
7 plus 9 is 16.

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And so everything we have
simplifies to 7 times 2 plus

00:03:32.280 --> 00:03:38.140
16 divided by 4 times 2.

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Now we don't have any
parentheses left, so we

00:03:40.103 --> 00:03:42.910
don't have to worry
about the P in PEMDAS.

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We have no E, no
exponents in this.

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So then we go straight to
multiplication and division.

00:03:48.650 --> 00:03:51.970
We have a multiplication--
we have some multiplication

00:03:51.970 --> 00:03:53.200
going on there.

00:03:53.200 --> 00:03:55.200
We have some division
going on here, and a

00:03:55.200 --> 00:03:56.800
multiplication there.

00:03:56.800 --> 00:04:00.210
So we should do these
next, before we do this

00:04:00.210 --> 00:04:01.670
addition right there.

00:04:01.670 --> 00:04:03.830
So we could do this
multiplication.

00:04:03.830 --> 00:04:05.260
We could do that
multiplication.

00:04:05.260 --> 00:04:08.480
7 times 2 is 14.

00:04:08.480 --> 00:04:10.015
We're going to wait
to do that addition.

00:04:12.730 --> 00:04:16.070
And then here we have a
16 divided by 4 times 2.

00:04:16.070 --> 00:04:17.770
That gets priority of the
addition, so we're going to do

00:04:17.770 --> 00:04:19.160
that before we do the addition.

00:04:19.160 --> 00:04:21.370
But how do we evaluate that?

00:04:21.370 --> 00:04:23.810
Do we do the division first,
or the multiplication first?

00:04:23.810 --> 00:04:26.340
And remember, I told you in the
last video, when you have 2--

00:04:26.340 --> 00:04:28.760
when you have multiple
operations of the same level--

00:04:28.760 --> 00:04:31.190
in this case, division and
multiplication-- they're

00:04:31.190 --> 00:04:32.220
at the same level.

00:04:32.220 --> 00:04:34.460
You're safest going
left to right.

00:04:34.460 --> 00:04:36.360
Or you should go left to right.

00:04:36.360 --> 00:04:39.950
So you do 16 divided by 4 is 4.

00:04:39.950 --> 00:04:42.180
So this thing right here--
simplify 16 divided

00:04:42.180 --> 00:04:43.680
by 4 times 2.

00:04:43.680 --> 00:04:49.420
It simplifies to 4 times 2.

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That's this thing in
green right there.

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And then we're going to want to
do the multiplication next.

00:04:54.180 --> 00:04:58.050
So this is going to simplify
to-- because multiplication

00:04:58.050 --> 00:05:01.930
takes priority over addition--
this simplifies to 8.

00:05:01.930 --> 00:05:08.740
And so you get 14-- this
14 right here-- plus 8.

00:05:08.740 --> 00:05:10.730
And what's 14 plus 8?

00:05:10.730 --> 00:05:13.190
That is 22.

00:05:13.190 --> 00:05:16.450
That is equal to 22.

00:05:16.450 --> 00:05:18.990
And we are done.


Introduction to order of operations

WEBVTT
Kind: captions
Language: en

00:00:00.380 --> 00:00:02.640
In this video we're going
to talk a little bit about

00:00:02.640 --> 00:00:04.720
order of operations.

00:00:04.720 --> 00:00:07.310
And I want you to pay close
attention because really

00:00:07.310 --> 00:00:09.570
everything else that you're
going to do in mathematics

00:00:09.570 --> 00:00:12.630
is going to be based on you
having a solid grounding

00:00:12.630 --> 00:00:14.150
in order of operations.

00:00:14.150 --> 00:00:17.140
So what do we even mean when
we say order of operations?

00:00:17.140 --> 00:00:18.640
So let me give you an example.

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The whole point is so that we
have one way to interpret

00:00:21.780 --> 00:00:23.380
a mathematical statement.

00:00:23.380 --> 00:00:25.660
So let's say I have the
mathematical statement

00:00:25.660 --> 00:00:32.100
7 plus 3 times 5.

00:00:32.100 --> 00:00:36.495
Now if we didn't all agree on
order of operations, there

00:00:36.495 --> 00:00:39.180
would be two ways of
interpreting this statement.

00:00:39.180 --> 00:00:42.400
You could just read it left to
right, so you could say well,

00:00:42.400 --> 00:00:47.030
let me just take 7 plus 3, you
could say 7 plus 3 and then

00:00:47.030 --> 00:00:49.510
multiply that times 5.

00:00:49.510 --> 00:00:56.090
And 7 plus 3 is 10, and then
you multiply that by 5.

00:00:56.090 --> 00:00:59.500
10 times 5, it
would get you 50.

00:00:59.500 --> 00:01:02.180
So that's one way you would
interpret it if we didn't agree

00:01:02.180 --> 00:01:03.310
on an order of operations.

00:01:03.310 --> 00:01:03.925
Maybe it's a natural way.

00:01:03.925 --> 00:01:05.530
You just go left to right.

00:01:05.530 --> 00:01:07.540
Another way you could interpret
it you say, I like to do

00:01:07.540 --> 00:01:10.210
multiplication before
I do addition.

00:01:10.210 --> 00:01:13.980
So you might interpret it as --
I'll try to color code it --

00:01:13.980 --> 00:01:18.070
7 plus -- and you do
the 3 times 5 first.

00:01:18.070 --> 00:01:26.550
7 plus 3 times 5, which would
be 7 plus 3 times 5 is

00:01:26.550 --> 00:01:30.040
15, and 7 plus 15 is 22.

00:01:32.560 --> 00:01:34.860
So notice, we interpreted
this statement in

00:01:34.860 --> 00:01:35.600
two different ways.

00:01:35.600 --> 00:01:38.140
This was just straight left
to right doing addition

00:01:38.140 --> 00:01:39.680
then the multiplication.

00:01:39.680 --> 00:01:42.143
This way we did the
multiplication first then the

00:01:42.143 --> 00:01:44.540
addition, we got two different
answers, and that's just not

00:01:44.540 --> 00:01:45.880
cool in mathematics.

00:01:45.880 --> 00:01:49.910
If this was part of some effort
to send something to the moon

00:01:49.910 --> 00:01:52.070
because two people interpreted
it a different way or another

00:01:52.070 --> 00:01:54.170
one computer interpreted one
way and another computer

00:01:54.170 --> 00:01:57.000
interpreted it another way, the
satellite might go to mars.

00:01:57.000 --> 00:01:59.730
So this is just completely
unacceptable, and that's why

00:01:59.730 --> 00:02:02.920
we have to have an agreed
upon order of operations.

00:02:02.920 --> 00:02:06.540
An agreed upon way to
interpret this statement.

00:02:06.540 --> 00:02:10.750
So the agreed upon order of
operations is to do parentheses

00:02:10.750 --> 00:02:19.600
first -- let me write it over
here -- then do exponents.

00:02:19.600 --> 00:02:21.140
If you don't know what
exponents are don't worry

00:02:21.140 --> 00:02:24.220
about it right now.

00:02:24.220 --> 00:02:26.550
In this video we're not going
to have any exponents in our

00:02:26.550 --> 00:02:28.940
examples, so you don't
really have to worry about

00:02:28.940 --> 00:02:30.020
them for this video.

00:02:30.020 --> 00:02:34.140
Then you do multiplication --
I'll just right mult, short for

00:02:34.140 --> 00:02:37.250
multiplication -- then you do
multiplication and division

00:02:37.250 --> 00:02:40.260
next, they kind of have the
same level of priority.

00:02:40.260 --> 00:02:42.410
And then finally you do
addition and subtraction.

00:02:47.280 --> 00:02:50.230
So what does this order of
operations -- let me label

00:02:50.230 --> 00:02:52.990
it -- this right here,
that is the agreed upon

00:02:52.990 --> 00:02:55.910
order of operations.

00:02:55.910 --> 00:02:58.740
If we follow these order of
operations we should always

00:02:58.740 --> 00:03:01.400
get to the same answer
for a given statement.

00:03:01.400 --> 00:03:02.390
So what does this tell us?

00:03:02.390 --> 00:03:05.070
What is the best way to
interpret this up here?

00:03:05.070 --> 00:03:08.510
Well we have no parentheses --
parentheses look like that.

00:03:08.510 --> 00:03:10.200
Those little curly
things around numbers.

00:03:10.200 --> 00:03:11.870
We don't have any
parentheses here.

00:03:11.870 --> 00:03:14.690
I'll do some examples that
do have parentheses.

00:03:14.690 --> 00:03:16.380
We don't have any
exponents here.

00:03:16.380 --> 00:03:18.670
But we do have some
multiplication and division

00:03:18.670 --> 00:03:20.440
or we actually just have
some multiplication.

00:03:20.440 --> 00:03:23.820
So we'll order of operations,
do the multiplication

00:03:23.820 --> 00:03:25.420
and division first.

00:03:25.420 --> 00:03:27.970
So it says do the
multiplication first.

00:03:27.970 --> 00:03:29.080
That's a multiplication.

00:03:29.080 --> 00:03:32.010
So it says do this
operation first.

00:03:32.010 --> 00:03:36.500
It gets priority over
addition or subtraction.

00:03:36.500 --> 00:03:39.390
So if we do this first we
get the 3 times 5, which is

00:03:39.390 --> 00:03:41.870
15, and then we add the 7.

00:03:41.870 --> 00:03:44.570
The addition or subtraction --
I'll do it here, addition,

00:03:44.570 --> 00:03:46.470
we just have addition.

00:03:46.470 --> 00:03:47.370
Just like that.

00:03:47.370 --> 00:03:49.840
So we do the multiplication
first, get 15, then

00:03:49.840 --> 00:03:51.920
add the 7, 22.

00:03:51.920 --> 00:03:56.520
So based upon the agreed order
of operations, this right

00:03:56.520 --> 00:03:58.770
here is the correct answer.

00:03:58.770 --> 00:04:01.620
The correct way to
interpret this statement.

00:04:01.620 --> 00:04:03.040
Let's do another example.

00:04:03.040 --> 00:04:07.550
I think it'll make things a
little bit more clear, and

00:04:07.550 --> 00:04:09.620
I'll do the example in pink.

00:04:09.620 --> 00:04:19.063
So let's say I have 7 plus 3 --
I'll put some parentheses there

00:04:19.063 --> 00:04:29.980
-- times 4 divided by
2 minus 5 times 6.

00:04:29.980 --> 00:04:32.890
So there's all sorts of crazy
things here, but if you just

00:04:32.890 --> 00:04:36.670
follow the order of operations
you'll simplify it in a very

00:04:36.670 --> 00:04:38.960
clean way and hopefully we'll
all get the same answer.

00:04:38.960 --> 00:04:41.330
So let's just follow the
order of operations.

00:04:41.330 --> 00:04:43.320
The first thing we have to
do is look for parentheses.

00:04:43.320 --> 00:04:44.540
Are there parentheses here?

00:04:44.540 --> 00:04:45.840
Yes, there are.

00:04:45.840 --> 00:04:48.440
There's parentheses
around the 7 plus 3.

00:04:48.440 --> 00:04:51.050
So it says let's do that first.

00:04:51.050 --> 00:04:53.620
So 7 plus 3 is 10.

00:04:53.620 --> 00:04:55.740
So this we can simplify,
just looking at this

00:04:55.740 --> 00:04:59.280
order operations, to
10 times all of that.

00:04:59.280 --> 00:05:00.900
Let me copy and paste
that so I don't have

00:05:00.900 --> 00:05:02.310
to keep re-writing it.

00:05:06.610 --> 00:05:09.310
So that simplifies to
10 times all of that.

00:05:09.310 --> 00:05:12.000
We did our parentheses first.

00:05:12.000 --> 00:05:13.030
Then what do we do?

00:05:13.030 --> 00:05:15.370
There are no more parentheses
in this expression.

00:05:15.370 --> 00:05:16.630
Then we should do exponents.

00:05:16.630 --> 00:05:19.105
I don't see any exponents here,
and if you're curious what

00:05:19.105 --> 00:05:20.650
exponents look like, an
exponent would look

00:05:20.650 --> 00:05:22.890
like 7 squared.

00:05:22.890 --> 00:05:25.240
You'd see these little small
numbers up in the top right.

00:05:25.240 --> 00:05:26.880
We don't have any exponents
here so we don't have

00:05:26.880 --> 00:05:28.130
to worry about it.

00:05:28.130 --> 00:05:30.160
Then it says to do
multiplication and

00:05:30.160 --> 00:05:32.400
division next.

00:05:32.400 --> 00:05:33.890
So where do we see
multiplication?

00:05:33.890 --> 00:05:36.470
We have a multiplication,
a division, a

00:05:36.470 --> 00:05:38.520
multiplication again.

00:05:38.520 --> 00:05:43.090
Now, when you have multiple
operations at the same level,

00:05:43.090 --> 00:05:45.760
when our order of operations,
multiplication and division

00:05:45.760 --> 00:05:50.230
are the same level, then
you do left to right.

00:05:50.230 --> 00:05:54.010
So in this situation you're
going to multiply by 4

00:05:54.010 --> 00:05:55.500
and then divide by 2.

00:05:55.500 --> 00:05:59.710
You won't multiply
by 4 divided by 2.

00:05:59.710 --> 00:06:02.200
Then we'll do the 5 times
6 before we do the

00:06:02.200 --> 00:06:03.950
subtraction right here.

00:06:03.950 --> 00:06:06.070
So let's figure
out what this is.

00:06:06.070 --> 00:06:10.040
So we'll do this
multiplication first.

00:06:10.040 --> 00:06:12.500
We could simultaneously do this
multiplication because it's

00:06:12.500 --> 00:06:13.360
not going to change things.

00:06:13.360 --> 00:06:15.570
But I'll do things
one step at a time.

00:06:15.570 --> 00:06:19.840
So the next step we're going
to do is this 10 times 4.

00:06:19.840 --> 00:06:23.160
10 times 4 is 40.

00:06:23.160 --> 00:06:32.430
10 times 4 is 40, then you
have 40 divided by 2 and it

00:06:32.430 --> 00:06:34.250
simplifies to that right there.

00:06:34.250 --> 00:06:36.560
Remember, multiplication and
division, they're at the exact

00:06:36.560 --> 00:06:39.520
same level so we're going
to do it left to right.

00:06:39.520 --> 00:06:42.770
You could also express this as
multiplying by 1/2 and then it

00:06:42.770 --> 00:06:43.820
wouldn't matter the order.

00:06:43.820 --> 00:06:46.530
But for simplicity,
multiplication and division

00:06:46.530 --> 00:06:48.960
go left to right.

00:06:48.960 --> 00:06:52.860
So then you have 40 divided
by 2 minus 5 times 6.

00:06:52.860 --> 00:06:55.100
So, division, you just
have one division here,

00:06:55.100 --> 00:06:57.690
you want to do that.

00:06:57.690 --> 00:07:00.000
You have this division and you
have this multiplication,

00:07:00.000 --> 00:07:01.550
they're not together so you
can actually kind of do

00:07:01.550 --> 00:07:03.020
them simultaneously.

00:07:03.020 --> 00:07:05.600
And to make it clear that you
do this before you do the

00:07:05.600 --> 00:07:08.460
subtraction because
multiplication and division

00:07:08.460 --> 00:07:11.550
take priority over addition and
subtraction, we could put

00:07:11.550 --> 00:07:13.940
parentheses around them to say
look, we're going to do that

00:07:13.940 --> 00:07:18.490
and that first before I do that
subtraction, because

00:07:18.490 --> 00:07:22.040
multiplication and
division have priority.

00:07:22.040 --> 00:07:25.230
So 40 divided by 2 is 20.

00:07:25.230 --> 00:07:30.770
We're going to have that minus
sign, minus 5 times 6 is 30.

00:07:30.770 --> 00:07:35.120
20 minus 30 is equal
to negative 10.

00:07:35.120 --> 00:07:38.300
And that is the correct
interpretation of that.

00:07:38.300 --> 00:07:41.010
So I want to make something
very, very, very clear.

00:07:41.010 --> 00:07:46.190
If you have things at the same
level, so if you have 1 plus

00:07:46.190 --> 00:07:51.010
2 minus 3 plus 4 minus 1.

00:07:51.010 --> 00:07:53.830
So addition and subtraction are
all the same level in order of

00:07:53.830 --> 00:07:57.200
operations, you should
go left to right.

00:07:57.200 --> 00:08:02.240
So you should interpret this as
1 plus 2 is 3, so this is the

00:08:02.240 --> 00:08:05.680
same thing as 3 minus
3 plus 4 minus 1.

00:08:05.680 --> 00:08:09.830
Then you do 3 minus 3
is 0 plus 4 minus 1.

00:08:09.830 --> 00:08:13.070
Or this is the same thing
as 4 minus 1, which is

00:08:13.070 --> 00:08:13.945
the same thing as 3.

00:08:13.945 --> 00:08:16.060
You just go left to right.

00:08:16.060 --> 00:08:20.620
Same thing if you have
multiplication and division,

00:08:20.620 --> 00:08:21.930
they're at the same level.

00:08:21.930 --> 00:08:29.200
So if you have 4 times 2
divided by 3 times 2,

00:08:29.200 --> 00:08:34.820
you do 4 times 2 is 8
divided by 3 times 2.

00:08:34.820 --> 00:08:39.120
And you say 8 divided by 3 is,
well, we got a fraction there.

00:08:39.120 --> 00:08:40.510
It would be 8/3.

00:08:40.510 --> 00:08:44.340
So this would be 8/3 times 2.

00:08:44.340 --> 00:08:50.430
And then 8/3 times to
is equal to 16 over 3.

00:08:50.430 --> 00:08:51.580
That's how you interpret it.

00:08:51.580 --> 00:08:54.460
You don't do this
multiplication first or divide

00:08:54.460 --> 00:08:56.140
the 2 by that and all of that.

00:08:56.140 --> 00:08:59.000
Now the one time where you can
be loosey-goosey with order of

00:08:59.000 --> 00:09:02.840
operations, if you have all
addition or all multiplication.

00:09:02.840 --> 00:09:09.110
So if you have 1 plus 5 plus 7
plus 3 plus 2, it does not

00:09:09.110 --> 00:09:10.590
matter what order you do it in.

00:09:10.590 --> 00:09:12.690
You can do the 2 plus 3, you
can go from the right to the

00:09:12.690 --> 00:09:14.360
left, you can go from the
left to the right, you could

00:09:14.360 --> 00:09:15.610
start some place in between.

00:09:15.610 --> 00:09:17.470
If it's only all addition.

00:09:17.470 --> 00:09:20.030
And the same thing is true if
you have all multiplication.

00:09:20.030 --> 00:09:24.495
It's 1 times 5 times
7 times 3 times 2.

00:09:24.495 --> 00:09:27.280
It does not matter what
order you're doing it.

00:09:27.280 --> 00:09:31.620
But it's only with all
multiplication or all addition.

00:09:31.620 --> 00:09:33.290
If there was some division
in here, if there's some

00:09:33.290 --> 00:09:36.460
subtraction in here, you're
best off just going

00:09:36.460 --> 00:09:38.420
left to right.


Example of calculating a surface integral part 3

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:04.280
In the last couple of videos
we've been slowly moving

00:00:04.280 --> 00:00:07.580
towards our goal of figuring
out the surface area

00:00:07.580 --> 00:00:09.210
of this torus.

00:00:09.210 --> 00:00:11.880
And we did it by evaluating a
surface integral, and in order

00:00:11.880 --> 00:00:14.300
to evaluate a surface integral
we had to take the

00:00:14.300 --> 00:00:17.530
parameterization-- take its
partial with respect to s and

00:00:17.530 --> 00:00:18.830
with respect to t.

00:00:18.830 --> 00:00:20.590
We did that in the first video.

00:00:20.590 --> 00:00:22.410
Then we had to take
its cross product.

00:00:22.410 --> 00:00:24.080
We did that in the
second video.

00:00:24.080 --> 00:00:29.050
Now, we're ready to take the
magnitude of the cross product.

00:00:29.050 --> 00:00:31.650
And then we can evaluate it
inside of a double integral and

00:00:31.650 --> 00:00:35.120
we will have solved or we would
have computed an actual surface

00:00:35.120 --> 00:00:37.610
integral-- something you see
very few times in your

00:00:37.610 --> 00:00:38.370
education career.

00:00:38.370 --> 00:00:40.570
So this is kind of exciting.

00:00:40.570 --> 00:00:43.420
So this was the cross
product right here.

00:00:43.420 --> 00:00:46.200
Now, let's take the
magnitude of this thing.

00:00:46.200 --> 00:00:49.710
And you might remember, the
magnitude of any vector is kind

00:00:49.710 --> 00:00:50.980
of a Pythagorean theorem.

00:00:50.980 --> 00:00:53.380
And in this case it's going
to be kind of the distance

00:00:53.380 --> 00:00:56.400
formula to the Pythagorean
theorem n 3 dimensions.

00:00:56.400 --> 00:01:01.760
So the magnitude-- this is
equal to, just as a reminder,

00:01:01.760 --> 00:01:03.180
is equal to this right here.

00:01:03.180 --> 00:01:08.320
It's equal to the partial of
r with respect to s cross

00:01:08.320 --> 00:01:11.250
with the partial of
r with respect to t.

00:01:11.250 --> 00:01:13.550
Let me copy it and paste it.

00:01:13.550 --> 00:01:15.980
That is equal to
that right there.

00:01:15.980 --> 00:01:17.890
Put an equal sign.

00:01:17.890 --> 00:01:19.690
These two quantities are equal.

00:01:19.690 --> 00:01:22.260
Now we want to figure
out the magnitude.

00:01:22.260 --> 00:01:29.860
So if we want to take the
magnitude of this thing, that's

00:01:29.860 --> 00:01:32.450
going to be equal to-- well,
this is just a scalar that's

00:01:32.450 --> 00:01:33.570
multiply everything.

00:01:33.570 --> 00:01:36.040
So let's just write
the scalar out there.

00:01:36.040 --> 00:01:42.540
So b plus a cosine of s
times the magnitude of

00:01:42.540 --> 00:01:44.310
this thing right here.

00:01:44.310 --> 00:01:49.950
And the magnitude of this thing
right here is going to be the

00:01:49.950 --> 00:01:53.280
sum, of-- you can imagine,
it's the square root of this

00:01:53.280 --> 00:01:55.510
thing dotted with itself.

00:01:55.510 --> 00:01:58.500
Or you could say it's the sum
of the squares of each of

00:01:58.500 --> 00:02:01.030
these terms to the 1/2 power.

00:02:01.030 --> 00:02:03.800
So let me write it like that.

00:02:03.800 --> 00:02:06.330
Let me write the sum
of the squares.

00:02:06.330 --> 00:02:14.160
So if you square this you get
a squared cosine squared

00:02:14.160 --> 00:02:17.540
of s, sine squared of t.

00:02:17.540 --> 00:02:19.170
That's that term.

00:02:19.170 --> 00:02:20.700
Plus-- let me color code it.

00:02:20.700 --> 00:02:23.530
That's that term.

00:02:23.530 --> 00:02:24.930
I'll do the magenta.

00:02:24.930 --> 00:02:27.200
Plus that term squared.

00:02:27.200 --> 00:02:36.230
Plus a squared cosine squared
of s, cosine squared of t.

00:02:36.230 --> 00:02:37.500
That's that term.

00:02:37.500 --> 00:02:40.580
And then finally-- I'll
do another color--

00:02:40.580 --> 00:02:41.450
this term squared.

00:02:41.450 --> 00:02:47.900
So plus a squared
sine squared of s.

00:02:47.900 --> 00:02:52.480
And it's going to be all of
this business to the 1/2 power.

00:02:52.480 --> 00:02:57.210
This right here is the same
thing as the magnitude

00:02:57.210 --> 00:02:58.310
of this right here.

00:02:58.310 --> 00:03:01.170
This is just a scalar
that's multiplying by

00:03:01.170 --> 00:03:02.140
both of these terms.

00:03:02.140 --> 00:03:04.590
So let's see if we can do
anything interesting here.

00:03:04.590 --> 00:03:06.080
If this can be
simplified in any way.

00:03:06.080 --> 00:03:08.590
We have a squared
cosine squared of s.

00:03:08.590 --> 00:03:11.830
We have an a squared cosine
squared of s here, so let's

00:03:11.830 --> 00:03:15.360
factor that out from both
of these terms and

00:03:15.360 --> 00:03:17.280
see what happens.

00:03:17.280 --> 00:03:19.220
I'm just going to rewrite
this second part.

00:03:19.220 --> 00:03:23.950
So this is going to be a
squared cosine squared of s

00:03:23.950 --> 00:03:34.100
times sine squared of t-- put
a parentheses-- plus cosine--

00:03:34.100 --> 00:03:37.790
oh, I want to do it in that
magenta color, not orange.

00:03:37.790 --> 00:03:40.970
Plus cosine squared of t.

00:03:44.900 --> 00:03:50.330
And then you're going to
have this plus a squared

00:03:50.330 --> 00:03:52.590
sine squared of s.

00:03:52.590 --> 00:03:59.810
And of course, all of that
is to the 1/2 power.

00:03:59.810 --> 00:04:00.790
Now what is this?

00:04:00.790 --> 00:04:03.840
Well, we have sine squared of
t plus cosine squared of t.

00:04:03.840 --> 00:04:04.180
That's nice.

00:04:04.180 --> 00:04:08.320
That's equal to 1, the most
basic of trig identities.

00:04:08.320 --> 00:04:14.170
So this expression right here
simplifies to a squared cosine

00:04:14.170 --> 00:04:21.340
squared of s plus this
over here: a squared

00:04:21.340 --> 00:04:23.780
sine squared of s.

00:04:23.780 --> 00:04:26.450
And all of that to
the 1/2 power.

00:04:26.450 --> 00:04:28.250
You might immediately
recognize you can factor

00:04:28.250 --> 00:04:29.570
out an a squared.

00:04:29.570 --> 00:04:34.160
This is equal to a squared
times cosine squared of s

00:04:34.160 --> 00:04:40.880
plus sine squared of s.

00:04:40.880 --> 00:04:43.660
And all of that to
the 1/2 power.

00:04:43.660 --> 00:04:45.340
I'm just focusing on
this term right here.

00:04:45.340 --> 00:04:46.810
I'll write this in a second.

00:04:46.810 --> 00:04:49.520
But once again, cosine squared
plus sine squared of anything

00:04:49.520 --> 00:04:51.940
is going to be equal 1 as long
as it's the same anything

00:04:51.940 --> 00:04:52.930
it's equal to 1.

00:04:52.930 --> 00:04:56.080
So this term is a squared
to the 1/2 power.

00:04:56.080 --> 00:04:58.850
Or the square root of a
squared, which is just

00:04:58.850 --> 00:05:00.380
going to be equal to a.

00:05:00.380 --> 00:05:04.220
So all of this-- all that crazy
business right here just

00:05:04.220 --> 00:05:08.340
simplifies, all of that
just simplifies to a.

00:05:08.340 --> 00:05:12.580
So this cross product here
simplifies to this times

00:05:12.580 --> 00:05:16.260
a, which is a pretty neat
and clean simplification.

00:05:16.260 --> 00:05:17.395
So let me rewrite this.

00:05:21.090 --> 00:05:30.040
That simplifies, it
simplifies to a times that.

00:05:30.040 --> 00:05:33.100
And what's that? a
times b, so it's ab.

00:05:33.100 --> 00:05:36.520
ab plus a squared cosine of s.

00:05:40.550 --> 00:05:43.020
So already, we've gotten pretty
far and it's nice when you do

00:05:43.020 --> 00:05:45.230
something so beastly and
eventually it gets to

00:05:45.230 --> 00:05:48.590
something reasonably simple.

00:05:48.590 --> 00:05:51.850
And just to review what we had
to do, what our mission was

00:05:51.850 --> 00:05:55.710
several videos ago, is we want
to evaluate what this thing is

00:05:55.710 --> 00:06:00.110
over the region from s-- over
the region over with the

00:06:00.110 --> 00:06:01.090
surface is defined.

00:06:01.090 --> 00:06:06.460
So s going from 0 to 2 pi
and t going from 0 to 2 pi.

00:06:06.460 --> 00:06:07.890
Over this region.

00:06:07.890 --> 00:06:10.860
So we want to integrate
this over that region.

00:06:10.860 --> 00:06:16.720
So that region we're going
to vary s from 0 to 2 pi.

00:06:16.720 --> 00:06:18.130
So ds.

00:06:18.130 --> 00:06:23.870
And then we're going to vary
t from 0 to 2 pi-- dt.

00:06:23.870 --> 00:06:25.460
And this is what
we're evaluating.

00:06:25.460 --> 00:06:28.370
We're evaluating the magnitude
of the cross product of these

00:06:28.370 --> 00:06:32.060
two partial derivatives of our
original parameterization.

00:06:32.060 --> 00:06:33.920
So this is what we
can put in there.

00:06:33.920 --> 00:06:36.010
Things are getting simple all
of a sudden, or simpler.

00:06:36.010 --> 00:06:41.180
ab plus a squared cosine of s.

00:06:41.180 --> 00:06:43.080
And what is this equal to?

00:06:43.080 --> 00:06:47.430
So this is going to be equal
to-- well, we just take

00:06:47.430 --> 00:06:51.610
antiderivative of the
inside with respect to s.

00:06:51.610 --> 00:06:53.800
So the antiderivative--
so let me do the outside

00:06:53.800 --> 00:06:54.830
of our integral.

00:06:54.830 --> 00:06:56.690
So we're still going to have
to deal with the 0 to 2

00:06:56.690 --> 00:07:00.590
pi and our dt right here.

00:07:00.590 --> 00:07:03.900
But the antiderivative with
respect to s right here is

00:07:03.900 --> 00:07:06.630
going to be-- ab is just a
constant, so it's going to

00:07:06.630 --> 00:07:11.920
be abs plus-- what's the
antiderivative of cosine of s?

00:07:11.920 --> 00:07:12.840
It's sine of s.

00:07:12.840 --> 00:07:17.540
So plus a squared sine of s.

00:07:17.540 --> 00:07:21.860
And we're going to evaluate
it from 0 to 2 pi.

00:07:21.860 --> 00:07:24.270
And what is this going
to be equal to?

00:07:24.270 --> 00:07:27.900
Let's put our boundaries out
again or the t integral that

00:07:27.900 --> 00:07:33.800
we're going to have to do in
a second-- 0 the 2 pi d t.

00:07:33.800 --> 00:07:37.480
When you put 2 pi here you're
going to get ab times

00:07:37.480 --> 00:07:39.240
2 pi or 2 pi ab.

00:07:39.240 --> 00:07:45.840
So you're going to have 2 pi ab
plus a squared sine of 2 pi.

00:07:45.840 --> 00:07:48.670
Sine of 2 pi is 0, so there's
not going to be any term there.

00:07:48.670 --> 00:07:54.750
And then minus 0 times
ab, which is 0.

00:07:54.750 --> 00:07:57.220
And then you're going to
have minus a squared sine

00:07:57.220 --> 00:07:58.620
of 0, which is also 0.

00:07:58.620 --> 00:08:00.300
So all of the other
terms are all 0's.

00:08:00.300 --> 00:08:03.140
So that's what we're left with
it, it simplified nicely.

00:08:03.140 --> 00:08:05.140
So now we just have to take
the antiderivative of

00:08:05.140 --> 00:08:08.710
this with respect to t.

00:08:08.710 --> 00:08:11.620
And this is a constant in t, so
this is going to be equal to--

00:08:11.620 --> 00:08:17.750
take the antiderivative with
respect to t-- 2 pi abt and we

00:08:17.750 --> 00:08:21.950
need to evaluate that from 0 to
2 pi, which is equal to--

00:08:21.950 --> 00:08:23.570
so we put 2 pi in there.

00:08:23.570 --> 00:08:27.340
You have a 2 pi for t, it'll
be a 2 pi times 2 pi ab.

00:08:27.340 --> 00:08:33.250
Or we should say, 2 pi
squared times ab minus

00:08:33.250 --> 00:08:34.400
0 times this thing.

00:08:34.400 --> 00:08:35.880
Well, that's just going to
be 0, so we don't even

00:08:35.880 --> 00:08:37.230
have to write it down.

00:08:37.230 --> 00:08:37.860
So we're done.

00:08:37.860 --> 00:08:40.640
This is the surface
area of the torus.

00:08:40.640 --> 00:08:41.410
This is exciting.

00:08:41.410 --> 00:08:43.100
It just kind of snuck up on us.

00:08:43.100 --> 00:08:49.680
This is equal to 4 pi squared
ab, which is kind of a neat

00:08:49.680 --> 00:08:51.750
formula because it's
very neat and clean.

00:08:51.750 --> 00:08:53.730
You know, it has a 2 pi,
which is kind of the

00:08:53.730 --> 00:08:55.720
diameter of a circle.

00:08:55.720 --> 00:08:57.860
We're squaring it, which kind
of makes sense because we're

00:08:57.860 --> 00:09:03.630
taking the product of-- you can
kind of imagine the product

00:09:03.630 --> 00:09:04.700
of these 2 circles.

00:09:04.700 --> 00:09:07.150
I'm speaking in very abstract,
general terms, but that

00:09:07.150 --> 00:09:08.380
kind of feels good.

00:09:08.380 --> 00:09:09.910
And then we're taking just
the product of those

00:09:09.910 --> 00:09:11.340
two radiuses, remember.

00:09:11.340 --> 00:09:14.310
Let me just copy this
thing down here.

00:09:14.310 --> 00:09:17.160
Actually, let me copy this
thing because this is our new--

00:09:17.160 --> 00:09:19.570
this is our exciting result.

00:09:19.570 --> 00:09:20.710
Let me copy this.

00:09:20.710 --> 00:09:23.110
So copy.

00:09:23.110 --> 00:09:27.350
So all of this work that we
did simplified to this,

00:09:27.350 --> 00:09:30.460
which is exciting.

00:09:30.460 --> 00:09:33.990
We now know that if you have a
torus where the radius of the

00:09:33.990 --> 00:09:40.440
cross section is a, and the
radius from the center of

00:09:40.440 --> 00:09:44.130
the torus to the middle of
the cross sections is b.

00:09:44.130 --> 00:09:47.820
That the surface area of that
torus is going to be 4 pi

00:09:47.820 --> 00:09:50.380
squared times a times b.

00:09:50.380 --> 00:09:53.690
Which I think is a
pretty neat outcome.


Example of calculating a surface integral part 1

WEBVTT
Kind: captions
Language: en

00:00:00.490 --> 00:00:05.380
We saw several videos ago that
we can parameterize a torus or

00:00:05.380 --> 00:00:12.670
a doughnut shape as a position
vector-valued function

00:00:12.670 --> 00:00:13.980
of two parameters.

00:00:13.980 --> 00:00:15.820
And this is the
outcome that we had.

00:00:15.820 --> 00:00:17.000
I think I did it over
several videos because

00:00:17.000 --> 00:00:18.050
it was a bit hairy.

00:00:18.050 --> 00:00:23.500
And I'll write our position
vector-valued function first.

00:00:23.500 --> 00:00:26.220
So we have r as a function of
our two parameters s and t.

00:00:26.220 --> 00:00:28.080
And then I'll review a little
bit of what all the terms--

00:00:28.080 --> 00:00:31.430
what the s, the t, and the
a's and the b's represent.

00:00:31.430 --> 00:00:37.550
But it's equal to b
plus a cosine of s.

00:00:37.550 --> 00:00:40.980
And once again, we saw
this several videos ago.

00:00:40.980 --> 00:00:45.420
So you might want to watch the
videos on parameterizing

00:00:45.420 --> 00:00:49.950
surfaces with two parameters to
figure out how we got here.

00:00:49.950 --> 00:00:51.660
Times the sine of t.

00:00:51.660 --> 00:00:54.070
I'm going to put the s
terms and the t terms

00:00:54.070 --> 00:00:55.440
in different colors.

00:00:55.440 --> 00:00:56.860
Times our i unit vector.

00:00:56.860 --> 00:00:58.960
I'll put the vectors,
or the unit vectors

00:00:58.960 --> 00:01:00.640
in this orange color.

00:01:00.640 --> 00:01:03.470
Plus-- I'll do it in
the same yellow.

00:01:03.470 --> 00:01:16.490
Plus b plus a cosine of s times
cosine of t times the j unit

00:01:16.490 --> 00:01:20.270
vector-- the unit factor
in the y direction.

00:01:20.270 --> 00:01:29.970
Plus a sine of s times the
k unit vector or the unit

00:01:29.970 --> 00:01:32.940
vector in the z direction.

00:01:32.940 --> 00:01:36.570
And in order to generate the
torus or the doughnut shape,

00:01:36.570 --> 00:01:40.320
this is true for our
parameters-- so we don't wrap

00:01:40.320 --> 00:01:49.820
multiple times around the
torus-- for s being between 0

00:01:49.820 --> 00:01:55.940
and 2 pi and for t being
between 0 and 2 pi.

00:01:55.940 --> 00:01:58.680
And just as a bit of review,
we're all of this came from--

00:01:58.680 --> 00:02:01.030
and I'm going to have to do
what my plan is for this

00:02:01.030 --> 00:02:02.280
video over several videos.

00:02:02.280 --> 00:02:05.510
But let's review where
all of this came from.

00:02:05.510 --> 00:02:07.640
Let me draw a doughnut.

00:02:07.640 --> 00:02:11.360
My best effort at a
doughnut right here.

00:02:11.360 --> 00:02:14.850
That looks like a
doughnut or a torus.

00:02:14.850 --> 00:02:18.720
And you can imagine a torus, or
this doughnut shape is kind of

00:02:18.720 --> 00:02:20.130
the product of two circles.

00:02:20.130 --> 00:02:22.790
You have the circle that's
kind of the cross section of

00:02:22.790 --> 00:02:23.900
the doughnut at any point.

00:02:23.900 --> 00:02:25.040
You could take it there.

00:02:25.040 --> 00:02:27.370
You could take it over there.

00:02:27.370 --> 00:02:30.320
And then you have the circle
that kind of wraps around all

00:02:30.320 --> 00:02:33.930
of these other circles or these
other circles wrap around it.

00:02:33.930 --> 00:02:39.030
And so, when we derived this
formula up here or this

00:02:39.030 --> 00:02:42.790
parameterization, a was the
radius of these cross

00:02:42.790 --> 00:02:44.230
sectional circles.

00:02:44.230 --> 00:02:45.510
That's a.

00:02:45.510 --> 00:02:47.890
That's what these a terms were.

00:02:47.890 --> 00:02:55.340
And b was the distance from the
center of our torus out to the

00:02:55.340 --> 00:02:57.520
center of these cross sections.

00:02:57.520 --> 00:02:59.080
So this was b.

00:02:59.080 --> 00:03:03.530
So you can imagine that b is
kind of the radius of the big

00:03:03.530 --> 00:03:07.170
circle up to the midpoint of
the, I guess, cross section.

00:03:07.170 --> 00:03:11.760
And a is the radius of the
cross sectional circles.

00:03:11.760 --> 00:03:15.410
And when we parameterized it,
the parameter s was essentially

00:03:15.410 --> 00:03:19.810
telling us how far-- s was
telling us how far or where are

00:03:19.810 --> 00:03:21.570
we wrapping around this circle.

00:03:21.570 --> 00:03:24.450
So it's an angle from 0
to 2 pi to say where

00:03:24.450 --> 00:03:25.620
we are on that circle.

00:03:25.620 --> 00:03:28.680
And t tells us how much
we've rotated around

00:03:28.680 --> 00:03:29.475
the larger circle.

00:03:33.720 --> 00:03:37.650
So if you think about it, you
can specify any point on this

00:03:37.650 --> 00:03:41.210
doughnut or on this surface or
on this torus by telling

00:03:41.210 --> 00:03:42.390
you an s or a t.

00:03:42.390 --> 00:03:46.870
And so that's why we picked
that as the parameterization.

00:03:46.870 --> 00:03:50.470
Now, the whole reason why I'm
even revisiting this stuff that

00:03:50.470 --> 00:03:53.500
we saw several videos ago is
we're going to actually use it

00:03:53.500 --> 00:03:55.900
to compute an actual
surface integral.

00:03:55.900 --> 00:03:58.480
And the surface integral we're
going to compute will tell us

00:03:58.480 --> 00:04:02.140
the surface area of this torus.

00:04:02.140 --> 00:04:08.340
So this surface right here is
sigma, like that and it's being

00:04:08.340 --> 00:04:11.435
represented by this position
vector-valued function.

00:04:14.400 --> 00:04:18.200
That is parameterized by these
two parameters right there.

00:04:18.200 --> 00:04:20.540
And if we wanted to figure out
the surface area, if we just

00:04:20.540 --> 00:04:23.270
kind of set it as the surface
integral we saw in, I think,

00:04:23.270 --> 00:04:25.670
the last video at least the
last vector calculus video I

00:04:25.670 --> 00:04:30.380
did that this is a surface
integral over the surface.

00:04:30.380 --> 00:04:33.460
Here this capital Sigma does
not represent a sum, it

00:04:33.460 --> 00:04:38.270
represents a surface of a bunch
of the little d sigmas--

00:04:38.270 --> 00:04:40.270
a bunch of the little
chunks of the surface.

00:04:40.270 --> 00:04:43.240
And just as a review, you
can imagine each d sigma

00:04:43.240 --> 00:04:45.880
is a little patch of the
surface right there.

00:04:45.880 --> 00:04:49.290
That is a d sigma.

00:04:49.290 --> 00:04:51.400
It's a double integral here
because we want to add up

00:04:51.400 --> 00:04:53.830
all of the d sigmas
in 2 directions.

00:04:53.830 --> 00:04:57.000
You can imagine one kind of
rotating this way around the

00:04:57.000 --> 00:05:00.190
torus and then the other
direction is going in the other

00:05:00.190 --> 00:05:01.440
direction around the torus.

00:05:01.440 --> 00:05:04.050
So that's why it's
a double integral.

00:05:04.050 --> 00:05:06.230
And this is just going to give
you the surface area, which is

00:05:06.230 --> 00:05:08.310
the whole point of this video
and probably the next

00:05:08.310 --> 00:05:09.580
one or two videos.

00:05:09.580 --> 00:05:12.150
But if you wanted to also
multiply these sigmas times

00:05:12.150 --> 00:05:15.550
some other value-- there's some
scalar field that this is in

00:05:15.550 --> 00:05:17.500
that you cared about-- you
could put that other

00:05:17.500 --> 00:05:18.560
value right there.

00:05:18.560 --> 00:05:21.330
But here we're just
multiplying it by 1.

00:05:21.330 --> 00:05:24.060
And we saw in the last video
that it's a way of expressing

00:05:24.060 --> 00:05:27.100
an idea, but you really can't
do much computation with this.

00:05:27.100 --> 00:05:29.750
But a way that you can express
this so that you can actually

00:05:29.750 --> 00:05:32.290
take the integral, you say this
is the same thing-- and we saw

00:05:32.290 --> 00:05:35.510
this in the last
several videos.

00:05:35.510 --> 00:05:39.080
This is the same thing as the
double integral over the

00:05:39.080 --> 00:05:41.610
region over which our
parameters are defined.

00:05:41.610 --> 00:05:45.690
So it's this region over here
where s and t go from 0 to 2 pi

00:05:45.690 --> 00:05:47.790
of whatever function this is.

00:05:47.790 --> 00:05:50.760
We just have a 1 here, so we
could just write a 1 if we

00:05:50.760 --> 00:05:52.530
like; it doesn't change much.

00:05:52.530 --> 00:05:55.490
Times-- and this is
what we learned.

00:05:55.490 --> 00:05:59.150
Times the magnitude of the
partial derivative of

00:05:59.150 --> 00:06:03.310
r with respect to s.

00:06:03.310 --> 00:06:07.010
The magnitude of that crossed
with the partial derivative

00:06:07.010 --> 00:06:13.370
of r with respect to t ds.

00:06:13.370 --> 00:06:17.250
You could take it in
either order, but ds dt.

00:06:17.250 --> 00:06:18.460
So we saw this in
the last video.

00:06:18.460 --> 00:06:21.570
What we're going to do now
is actually compute this.

00:06:21.570 --> 00:06:23.060
That's the whole
point of this video.

00:06:23.060 --> 00:06:27.340
We're going to take the cross
product of these two vectors.

00:06:27.340 --> 00:06:28.580
So let's figure out
these vectors.

00:06:28.580 --> 00:06:30.020
Then in the next video
we're going to take

00:06:30.020 --> 00:06:30.780
the cross product.

00:06:30.780 --> 00:06:33.520
And then the video after that
we'll actually evaluate

00:06:33.520 --> 00:06:34.570
this double integral.

00:06:34.570 --> 00:06:36.900
And you're going to see it's a
pretty hairy problem and this

00:06:36.900 --> 00:06:39.745
is probably the reason that
very few people ever see an

00:06:39.745 --> 00:06:42.330
actual surface integral
get computed.

00:06:42.330 --> 00:06:44.470
But let's do it anyway.

00:06:44.470 --> 00:06:49.940
So the partial derivative of
r with respect to s-- so

00:06:49.940 --> 00:06:51.040
this term right here.

00:06:51.040 --> 00:06:53.180
We'll do the cross product
in the next video.

00:06:53.180 --> 00:06:54.030
This term is what?

00:06:54.030 --> 00:06:56.240
We just want to hold t
constant and took the partial

00:06:56.240 --> 00:06:59.440
with respect to just s.

00:06:59.440 --> 00:07:03.060
So this up here, if you
distributed the sine of t times

00:07:03.060 --> 00:07:04.940
b-- that's just going to be a
constant in terms of s,

00:07:04.940 --> 00:07:06.160
so we can ignore that.

00:07:06.160 --> 00:07:08.350
Then you have sine of t
times this over here.

00:07:08.350 --> 00:07:10.570
So sine of t and
a is a constant.

00:07:10.570 --> 00:07:12.680
And you take the derivative
of cosine of s.

00:07:12.680 --> 00:07:16.330
That's negative sine of s.

00:07:16.330 --> 00:07:19.210
So the derivative of this with
respect to s or the partial

00:07:19.210 --> 00:07:24.580
with respect to s is going to
be minus a-- I'll write in

00:07:24.580 --> 00:07:27.526
green the sine of t, so you
know that's where it came from.

00:07:27.526 --> 00:07:32.870
Sine of t and then sine of s.

00:07:32.870 --> 00:07:34.800
The derivative of this
is negative sine of s.

00:07:34.800 --> 00:07:36.070
That's where that
negative came from.

00:07:36.070 --> 00:07:37.790
And then I'll write the
sine of s right there.

00:07:41.090 --> 00:07:44.180
Times the unit vector i.

00:07:44.180 --> 00:07:48.310
That's the partial of just this
x term with respect to s.

00:07:48.310 --> 00:07:50.050
And then we'll do the
same thing with the y

00:07:50.050 --> 00:07:52.270
term or the j term.

00:07:52.270 --> 00:07:56.230
So plus-- same logic-- b times
cosine of t with respect to s.

00:07:56.230 --> 00:07:57.240
When you take the
partial [INAUDIBLE]

00:07:57.240 --> 00:08:01.750
becomes 0 so you're left with
a-- well, it's going to be a

00:08:01.750 --> 00:08:05.460
minus a again because when you
take the derivative of the

00:08:05.460 --> 00:08:09.110
cosine of s it's going to
be negative sine of s.

00:08:09.110 --> 00:08:09.890
Let me do it.

00:08:09.890 --> 00:08:12.660
You're going to have a minus a.

00:08:12.660 --> 00:08:14.810
This cosine of t.

00:08:14.810 --> 00:08:16.960
Minus a cosine of t.

00:08:16.960 --> 00:08:18.490
That's the constant terms.

00:08:18.490 --> 00:08:21.260
Sine of s.

00:08:21.260 --> 00:08:23.470
Just taking partial
derivatives.

00:08:23.470 --> 00:08:24.945
Sine of s j.

00:08:28.730 --> 00:08:30.790
And then finally, we take
the derivative of this

00:08:30.790 --> 00:08:32.810
with respect to s.

00:08:32.810 --> 00:08:33.980
And that's pretty
straightforward.

00:08:33.980 --> 00:08:37.000
It's just going to
be a cosine of s.

00:08:37.000 --> 00:08:47.362
So plus a cosine of s k.

00:08:47.362 --> 00:08:49.490
Now hopefully you didn't
find this confusing.

00:08:49.490 --> 00:08:51.810
The negative sines because
the derivative of cosines

00:08:51.810 --> 00:08:53.090
are negative sines.

00:08:53.090 --> 00:08:54.690
So negative sine of s.

00:08:54.690 --> 00:08:57.290
That's why it's negative sine
of s times the constant.

00:08:57.290 --> 00:09:00.290
Negative sine of s times the
constant-- the constant

00:09:00.290 --> 00:09:01.163
cosine of t sine of t.

00:09:01.163 --> 00:09:03.140
So hopefully this makes some
sense just as a review of

00:09:03.140 --> 00:09:04.710
taking a partial derivative.

00:09:04.710 --> 00:09:07.795
Now let's do the same
thing with respect to t.

00:09:10.990 --> 00:09:12.240
And I'll do that in
a different color.

00:09:12.240 --> 00:09:15.750
So we're now going to take the
partial of r with respect to t.

00:09:15.750 --> 00:09:20.530
So the partial of r with
respect to t is equal to-- so

00:09:20.530 --> 00:09:23.690
now this whole term over here
is a constant, and so it's

00:09:23.690 --> 00:09:25.630
going to be that whole term
times the derivative of this

00:09:25.630 --> 00:09:28.120
with respect to t, which
is just cosine of t.

00:09:28.120 --> 00:09:35.910
So it's going to be b
plus a cosine of s

00:09:35.910 --> 00:09:43.130
times cosine of t i.

00:09:43.130 --> 00:09:45.980
And then, plus-- and it's
actually going to be a minus

00:09:45.980 --> 00:09:47.400
because when you take the
derivative of this with

00:09:47.400 --> 00:09:49.670
respect to t it's going
to be minus sine of t.

00:09:49.670 --> 00:09:54.590
So it's going to be negative
and then let me leave some

00:09:54.590 --> 00:09:55.810
space for this term right here.

00:09:55.810 --> 00:09:59.150
Negative sine of t.

00:09:59.150 --> 00:10:01.340
And you're going to have
this constant out there.

00:10:01.340 --> 00:10:03.790
That's a constant in t.

00:10:03.790 --> 00:10:07.230
b plus a cosine of s.

00:10:07.230 --> 00:10:08.730
That's just that
term right there.

00:10:08.730 --> 00:10:15.780
Derivative of cosine t is
negative sine of t times j.

00:10:15.780 --> 00:10:18.150
And then the partial of this
with respect to t-- this

00:10:18.150 --> 00:10:19.340
is just a constant in t.

00:10:19.340 --> 00:10:20.910
So the partial's going to be 0.

00:10:20.910 --> 00:10:22.195
So I'll write plus 0k.

00:10:25.970 --> 00:10:28.130
Let me do all my vectors
in that same color.

00:10:28.130 --> 00:10:32.290
Plus 0 times the unit vector k.

00:10:32.290 --> 00:10:34.240
So that gives us our
partial derivatives.

00:10:34.240 --> 00:10:36.770
Now we have to take their cross
product, then find the

00:10:36.770 --> 00:10:39.930
magnitude of the cross product,
and then evaluate this

00:10:39.930 --> 00:10:40.930
double integral.

00:10:40.930 --> 00:10:44.460
And I'll do that in the
next couple of videos.


Example of calculating a surface integral part 2

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.140
Where we left off in the last
video, we were finding the

00:00:03.140 --> 00:00:06.580
surface area of a torus,
or a doughnut shape.

00:00:06.580 --> 00:00:09.210
And we were doing it by
taking a surface integral.

00:00:09.210 --> 00:00:11.920
And in order to take a surface
integral, we had to find the

00:00:11.920 --> 00:00:15.310
partial of our parameterization
with respect to s, and the

00:00:15.310 --> 00:00:18.920
partial with respect to t, and
now we're ready to take

00:00:18.920 --> 00:00:19.910
the cross product.

00:00:19.910 --> 00:00:21.940
And then we can take the
magnitude of the cross product.

00:00:21.940 --> 00:00:24.810
And then we can actually take
this double integral and

00:00:24.810 --> 00:00:26.820
figure out the surface area.

00:00:26.820 --> 00:00:28.670
So let's just do
it step by step.

00:00:28.670 --> 00:00:31.360
Here we could take the cross
product, which is not

00:00:31.360 --> 00:00:33.330
a non-hairy operation.

00:00:33.330 --> 00:00:36.150
This is why you don't see many
surface integrals actually get

00:00:36.150 --> 00:00:38.400
done, or many examples done.

00:00:38.400 --> 00:00:42.220
Let's take the cross product
of these two fellows.

00:00:42.220 --> 00:00:46.560
So the partial of r with
respect to s, crossed with--

00:00:46.560 --> 00:00:50.720
in magenta-- the partial
of r with respect to t.

00:00:50.720 --> 00:00:52.610
This will be a little
bit of review of cross

00:00:52.610 --> 00:00:53.390
products for you.

00:00:53.390 --> 00:00:55.070
You might remember this
is going to be equal

00:00:55.070 --> 00:00:57.660
to the determinant.

00:00:57.660 --> 00:01:00.360
I'm going to write the
unit vectors up here.

00:01:00.360 --> 00:01:05.240
The first row is i, j, and k.

00:01:05.240 --> 00:01:07.630
And then the next 2 rows are
going to be-- let me do that in

00:01:07.630 --> 00:01:13.760
that yellow color-- the next 2
rows are going to be the

00:01:13.760 --> 00:01:15.460
components of these guys.

00:01:15.460 --> 00:01:18.670
So let me copy and paste them.

00:01:18.670 --> 00:01:20.740
You have that right there.

00:01:20.740 --> 00:01:23.620
Copy and paste.

00:01:23.620 --> 00:01:26.050
Put that guy right there.

00:01:26.050 --> 00:01:28.930
Then you have this
fellow right there.

00:01:28.930 --> 00:01:30.140
Copy and paste.

00:01:30.140 --> 00:01:31.510
Put him right there.

00:01:31.510 --> 00:01:34.130
And then you got this
guy right here.

00:01:34.130 --> 00:01:35.350
This'll save us some time.

00:01:35.350 --> 00:01:37.160
Copy and paste.

00:01:37.160 --> 00:01:38.900
Put him right there.

00:01:38.900 --> 00:01:42.440
Then the last row is going to
be this guy's components.

00:01:42.440 --> 00:01:44.260
Copy and paste.

00:01:44.260 --> 00:01:46.720
Put him right here.

00:01:46.720 --> 00:01:48.680
Almost done.

00:01:48.680 --> 00:01:51.740
This guy-- copy and paste.

00:01:51.740 --> 00:01:52.870
Put him right there.

00:01:52.870 --> 00:01:54.890
Make sure we know that
these are separate terms.

00:01:54.890 --> 00:01:57.280
And finally, we don't have to
copy and paste it, but just

00:01:57.280 --> 00:01:59.430
since we did for all of the
other terms, I'll do it

00:01:59.430 --> 00:02:01.490
for that 0, as well.

00:02:01.490 --> 00:02:04.010
So the cross product of these
is literally the determinant

00:02:04.010 --> 00:02:06.275
of this matrix right here.

00:02:11.350 --> 00:02:13.600
And so, just as a bit of
a refresher of taking

00:02:13.600 --> 00:02:17.490
determinants, this is going to
be i times the subdeterminant

00:02:17.490 --> 00:02:20.700
right here, if you cross out
this column and that row.

00:02:20.700 --> 00:02:26.350
So it's going to be equal to
i-- you're not used to seeing

00:02:26.350 --> 00:02:28.770
the unit vector written first,
but we can switch the order

00:02:28.770 --> 00:02:32.330
later-- times i times the
submatrix right here.

00:02:32.330 --> 00:02:34.290
If you cross out this
column and that row.

00:02:34.290 --> 00:02:38.370
So it's going to be this term
times 0-- which is just

00:02:38.370 --> 00:02:42.490
0-- minus this term
times that term.

00:02:42.490 --> 00:02:45.930
So minus this term times this
term- the negative signs are

00:02:45.930 --> 00:02:47.520
going to cancel out, so
this'll be positive.

00:02:47.520 --> 00:02:50.510
So it's just going to be i
times this term times this

00:02:50.510 --> 00:02:52.240
term, without a negative
sign right there.

00:02:52.240 --> 00:02:58.850
So i times this term,
which is a cosine of s.

00:02:58.850 --> 00:03:01.630
It's really that term times
that term, minus that term

00:03:01.630 --> 00:03:03.230
times that term, but the
negatives cancel out.

00:03:03.230 --> 00:03:04.370
That times that is 0.

00:03:04.370 --> 00:03:05.100
So that's how we can do this.

00:03:05.100 --> 00:03:14.230
It's a cosine of s times b plus
a cosine of s-- I'll just all

00:03:14.230 --> 00:03:18.140
switch to the same
color-- sine of t.

00:03:18.140 --> 00:03:20.700
So we've got our i term
for the cross product.

00:03:20.700 --> 00:03:24.190
Now it's going to be minus j--
remember when you take the

00:03:24.190 --> 00:03:27.590
determinant, you actually have
this, kind of, you have to

00:03:27.590 --> 00:03:29.060
checker board of
switching sines.

00:03:29.060 --> 00:03:34.370
So now it's going to be minus j
times-- and you cross out that

00:03:34.370 --> 00:03:38.390
row and that column-- and it's
going to be this term times

00:03:38.390 --> 00:03:43.000
this term-- which is just
0-- minus this term

00:03:43.000 --> 00:03:44.160
times this term.

00:03:44.160 --> 00:03:46.770
And once again, when
you have-- oh, sorry.

00:03:46.770 --> 00:03:49.020
When you cross out this
column and that row.

00:03:49.020 --> 00:03:54.750
So it's going to be that
guy times that guy, minus

00:03:54.750 --> 00:03:58.230
this guy times this guy.

00:03:58.230 --> 00:04:00.690
So it's going to be minus this
guy times this guy-- so it's

00:04:00.690 --> 00:04:03.440
going to be-- let me
do it in yellow.

00:04:03.440 --> 00:04:11.980
So the negative times negative
that guy, b plus a cosine of s

00:04:11.980 --> 00:04:16.680
cosine of t times this
guy, a cosine of s.

00:04:19.210 --> 00:04:21.574
We'll clean it up
in a little bit.

00:04:21.574 --> 00:04:24.420
Well, we'll clean this up, and
you see this negative and that

00:04:24.420 --> 00:04:25.500
negative will cancel out.

00:04:25.500 --> 00:04:27.520
We're just multiplying
everything.

00:04:27.520 --> 00:04:29.800
And then finally, the k term.

00:04:29.800 --> 00:04:35.470
So plus-- I'll go to the next
line-- plus k times-- cross out

00:04:35.470 --> 00:04:38.800
that row, that column-- it's
going to be that times that,

00:04:38.800 --> 00:04:42.340
minus that times that.

00:04:42.340 --> 00:04:44.990
So that looks like a kind
of a beastly thing.

00:04:44.990 --> 00:04:46.520
But I think if we take
it step by step, it

00:04:46.520 --> 00:04:47.390
shouldn't be too bad.

00:04:47.390 --> 00:04:48.300
So that times that.

00:04:48.300 --> 00:04:51.255
The negatives are
going to cancel out.

00:04:51.255 --> 00:04:56.270
So this term right here
is going to be a sine

00:04:56.270 --> 00:04:59.770
of t, sine of s.

00:04:59.770 --> 00:05:05.960
And then this term
right here is b plus a

00:05:05.960 --> 00:05:09.375
cosine of s sine of t.

00:05:13.210 --> 00:05:15.490
So that's that times that-- and
the negatives canceled out,

00:05:15.490 --> 00:05:17.890
that's why I didn't put any
negatives here-- minus

00:05:17.890 --> 00:05:19.370
this times this.

00:05:19.370 --> 00:05:22.000
So this times this is going
to be a negative number.

00:05:22.000 --> 00:05:23.660
But if you take the negative
of it, it's going to

00:05:23.660 --> 00:05:24.480
be a positive value.

00:05:24.480 --> 00:05:31.470
So it's going to be plus
that a cosine of t

00:05:31.470 --> 00:05:34.780
sine of s times that.

00:05:34.780 --> 00:05:40.710
Times b plus a cosine
of s cosine of t.

00:05:40.710 --> 00:05:42.850
Now you see why you don't see
many examples of surface

00:05:42.850 --> 00:05:44.420
integrals being done.

00:05:44.420 --> 00:05:49.070
Let's see if we can clean this
up a little bit, especially if

00:05:49.070 --> 00:05:51.930
we can clean up this
last term a bit.

00:05:51.930 --> 00:05:54.660
So let's see what we
can do to simplify it.

00:05:54.660 --> 00:05:55.985
So our first term.

00:05:55.985 --> 00:05:58.040
So let's just multiply
it out, I guess is the

00:05:58.040 --> 00:05:59.510
easiest way to do it.

00:05:59.510 --> 00:06:02.770
Actually, the easiest first
step would just be factor out

00:06:02.770 --> 00:06:04.960
the b plus a cosine of s.

00:06:04.960 --> 00:06:07.680
Because that's in every
term. b plus a cosine of s.

00:06:07.680 --> 00:06:09.530
b plus a cosine of s.

00:06:09.530 --> 00:06:12.480
b plus a cosine of s.
b plus a cosine of s.

00:06:12.480 --> 00:06:13.710
So let's just factor that out.

00:06:13.710 --> 00:06:20.310
So this whole crazy thing can
be written as b plus a cosine

00:06:20.310 --> 00:06:24.370
of s-- so we factored
it out-- times--.

00:06:24.370 --> 00:06:27.140
I'll put in some brackets
here, so you don't multiply

00:06:27.140 --> 00:06:28.790
times every component.

00:06:28.790 --> 00:06:31.125
So the i component, when you
factor this guy out, is going

00:06:31.125 --> 00:06:34.310
to be a cosine of s sine of t.

00:06:34.310 --> 00:06:36.480
Let me write it in green.

00:06:36.480 --> 00:06:46.770
So it's going to be a cosine of
s sine of t times i-- you're

00:06:46.770 --> 00:06:48.700
not used to seeing the i
before, so I'm going to write

00:06:48.700 --> 00:06:53.290
the i here-- and then plus--.

00:06:53.290 --> 00:06:56.170
We're factoring this guy out,
so you're just going to be

00:06:56.170 --> 00:06:58.750
left with cosine of
t, a cosine of s.

00:06:58.750 --> 00:07:06.350
Or we can write it as a cosine
of s cosine of t-- that's that

00:07:06.350 --> 00:07:09.050
right there, just putting it in
the same order as that--

00:07:09.050 --> 00:07:12.110
times the unit vector j.

00:07:12.110 --> 00:07:15.590
And then when we factored this
guy out-- so we're not going

00:07:15.590 --> 00:07:21.290
to see that or that anymore.

00:07:21.290 --> 00:07:24.360
When you factor that out,
we can multiply this

00:07:24.360 --> 00:07:25.990
out, and what do we get?

00:07:25.990 --> 00:07:29.140
So in green, I'll write again.

00:07:29.140 --> 00:07:32.220
So if you multiply sine of t
times this thing over here--

00:07:32.220 --> 00:07:35.500
because that's all that we have
left after we factor out this

00:07:35.500 --> 00:07:44.550
thing-- we get a sine of s,
sine squared of t, right?

00:07:44.550 --> 00:07:46.360
We have sine of t
times sine of t.

00:07:50.250 --> 00:07:51.910
So that's that over there.

00:07:51.910 --> 00:07:54.460
Plus-- what do we
have over here?

00:07:54.460 --> 00:07:57.520
We have a sine of s times
cosine squared of t.

00:08:05.000 --> 00:08:08.450
And all of that times
the k unit vector.

00:08:12.510 --> 00:08:15.310
And so things are looking a
little bit more simplified,

00:08:15.310 --> 00:08:17.580
but you might see
something jump out at you.

00:08:17.580 --> 00:08:19.700
You have a sine squared
and a cosine squared.

00:08:19.700 --> 00:08:22.160
So somehow, if I can just make
that just sine squared plus

00:08:22.160 --> 00:08:24.240
cosine squared of t, those
will simplify to 1.

00:08:24.240 --> 00:08:25.580
And we can.

00:08:25.580 --> 00:08:29.280
And this term right here, we
can-- if we just focus on that

00:08:29.280 --> 00:08:32.220
term-- and this is all kind
of algebraic manipulation.

00:08:32.220 --> 00:08:36.120
If we just focus on that term,
this term right here can be

00:08:36.120 --> 00:08:42.170
rewritten as a sine of s-- if
we factor that out-- times sine

00:08:42.170 --> 00:08:47.400
squared of t plus cosine
squared of t times

00:08:47.400 --> 00:08:50.360
our unit vector, k.

00:08:50.360 --> 00:08:50.590
Right?

00:08:50.590 --> 00:08:53.480
I just factored out an a sine
of s from both of these terms.

00:08:53.480 --> 00:08:56.360
And this is our most
fundamental trig identity

00:08:56.360 --> 00:08:57.330
from the unit circle.

00:08:57.330 --> 00:08:59.950
This is equal to 1.

00:08:59.950 --> 00:09:05.110
So this last term simplifies
to a sine of s times k.

00:09:05.110 --> 00:09:07.920
So, so far we've
gotten pretty far.

00:09:07.920 --> 00:09:11.840
We were able to figure out the
cross product of these 2, I

00:09:11.840 --> 00:09:14.445
guess, partial derivatives of
the vector valued,

00:09:14.445 --> 00:09:16.710
or our original
parameterization there.

00:09:16.710 --> 00:09:20.820
We were able to figure out what
this thing right here, before

00:09:20.820 --> 00:09:24.070
we take the magnitude of it, it
translates to this

00:09:24.070 --> 00:09:25.370
thing right here.

00:09:25.370 --> 00:09:27.270
Let me rewrite it-- well, I
don't need to rewrite it.

00:09:27.270 --> 00:09:27.780
You know it.

00:09:27.780 --> 00:09:28.820
Well, I'll rewrite it.

00:09:28.820 --> 00:09:31.393
So that's equal to-- I'll
rewrite it neatly and we'll use

00:09:31.393 --> 00:09:38.000
this in the next video-- b plus
a cosine of s times open

00:09:38.000 --> 00:09:46.810
bracket a cosine of s sine of t
times i plus-- switch back to

00:09:46.810 --> 00:09:56.840
the blue-- plus a cosine of s
cosine of t times j plus--

00:09:56.840 --> 00:09:59.900
switch back to the blue-- this
thing-- plus-- this simplified

00:09:59.900 --> 00:10:04.870
nicely-- a sine of s times k.

00:10:04.870 --> 00:10:07.315
Times the unit vector k.

00:10:10.000 --> 00:10:15.020
This right here is this
expression right there.

00:10:15.020 --> 00:10:16.740
And I'll finish this
video, since I'm already

00:10:16.740 --> 00:10:17.970
over 10 minutes.

00:10:17.970 --> 00:10:18.900
And in the next video,
we're going to take

00:10:18.900 --> 00:10:20.180
the magnitude of it.

00:10:20.180 --> 00:10:22.540
And then, if we have
time, actually take

00:10:22.540 --> 00:10:23.470
this double integral.

00:10:23.470 --> 00:10:24.470
And we'll all be done.

00:10:24.470 --> 00:10:28.390
We'll figure out the surface
area of this torus.


Ponzi schemes

WEBVTT
Kind: captions
Language: en

00:00:00.480 --> 00:00:03.790
You've probably heard the term
Ponzi scheme before, let me

00:00:03.790 --> 00:00:09.810
write it down, and what we're
going to do in this video is

00:00:09.810 --> 00:00:11.360
explain what it really is.

00:00:11.360 --> 00:00:14.050
You might have a sense it's some
type of scam, that you're

00:00:14.050 --> 00:00:16.430
taking one person's money and
giving it to another, but

00:00:16.430 --> 00:00:17.800
we're going to do a tangible
example of

00:00:17.800 --> 00:00:19.290
how it actually works.

00:00:19.290 --> 00:00:23.700
And right here, I have two
pictures of the probably the

00:00:23.700 --> 00:00:27.530
two most famous perpetrators
of Ponzi schemes, this is

00:00:27.530 --> 00:00:30.270
Charles Ponzi right here.

00:00:30.270 --> 00:00:34.820
It was obviously named after him
and then, more recently,

00:00:34.820 --> 00:00:39.660
this is Bernie Madoff, who
pulled off probably the

00:00:39.660 --> 00:00:43.310
longest-lasting and largest
Ponzi scheme of all time.

00:00:43.310 --> 00:00:46.380
Who knows, maybe there's a
longer-lasting and larger one

00:00:46.380 --> 00:00:48.640
out there that we have still
haven't figured out yet, but

00:00:48.640 --> 00:00:51.130
this is the largest
one to date.

00:00:51.130 --> 00:00:54.450
And Ponzi wasn't the first
person to come up with the

00:00:54.450 --> 00:00:57.350
Ponzi scheme, but they decided
to name it after him because

00:00:57.350 --> 00:00:59.890
he was the first person to
really make it famous.

00:00:59.890 --> 00:01:04.420
This mugshot was taken in the
early 1900s when he was

00:01:04.420 --> 00:01:06.560
finally caught for perpetrating
his scheme.

00:01:06.560 --> 00:01:07.910
So how does it work?

00:01:07.910 --> 00:01:09.600
So let's start with some
investors here.

00:01:09.600 --> 00:01:13.040
We could get rid of pictures
of these two gentlemen.

00:01:13.040 --> 00:01:15.900
So let's say that I've got a
scheme, and what I'm going to

00:01:15.900 --> 00:01:21.320
do is I'm going to set
up my investors, so

00:01:21.320 --> 00:01:22.900
these are my investors.

00:01:22.900 --> 00:01:25.960
And then, let's say, we have
several time periods.

00:01:25.960 --> 00:01:28.930
So we can see why the investors
think, at least

00:01:28.930 --> 00:01:31.970
initially, that my scheme
is legitimate.

00:01:31.970 --> 00:01:34.290
So, let's say, that this is the
time period, so we have

00:01:34.290 --> 00:01:38.930
period one, maybe this is years,
year one, year two,

00:01:38.930 --> 00:01:44.570
year three, year four,
year five.

00:01:44.570 --> 00:01:49.100
What I'm going to do is I'm
going to write each investor--

00:01:49.100 --> 00:01:51.970
I'm going to write down how
much money they think they

00:01:51.970 --> 00:01:55.860
have with me, the person
operating the Ponzi scheme and

00:01:55.860 --> 00:01:59.210
then I'll show you exactly how
much money I have and how I

00:01:59.210 --> 00:02:03.140
can even get away with having
the first investors think that

00:02:03.140 --> 00:02:04.640
I'm legitimate.

00:02:04.640 --> 00:02:07.410
So let's say I have
investor A.

00:02:07.410 --> 00:02:11.810
So this is going to be their
investor's perceived-- let me

00:02:11.810 --> 00:02:16.720
do this in a different color--
investor's perceived holdings

00:02:16.720 --> 00:02:19.130
or perceived value.

00:02:19.130 --> 00:02:21.560
And over here, I'm going to
write total actual value.

00:02:29.220 --> 00:02:32.310
So you can imagine, this is the
actual amount of cash that

00:02:32.310 --> 00:02:36.000
I have. So we have the investor
A in blue, and let's

00:02:36.000 --> 00:02:39.250
say in year one, he
gives me $10,000.

00:02:39.250 --> 00:02:42.730
And I say I I've got a surefire
way of doubling his

00:02:42.730 --> 00:02:44.860
money in the second year.

00:02:44.860 --> 00:02:47.090
So in the second year, I
actually do nothing with the

00:02:47.090 --> 00:02:49.760
money and you know, I might
actually be spending it on my

00:02:49.760 --> 00:02:52.940
own yachts and you know, fancy
suits and whatnot.

00:02:52.940 --> 00:02:55.040
But let's just say I'm just
keeping it in a bank account

00:02:55.040 --> 00:02:59.520
so you know, he gives it to me,
$10,000 and I do nothing

00:02:59.520 --> 00:03:00.020
with that money.

00:03:00.020 --> 00:03:01.060
I don't even get
interest on it.

00:03:01.060 --> 00:03:02.383
It's not even in a
bank account, I

00:03:02.383 --> 00:03:03.490
stuff it into my mattress.

00:03:03.490 --> 00:03:08.200
So the reality is, after a year,
it's still only $10,000,

00:03:08.200 --> 00:03:11.360
but I promised him that I had
some type of a genius scheme

00:03:11.360 --> 00:03:14.490
that could double his money
in a year, so I send him a

00:03:14.490 --> 00:03:20.910
statement that says that his
$10,000 is now $20,000.

00:03:20.910 --> 00:03:23.610
And I feel good about it because
I know that he's going

00:03:23.610 --> 00:03:26.040
to be so excited that his money
doubled that he's going

00:03:26.040 --> 00:03:28.300
to want to keep his money with
me, because he'll hope that it

00:03:28.300 --> 00:03:29.400
can double again.

00:03:29.400 --> 00:03:31.580
And not only is he going to do
that, but he's going to go to

00:03:31.580 --> 00:03:34.310
the country club and show off
to all of his other friends

00:03:34.310 --> 00:03:36.690
how he was able to do way better
than they did with

00:03:36.690 --> 00:03:37.600
their investments.

00:03:37.600 --> 00:03:39.290
So he's going to essentially
convince other

00:03:39.290 --> 00:03:40.570
people to join in.

00:03:40.570 --> 00:03:43.410
So let's say he convinces
investor B to join in.

00:03:43.410 --> 00:03:46.830
Investor B looks at the
statement says, hey, this guy

00:03:46.830 --> 00:03:48.542
running this-- well he doesn't
know it's a scheme, let's say

00:03:48.542 --> 00:03:51.140
he says , Sal seems to know what
he's doing, he doubled

00:03:51.140 --> 00:03:53.620
investor A's money in a year.

00:03:53.620 --> 00:03:55.510
I'm going to give him a bunch of
money, let's say I'm going

00:03:55.510 --> 00:04:01.890
to give him $15,000
in year two.

00:04:01.890 --> 00:04:04.860
So how much total actual value
do I have in year two now?

00:04:04.860 --> 00:04:13.040
I have the $10,000 from investor
A plus this $15,000,

00:04:13.040 --> 00:04:17.089
so I have a total of $25,000.

00:04:17.089 --> 00:04:19.279
This is the actual amount that
I have in my bank account,

00:04:19.279 --> 00:04:20.720
assuming that I'm not
spending it on my

00:04:20.720 --> 00:04:23.330
yacht or my fancy suits.

00:04:23.330 --> 00:04:26.240
But the total perceived value,
let me write that down another

00:04:26.240 --> 00:04:34.940
line, if everyone actually
wanted the amount of money

00:04:34.940 --> 00:04:36.800
that they thought they
had back, I would

00:04:36.800 --> 00:04:38.540
have to pay out $35,000.

00:04:38.540 --> 00:04:40.520
But we know that's not going
to happen, because people

00:04:40.520 --> 00:04:42.560
think I'm such a good investor,
they want their

00:04:42.560 --> 00:04:44.650
money to ride as long
as possible.

00:04:44.650 --> 00:04:46.450
So you already see
this discrepancy.

00:04:46.450 --> 00:04:49.830
There's only $25,000 in this
little pile of money that I'm

00:04:49.830 --> 00:04:53.160
collecting, but people think
that there should be $35,000.

00:04:53.160 --> 00:04:55.770
This is their perceived value,
because the this guy thought

00:04:55.770 --> 00:04:59.050
his money doubled, although it
didn't, it just sat there.

00:04:59.050 --> 00:05:01.270
Now, let's say that the next
year I sent them statements

00:05:01.270 --> 00:05:03.710
that say, look, I made
super-awesome investments

00:05:03.710 --> 00:05:06.250
again, the money
doubled again.

00:05:06.250 --> 00:05:09.140
So this guy's money, his
perceived value, he gets a

00:05:09.140 --> 00:05:13.170
statement that says you
now have $40,000.

00:05:13.170 --> 00:05:16.690
This guy down here, investor B,
gets a statement that says

00:05:16.690 --> 00:05:19.920
you now have $30,000.

00:05:19.920 --> 00:05:22.990
And then they go back to the
country club, and they get

00:05:22.990 --> 00:05:24.120
investor C onboard.

00:05:24.120 --> 00:05:26.680
They're like, look, both of us
have tried out this guy, he's

00:05:26.680 --> 00:05:29.630
doubled our money two years in
a row for both of us, you

00:05:29.630 --> 00:05:31.190
probably want in on
this as well.

00:05:31.190 --> 00:05:33.610
And investor C is like yeah,
well you know, my two buddies,

00:05:33.610 --> 00:05:36.200
they look like legitimate
guys, let me

00:05:36.200 --> 00:05:38.720
put my money there.

00:05:38.720 --> 00:05:41.160
Let's say it gets bigger every
time because that's usually

00:05:41.160 --> 00:05:43.030
how these things go, you know,
you normally don't don't have

00:05:43.030 --> 00:05:45.330
just three investors, you'll
have hundreds of investors.

00:05:45.330 --> 00:05:48.810
And the more fake positive
returns you get, the more

00:05:48.810 --> 00:05:50.750
people that want to put
their money in.

00:05:50.750 --> 00:05:52.870
So investor C, let's
say he comes in

00:05:52.870 --> 00:05:57.320
and he puts in $20,000.

00:05:57.320 --> 00:05:58.750
So what's the reality?

00:05:58.750 --> 00:05:59.470
Let's focus on it.

00:05:59.470 --> 00:06:02.330
So there's a perception
on year 3, this guy

00:06:02.330 --> 00:06:03.840
just put his $20,000.

00:06:03.840 --> 00:06:07.840
A think he's got $40,000, B
thinks he has $30,000, so the

00:06:07.840 --> 00:06:11.170
perceived value here's $40,000
plus $30,000, $70,000 plus

00:06:11.170 --> 00:06:13.700
$20,000 is $90,000.

00:06:13.700 --> 00:06:15.490
That's the perceived value.

00:06:15.490 --> 00:06:18.200
But the actual value is just
going to be this $25,000 we

00:06:18.200 --> 00:06:20.990
had in period two, assuming we
didn't spend the money or do

00:06:20.990 --> 00:06:23.190
anything with it plus
the $20,000 that

00:06:23.190 --> 00:06:24.310
this guy just deposited.

00:06:24.310 --> 00:06:28.890
So the reality is $45,000.

00:06:28.890 --> 00:06:33.700
Now let's say that as we go from
period three to period

00:06:33.700 --> 00:06:35.900
four, or let's say right when
this guy gets his statement

00:06:35.900 --> 00:06:40.790
for $40,000, and it's the exact
same time period that

00:06:40.790 --> 00:06:43.200
this guy had put
in his $20,000.

00:06:43.200 --> 00:06:46.520
Let's say person A, he says, you
know what, I felt like my

00:06:46.520 --> 00:06:49.940
ride has gone long enough, I
don't want to test fate, let

00:06:49.940 --> 00:06:54.960
me take my money out and you
might say, oh you the scheme

00:06:54.960 --> 00:06:57.920
will be ruined, but it's going
to work because enough money

00:06:57.920 --> 00:07:02.190
is coming in from new investors
to pay this guy off.

00:07:02.190 --> 00:07:05.800
We now have $45,000 actual
value, even though the people

00:07:05.800 --> 00:07:07.510
think there's $90,000.

00:07:07.510 --> 00:07:10.710
So if this guy withdraws all
of his money, so if he

00:07:10.710 --> 00:07:14.930
withdraws all of his money so
it goes to zero, I have the

00:07:14.930 --> 00:07:15.680
cash to pay them.

00:07:15.680 --> 00:07:19.000
I have $40,000 even though
people think there's $90,000.

00:07:19.000 --> 00:07:22.720
So I subtract out $40,000 right
here, and there's only

00:07:22.720 --> 00:07:25.880
$5,000 dollars left in the bank
account and since this

00:07:25.880 --> 00:07:31.000
guy withdrew his money, the
perceived value-- this $40,000

00:07:31.000 --> 00:07:34.830
is no longer there, I gave the
guy the cash-- the perceived

00:07:34.830 --> 00:07:37.050
value now is $50,000.

00:07:37.050 --> 00:07:40.740
So I essentially owe people
$50,000, investors B and C

00:07:40.740 --> 00:07:44.220
think that they have $50,000
invested with me, but the

00:07:44.220 --> 00:07:47.790
reality is that I only have
$5,000 in my bank account.

00:07:47.790 --> 00:07:50.090
And probably a more realistic
reality is I was probably

00:07:50.090 --> 00:07:54.730
spending a lot of this money on
my own little luxuries the

00:07:54.730 --> 00:07:55.500
whole time.

00:07:55.500 --> 00:07:58.880
But let's continue another way,
once again this guy, not

00:07:58.880 --> 00:08:01.290
only did he double his money for
two years, they're all the

00:08:01.290 --> 00:08:03.610
same country club.

00:08:03.610 --> 00:08:05.820
This guy doubled his money
again, this guy just

00:08:05.820 --> 00:08:06.670
invested his money.

00:08:06.670 --> 00:08:10.140
And now this guy says, look,
the scheme is legitimate.

00:08:10.140 --> 00:08:13.740
This legitimized the scheme
This is legit.

00:08:13.740 --> 00:08:16.570
Because, look, I doubled my
money for two years and I was

00:08:16.570 --> 00:08:18.410
able to withdraw the money.

00:08:24.450 --> 00:08:26.540
So he was able to withdraw his
money, so when he goes back to

00:08:26.540 --> 00:08:30.080
the country club, he gives this
guy and that guy more

00:08:30.080 --> 00:08:32.320
conviction that this is
all on the up and up.

00:08:32.320 --> 00:08:34.840
And then investor D will
probably jump in too and say,

00:08:34.840 --> 00:08:37.640
wow, now that he's doubled money
two years in a row, I

00:08:37.640 --> 00:08:38.549
see it's legitimate.

00:08:38.549 --> 00:08:41.539
Investor A was able to withdraw
his money, I'll give

00:08:41.539 --> 00:08:46.290
even more money, I'll
give $100,000.

00:08:46.290 --> 00:08:48.140
Maybe this is a ton of
people who are now

00:08:48.140 --> 00:08:49.740
going to put in a $100,000.

00:08:49.740 --> 00:08:52.400
And then the next year, I
double the money again.

00:08:52.400 --> 00:08:54.290
And obviously, I won't make it
exactly double, I'll make it,

00:08:54.290 --> 00:08:56.790
you know 40% one year and 30%
percent the next year so

00:08:56.790 --> 00:08:58.460
doesn't look too suspicious.

00:08:58.460 --> 00:09:00.590
I want to make it look like real
returns, but for the sake

00:09:00.590 --> 00:09:03.190
of our math, let's say
I double it again.

00:09:03.190 --> 00:09:05.210
So now and we're in year four.

00:09:05.210 --> 00:09:07.840
And this guy withdrew all of his
money, but investor B now

00:09:07.840 --> 00:09:12.800
thinks he has $60,000.

00:09:12.800 --> 00:09:17.990
Investor C thinks
he has $40,000.

00:09:17.990 --> 00:09:23.890
And investor D thinks
he has $200,000.

00:09:23.890 --> 00:09:26.330
Oh, and I forgot to put investor
D's deposit here.

00:09:26.330 --> 00:09:30.230
So when he put a $100,000, I
only had $5,000 in my bank

00:09:30.230 --> 00:09:32.100
account, but then if I add
$100,000 I'll now have

00:09:32.100 --> 00:09:36.140
$105,000 in my bank account
after this guy comes in at,

00:09:36.140 --> 00:09:39.830
you know, at the end of period
three, we can imagine.

00:09:39.830 --> 00:09:45.130
And, with the perceived amount,
I owe is $150,000.

00:09:45.130 --> 00:09:48.480
So the green is what happens
after D comes in, so these are

00:09:48.480 --> 00:09:50.350
no longer valid.

00:09:50.350 --> 00:09:53.220
But you can see that as more
money comes in, I have more

00:09:53.220 --> 00:09:54.810
and more money to pay
out, even though

00:09:54.810 --> 00:09:55.870
I'm not doing anything.

00:09:55.870 --> 00:09:57.980
Even though all of these
returns are fake.

00:09:57.980 --> 00:10:01.050
So now I actually have $105,000
even though people

00:10:01.050 --> 00:10:02.810
think, well that's at the end of
period three, at the end of

00:10:02.810 --> 00:10:05.470
period four, what
do people think?

00:10:05.470 --> 00:10:09.760
People think that I have
$300,000 of holdings.

00:10:09.760 --> 00:10:11.010
Let me write that down.

00:10:15.850 --> 00:10:18.870
But the reality is I still only

00:10:18.870 --> 00:10:23.610
have $105,000 of holdings.

00:10:23.610 --> 00:10:25.610
This is the total
actual value.

00:10:25.610 --> 00:10:28.750
But notice, if this guy or this
guy, some of the early

00:10:28.750 --> 00:10:30.740
investors, wanted to pull out
some of their money, although

00:10:30.740 --> 00:10:32.910
they probably don't want to,
because where else can you

00:10:32.910 --> 00:10:36.310
double your money every year
and this guy already showed

00:10:36.310 --> 00:10:37.930
that I'm good for paying
back the money.

00:10:37.930 --> 00:10:40.070
But even if this guy or this guy
wanted to pull out their

00:10:40.070 --> 00:10:43.030
money, I would be able to give
it to them because I have at

00:10:43.030 --> 00:10:45.110
least enough for those
withdrawals.

00:10:45.110 --> 00:10:48.390
Now, everything would be
ruined if everyone gets

00:10:48.390 --> 00:10:52.720
freaked out or scared and if I
have mass withdrawals or if

00:10:52.720 --> 00:10:56.170
more people withdraw money than
there is in the bank plus

00:10:56.170 --> 00:10:58.240
the amount of money
that comes in.

00:10:58.240 --> 00:11:01.500
So in order for a Ponzi scheme
to keep going, and Bernie

00:11:01.500 --> 00:11:05.060
Madoff was able to do this for
very long time, you have to

00:11:05.060 --> 00:11:08.250
have good, believable,
legitimate returns.

00:11:08.250 --> 00:11:09.710
Although they're not legitimate,
they just need to

00:11:09.710 --> 00:11:10.660
look legitimate.

00:11:10.660 --> 00:11:19.970
So that you have more money
coming in that out.

00:11:19.970 --> 00:11:22.480
And the whole point of the doing
this Ponzi scheme, if

00:11:22.480 --> 00:11:28.050
you're a stylish criminal, isn't
just to keep the cash

00:11:28.050 --> 00:11:30.780
there, you know, the $10,000
from one period to the next.

00:11:30.780 --> 00:11:32.600
The whole point of it is to
take a lot of that for

00:11:32.600 --> 00:11:35.260
yourself, for you to live off
of and put into some Swiss

00:11:35.260 --> 00:11:36.870
bank account to be able
to escape the

00:11:36.870 --> 00:11:38.580
country at some point.

00:11:38.580 --> 00:11:41.610
Anyway, hopefully you found
that enjoyable.