Multiplying positive and negative numbers

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:04.360
We know that if we were
to multiply 2 times 3,

00:00:04.360 --> 00:00:06.046
that would give us positive 6.

00:00:06.046 --> 00:00:07.670
And since we're going
to start thinking

00:00:07.670 --> 00:00:09.430
about negative numbers
in this video, one way

00:00:09.430 --> 00:00:11.150
to think about it I
had a positive number

00:00:11.150 --> 00:00:14.150
times another positive
number, and that gave me

00:00:14.150 --> 00:00:15.630
a positive number.

00:00:15.630 --> 00:00:19.570
So if I have a positive
times a positive,

00:00:19.570 --> 00:00:22.644
that will give me
a positive number.

00:00:22.644 --> 00:00:24.060
Now, let's mix it
up a little bit,

00:00:24.060 --> 00:00:26.360
introduce some negative numbers.

00:00:26.360 --> 00:00:31.360
So what happens if I
had negative 2 times 3?

00:00:34.549 --> 00:00:36.590
Well, one way to think
about, and we'll talk more

00:00:36.590 --> 00:00:38.881
about the intuition in this
video and in future videos,

00:00:38.881 --> 00:00:41.110
is, well, you could
view this as negative 2

00:00:41.110 --> 00:00:42.820
repeatedly added three times.

00:00:42.820 --> 00:00:48.970
So this could be negative 2 plus
negative 2 plus negative 2--

00:00:48.970 --> 00:00:51.240
not negative 6--
plus negative 2,

00:00:51.240 --> 00:00:54.370
which would be equal to-- well,
negative 2 plus negative 2

00:00:54.370 --> 00:00:58.040
is negative 4 plus another
negative 2 is negative 6.

00:00:58.040 --> 00:01:01.940
So this would be
equal to negative 6.

00:01:01.940 --> 00:01:04.959
Or another way to think about
it is if I had 2 times 3,

00:01:04.959 --> 00:01:06.030
I would get 6.

00:01:06.030 --> 00:01:09.350
But because one of these
two numbers is negative,

00:01:09.350 --> 00:01:12.280
then my product is
going to be negative.

00:01:12.280 --> 00:01:19.820
So if I multiply a
negative times a positive,

00:01:19.820 --> 00:01:22.130
I'm going to get a negative.

00:01:22.130 --> 00:01:25.030
Now, what if we swap the
order in which we multiply?

00:01:25.030 --> 00:01:30.347
So if we were to multiply
3 times negative 2.

00:01:30.347 --> 00:01:31.430
Well, it shouldn't matter.

00:01:31.430 --> 00:01:34.142
The order in which
we multiply things

00:01:34.142 --> 00:01:35.350
shouldn't change the product.

00:01:35.350 --> 00:01:37.550
Whether we multiply 2
times 3, we'll get 6,

00:01:37.550 --> 00:01:40.660
or if we multiply 3
times 2, we'll get 6.

00:01:40.660 --> 00:01:43.090
And so we should have
the same property here.

00:01:43.090 --> 00:01:45.730
3 times negative 2 should
give us the same result.

00:01:45.730 --> 00:01:50.260
It's going to be
equal to negative 6.

00:01:50.260 --> 00:01:53.370
And once again, we say
3 times 2 would be 6.

00:01:53.370 --> 00:01:55.990
One of these two
numbers is negative.

00:01:55.990 --> 00:01:58.880
And so our product is
going to be negative.

00:01:58.880 --> 00:02:01.080
So we could write
a positive times

00:02:01.080 --> 00:02:04.460
a negative is also
going to be a negative.

00:02:04.460 --> 00:02:06.900
And both of these are
just the same thing

00:02:06.900 --> 00:02:10.110
with the order in which we're
multiplying switched around.

00:02:10.110 --> 00:02:11.910
But this is one
of the two numbers

00:02:11.910 --> 00:02:14.370
are negative, exactly one.

00:02:14.370 --> 00:02:22.200
So one negative, one positive
number is being multiplied.

00:02:22.200 --> 00:02:25.150
Then you will get
a negative product.

00:02:25.150 --> 00:02:27.020
Now let's think about
the third circumstance

00:02:27.020 --> 00:02:31.260
when both of the
numbers are negative.

00:02:31.260 --> 00:02:33.060
I'll just switch
colors for fun here.

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If I were to multiply
negative 2 times

00:02:36.290 --> 00:02:38.415
negative 3-- and this might
be the least intuitive

00:02:38.415 --> 00:02:39.040
for you of all.

00:02:39.040 --> 00:02:41.248
And here I'm just going to
introduce you to the rule.

00:02:41.248 --> 00:02:43.700
And in future videos,
we'll explore why this is

00:02:43.700 --> 00:02:48.380
and why this makes mathematics
more all fit together.

00:02:48.380 --> 00:02:51.140
But this is going to be,
you say, well, 2 times 3

00:02:51.140 --> 00:02:54.630
would be 6, and I have a
negative times a negative.

00:02:54.630 --> 00:02:56.150
And one way you
can think about it

00:02:56.150 --> 00:02:58.270
is that the
negatives cancel out.

00:02:58.270 --> 00:03:00.354
And so you will actually
end up with a positive 6.

00:03:00.354 --> 00:03:02.311
I actually don't have to
write a positive here,

00:03:02.311 --> 00:03:04.280
but I'll write it here
just to reemphasize.

00:03:04.280 --> 00:03:06.760
This right over here
is a positive 6.

00:03:06.760 --> 00:03:10.010
So we have another
rule of thumb here.

00:03:10.010 --> 00:03:14.720
If I have a negative
times a negative,

00:03:14.720 --> 00:03:16.680
the negatives are
going to cancel out.

00:03:16.680 --> 00:03:21.440
And that's going to give
me a positive number.

00:03:21.440 --> 00:03:23.640
Now, with these out
of the way, let's just

00:03:23.640 --> 00:03:24.700
do a bunch of examples.

00:03:24.700 --> 00:03:27.180
I encourage you to try
them out before I do them.

00:03:27.180 --> 00:03:29.010
Pause the video, try
them out, and see

00:03:29.010 --> 00:03:30.810
if you get the same answer.

00:03:30.810 --> 00:03:35.140
So let's try negative
1 times negative 1.

00:03:35.140 --> 00:03:39.160
Well, 1 times 1 would be 1,
and we have a negative times

00:03:39.160 --> 00:03:40.050
a negative.

00:03:40.050 --> 00:03:40.896
They cancel out.

00:03:40.896 --> 00:03:42.770
Negative times a negative
give me a positive,

00:03:42.770 --> 00:03:44.250
so this is going
to be positive 1.

00:03:44.250 --> 00:03:46.000
I could just write 1,
or I could literally

00:03:46.000 --> 00:03:47.665
write a plus sign
there to emphasize

00:03:47.665 --> 00:03:49.940
that this is a positive 1.

00:03:49.940 --> 00:03:53.284
What happens if I did
negative 1 times 0?

00:03:53.284 --> 00:03:55.200
Now, this might say,
wait, this doesn't really

00:03:55.200 --> 00:03:56.700
fit into any of
these circumstances.

00:03:56.700 --> 00:03:59.414
0 is neither positive
nor negative.

00:03:59.414 --> 00:04:00.830
And here you just
have to remember

00:04:00.830 --> 00:04:04.320
anything times 0
is going to be 0.

00:04:04.320 --> 00:04:07.270
So negative 1 times
0 is going to be 0.

00:04:07.270 --> 00:04:11.800
Or I could have said
0 times negative 783,

00:04:11.800 --> 00:04:14.085
that is also going to be 0.

00:04:17.130 --> 00:04:18.970
Let me do some interesting ones.

00:04:18.970 --> 00:04:25.470
What about-- I'll pick a new
color-- 12 times negative 4?

00:04:25.470 --> 00:04:30.200
Well, once again, 12 times
positive 4 would be 48.

00:04:30.200 --> 00:04:33.010
And we're in the circumstance
where one of these two numbers

00:04:33.010 --> 00:04:36.200
right over here is negative,
this one right over here.

00:04:36.200 --> 00:04:38.830
If exactly one of the
two numbers is negative,

00:04:38.830 --> 00:04:41.650
then the product is
going to be negative.

00:04:41.650 --> 00:04:45.250
We are in this circumstance
right over here.

00:04:45.250 --> 00:04:48.170
We have one negative, so
the product is negative.

00:04:48.170 --> 00:04:51.366
You could imagine this as
repeatedly adding negative 4

00:04:51.366 --> 00:04:52.410
twelve times.

00:04:52.410 --> 00:04:55.300
And so you would
get to negative 48.

00:04:55.300 --> 00:04:56.610
Let's do another one.

00:04:56.610 --> 00:05:00.410
What is 7 times 3?

00:05:00.410 --> 00:05:01.790
Well, this is a bit of a trick.

00:05:01.790 --> 00:05:03.248
There are no negative
numbers here.

00:05:03.248 --> 00:05:06.732
This is just going to be 7 times
3, positive 7 times positive 3,

00:05:06.732 --> 00:05:08.440
the first circumstance,
which you already

00:05:08.440 --> 00:05:10.250
knew how to do
before this video.

00:05:10.250 --> 00:05:12.950
This would just be equal to 21.

00:05:12.950 --> 00:05:14.220
Let's do one more.

00:05:14.220 --> 00:05:18.520
So if I were to say negative
5 times negative 10,

00:05:18.520 --> 00:05:20.410
well, once again,
negative times a negative,

00:05:20.410 --> 00:05:21.990
the negatives cancel out.

00:05:21.990 --> 00:05:23.960
Then you're just left
with a positive product.

00:05:23.960 --> 00:05:25.310
So it's going to be 5 times 10.

00:05:25.310 --> 00:05:26.430
It's going to be 50.

00:05:26.430 --> 00:05:28.180
The negative and the
negatives cancel out.

00:05:28.180 --> 00:05:30.120
Your product is
going to be positive.

00:05:30.120 --> 00:05:33.430
That's this situation
right over there.


Orienting boundary with surface

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:02.420
I've restated Stokes' theorem.

00:00:02.420 --> 00:00:03.960
And what I want to
do in this video

00:00:03.960 --> 00:00:07.210
is make sure that we get
our orientation right.

00:00:07.210 --> 00:00:12.980
Because when we think about
a normal vector to a surface

00:00:12.980 --> 00:00:14.977
there are actually
two normal vectors.

00:00:14.977 --> 00:00:16.560
There-- based on the
way I've drawn it

00:00:16.560 --> 00:00:19.350
right over here, there could be
the one that might pop outward,

00:00:19.350 --> 00:00:20.360
like this.

00:00:20.360 --> 00:00:23.800
Or there might be the one that
pops inward, just like that.

00:00:23.800 --> 00:00:26.760
Both of those would be normal
to the surface right there.

00:00:26.760 --> 00:00:28.890
And also, when we
think about a path that

00:00:28.890 --> 00:00:30.760
goes around the
boundary of a surface,

00:00:30.760 --> 00:00:32.960
there's two ways to
think about that path.

00:00:32.960 --> 00:00:35.900
We could be going-- based on
how I've oriented it right now,

00:00:35.900 --> 00:00:39.610
we could go in a
counterclockwise orientation,

00:00:39.610 --> 00:00:40.510
or direction.

00:00:40.510 --> 00:00:45.370
Or we could go in a clockwise
orientation, or direction.

00:00:45.370 --> 00:00:48.110
So in order to make sure
we're using Stokes' theorem

00:00:48.110 --> 00:00:50.620
correctly, we need to
make sure we understand

00:00:50.620 --> 00:00:53.130
which each convention
it is using.

00:00:53.130 --> 00:00:55.980
And the way we think
about it is, whatever

00:00:55.980 --> 00:00:58.540
the normal direction
we pick-- and so let's

00:00:58.540 --> 00:01:01.080
say we pick this normal
direction right over here,

00:01:01.080 --> 00:01:02.840
the one I am drawing in yellow.

00:01:02.840 --> 00:01:05.205
So if we pick this
as our normal vector.

00:01:05.205 --> 00:01:06.580
So we're essentially
saying maybe

00:01:06.580 --> 00:01:08.740
that's the top, one way
of doing it is, that's

00:01:08.740 --> 00:01:13.080
the top of our surface, then
the positive orientation that we

00:01:13.080 --> 00:01:16.390
need to traverse the path in is
the one that if your head was

00:01:16.390 --> 00:01:19.590
pointed in the direction
of the normal vector,

00:01:19.590 --> 00:01:22.960
and you were to walk along
that path, the inside,

00:01:22.960 --> 00:01:27.180
or the surface itself
would be to your left.

00:01:27.180 --> 00:01:30.070
And so, if my head is
pointed in the direction

00:01:30.070 --> 00:01:33.420
of the normal vector-- so
this is me right over here--

00:01:33.420 --> 00:01:35.080
my head is pointed
in the direction

00:01:35.080 --> 00:01:37.320
of the normal vector-- I'm
wearing a big arrow hat

00:01:37.320 --> 00:01:40.590
right over there-- and if I'm
walking around the boundary,

00:01:40.590 --> 00:01:42.980
the actual surface
needs to be to my left.

00:01:42.980 --> 00:01:48.270
So I need to be-- this is me
walking right over here-- I

00:01:48.270 --> 00:01:52.010
need to be walking in the
counterclockwise direction

00:01:52.010 --> 00:01:53.194
just like that.

00:01:53.194 --> 00:01:54.610
Then that's the
convention that we

00:01:54.610 --> 00:01:58.040
use when we're thinking
about Stokes' theorem.

00:01:58.040 --> 00:02:00.280
If oriented this
thing differently,

00:02:00.280 --> 00:02:02.260
or if we said that
no, no, no, no, no,

00:02:02.260 --> 00:02:03.560
this is not the normal vector.

00:02:03.560 --> 00:02:07.560
This is not the, essentially,
the top that we want to pick.

00:02:07.560 --> 00:02:09.470
If we wanted to pick
it the other way,

00:02:09.470 --> 00:02:11.830
if we wanted to go
in that direction,

00:02:11.830 --> 00:02:15.330
If we wanted that to be
our normal vector, in order

00:02:15.330 --> 00:02:17.660
to be consistent, we would
have to now do the opposite.

00:02:17.660 --> 00:02:20.490
I would now have to have my
head going in that direction.

00:02:23.870 --> 00:02:26.135
And then I would have
to walk, once again,

00:02:26.135 --> 00:02:28.260
and this might be a little
bit harder to visualize.

00:02:28.260 --> 00:02:30.440
I would have to walk
in the direction

00:02:30.440 --> 00:02:32.057
that the surface is to my left.

00:02:32.057 --> 00:02:33.640
And now, in this
situation, instead of

00:02:33.640 --> 00:02:35.223
the surface looking
like a hill to me,

00:02:35.223 --> 00:02:37.470
the surface would look
like some type of a bowl,

00:02:37.470 --> 00:02:41.359
or some type of a valley or
something like that to me.

00:02:41.359 --> 00:02:43.150
And the way that I
would have to do it now,

00:02:43.150 --> 00:02:44.774
and it's a little
bit hard to visualize

00:02:44.774 --> 00:02:47.280
the upside down Sal,
but the upside down Sal

00:02:47.280 --> 00:02:53.050
would have to walk in
this direction in order

00:02:53.050 --> 00:02:57.570
for the bowl, or the
dip, to be to my left.

00:02:57.570 --> 00:03:00.390
So that's just important
to keep this in mind

00:03:00.390 --> 00:03:04.170
in order for this to be
consistent with this right

00:03:04.170 --> 00:03:04.780
over here.

00:03:04.780 --> 00:03:08.140
Put your head in the direction
of the normal vector.

00:03:08.140 --> 00:03:11.110
Or you can kind of view that
as the top of the direction

00:03:11.110 --> 00:03:14.900
that the top of the
surface is going in.

00:03:14.900 --> 00:03:18.290
And then the contour,
or the direction

00:03:18.290 --> 00:03:20.910
that you would have to traverse
the boundary in order for this

00:03:20.910 --> 00:03:24.280
to be true, is the
direction with which

00:03:24.280 --> 00:03:27.457
the surface is to your left.


Green's and Stokes' theorem relationship

WEBVTT
Kind: captions
Language: en

00:00:00.520 --> 00:00:03.242
In the last video, we began
to explore Stokes' theorem.

00:00:03.242 --> 00:00:04.700
And what I want to
do in this video

00:00:04.700 --> 00:00:07.060
is to see whether it's
consistent with some

00:00:07.060 --> 00:00:09.050
of what we have already seen.

00:00:09.050 --> 00:00:12.190
And to do that, let's imagine--
so let me draw my axes.

00:00:12.190 --> 00:00:14.340
So that's my z-axis.

00:00:14.340 --> 00:00:16.680
That is my x-axis.

00:00:16.680 --> 00:00:19.610
And then that is my y-axis.

00:00:19.610 --> 00:00:23.430
And now let's imagine a
region in the xy plane.

00:00:23.430 --> 00:00:25.770
So let me draw it like this.

00:00:25.770 --> 00:00:30.670
So let's say this is my
region in the xy plane.

00:00:30.670 --> 00:00:34.850
I will call that
region R. And I also

00:00:34.850 --> 00:00:36.580
have a boundary of that region.

00:00:36.580 --> 00:00:39.470
And let's say we care
about the direction

00:00:39.470 --> 00:00:40.720
that we traverse the boundary.

00:00:40.720 --> 00:00:41.650
And let's say we're
going to traverse it

00:00:41.650 --> 00:00:43.180
in a counterclockwise direction.

00:00:43.180 --> 00:00:47.150
So we have this path that
goes around this region.

00:00:47.150 --> 00:00:49.890
We can call that c.

00:00:49.890 --> 00:00:51.950
So we'll call that
c, and we're going

00:00:51.950 --> 00:00:57.010
to traverse it in the
counterclockwise direction.

00:00:57.010 --> 00:01:02.310
And let's say that we also
have a vector field f.

00:01:02.310 --> 00:01:05.360
That essentially its
i component is just

00:01:05.360 --> 00:01:08.030
going to be a
function of x and y.

00:01:08.030 --> 00:01:10.310
And its j component
is only going

00:01:10.310 --> 00:01:12.530
to be a function of x and y.

00:01:12.530 --> 00:01:14.780
And let's say it
has no k component.

00:01:14.780 --> 00:01:17.230
So the vector field
on this region,

00:01:17.230 --> 00:01:18.750
it might look
something like this.

00:01:18.750 --> 00:01:20.422
I'm just drawing random things.

00:01:20.422 --> 00:01:21.880
And then if you go
off that region,

00:01:21.880 --> 00:01:23.350
if you go in the
z direction, it's

00:01:23.350 --> 00:01:25.700
just going to look the same
as you go higher and higher.

00:01:25.700 --> 00:01:27.930
So that vector,
it wouldn't change

00:01:27.930 --> 00:01:29.660
as you change your z component.

00:01:29.660 --> 00:01:31.450
And all of the vectors
would essentially

00:01:31.450 --> 00:01:35.790
be parallel to, or
if z is 0, actually

00:01:35.790 --> 00:01:39.100
sitting on the xy plane.

00:01:39.100 --> 00:01:41.480
Now given this, let's
think about what

00:01:41.480 --> 00:01:46.010
Stokes' theorem would tell us
about the value of the line

00:01:46.010 --> 00:01:48.980
integral over the
contour-- let me

00:01:48.980 --> 00:01:51.470
draw that a little
bit neater-- the line

00:01:51.470 --> 00:02:00.960
integral over the
contour c of f dot

00:02:00.960 --> 00:02:05.960
dr, f dot lowercase dr,
Where dr is obviously

00:02:05.960 --> 00:02:08.280
going along the contour.

00:02:08.280 --> 00:02:11.470
So if we take Stokes'
theorem, then this quantity

00:02:11.470 --> 00:02:13.850
right over here should
be equal to this quantity

00:02:13.850 --> 00:02:14.610
right over here.

00:02:14.610 --> 00:02:18.850
It should be equal to the double
integral over the surface.

00:02:18.850 --> 00:02:21.270
Well this region is
really just a surface

00:02:21.270 --> 00:02:23.450
that's sitting in the xy plane.

00:02:23.450 --> 00:02:26.077
So it should really just
be the double integral--

00:02:26.077 --> 00:02:27.660
let me write that
in that same-- it'll

00:02:27.660 --> 00:02:31.310
be the double integral over
our region, which is really

00:02:31.310 --> 00:02:35.110
just the same thing
as our surface,

00:02:35.110 --> 00:02:37.840
of the curl of f dot n.

00:02:37.840 --> 00:02:40.437
So let's just think about
what the curl of f dot n is.

00:02:40.437 --> 00:02:42.270
And then d of s would
just be a little chunk

00:02:42.270 --> 00:02:45.510
of our region, a little chunk
of our flattened surface

00:02:45.510 --> 00:02:46.220
right over there.

00:02:46.220 --> 00:02:50.180
So instead of ds,
I'll just write da.

00:02:50.180 --> 00:02:54.000
But let's think of what curl
of f dot n would actually be.

00:02:54.000 --> 00:02:56.060
So let's work on
curl of f first.

00:02:56.060 --> 00:02:58.910
So the curl of f--
and the way I always

00:02:58.910 --> 00:03:00.810
remember it is
we're going to take

00:03:00.810 --> 00:03:06.850
the determinant of
this ijk partial

00:03:06.850 --> 00:03:10.820
with respect to x,
partial with respect to y,

00:03:10.820 --> 00:03:12.330
partial with respect to z.

00:03:12.330 --> 00:03:14.450
This is just the definition
of taking the curl.

00:03:14.450 --> 00:03:16.820
We're figuring out how
much this vector field

00:03:16.820 --> 00:03:18.910
would cause something to spin.

00:03:18.910 --> 00:03:20.980
And then we want the
i component, which

00:03:20.980 --> 00:03:24.416
is our function p, which is
just a function of x and y,

00:03:24.416 --> 00:03:26.990
j component, which is
just the function q.

00:03:26.990 --> 00:03:30.720
And there was no z
component over here, so 0.

00:03:30.720 --> 00:03:32.892
And so this is going
to be equal to-- well

00:03:32.892 --> 00:03:34.350
if we look at the
i component, it's

00:03:34.350 --> 00:03:35.960
going to be the
partial of y of 0.

00:03:35.960 --> 00:03:42.570
That's just going to be 0, minus
the partial of q with respect

00:03:42.570 --> 00:03:43.450
to z.

00:03:43.450 --> 00:03:46.000
Well what's the partial
of q with respect to z?

00:03:46.000 --> 00:03:48.190
Well q isn't a
function of z at all.

00:03:48.190 --> 00:03:50.470
So that's also going to be
0-- let me write this out

00:03:50.470 --> 00:03:52.330
just so it's not too confusing.

00:03:52.330 --> 00:03:56.180
So our i component, it's
going to be partial of 0

00:03:56.180 --> 00:03:57.130
with respect to y.

00:03:57.130 --> 00:04:01.000
Well that's just going to
be 0 minus the partial of q

00:04:01.000 --> 00:04:02.290
with respect to z.

00:04:02.290 --> 00:04:04.340
Well the partial of
q with respect to z

00:04:04.340 --> 00:04:05.700
is just going to be 0.

00:04:05.700 --> 00:04:07.500
So we have a 0 i component.

00:04:07.500 --> 00:04:10.260
And then we want to
subtract the j component.

00:04:10.260 --> 00:04:16.700
And then the j component partial
of 0 with respect to x is 0.

00:04:16.700 --> 00:04:20.079
And then from that you're going
to subtract the partial of p

00:04:20.079 --> 00:04:22.180
with respect to z.

00:04:22.180 --> 00:04:25.590
Well once again, p is not
a function of z at all.

00:04:25.590 --> 00:04:28.160
So that's going to be 0 again.

00:04:28.160 --> 00:04:33.911
And then you have plus k times
the partial of q with respect

00:04:33.911 --> 00:04:34.410
to x.

00:04:34.410 --> 00:04:36.320
Remember this is just the
partial derivative operator.

00:04:36.320 --> 00:04:38.180
So the partial of q
with respect to x.

00:04:41.160 --> 00:04:43.450
And from that we're going
to subtract the partial of p

00:04:43.450 --> 00:04:44.570
with respect to y.

00:04:49.690 --> 00:04:56.150
So the curl of f just simplifies
to this right over here.

00:04:56.150 --> 00:04:58.880
Now what is n?

00:04:58.880 --> 00:05:02.250
What is the unit normal vector.

00:05:02.250 --> 00:05:04.300
Well we're in the xy plane.

00:05:04.300 --> 00:05:05.930
So the unit normal
vector is just

00:05:05.930 --> 00:05:07.940
going to be straight
up in the z direction.

00:05:07.940 --> 00:05:10.390
It's going to have
a magnitude of 1.

00:05:10.390 --> 00:05:12.450
So in this case, our
unit normal vector

00:05:12.450 --> 00:05:14.660
is just going to
be the k vector.

00:05:14.660 --> 00:05:18.490
So we're essentially just going
to take-- so curl of f is this.

00:05:18.490 --> 00:05:21.880
And our unit normal
vector is just

00:05:21.880 --> 00:05:24.510
going to be equal to the k.

00:05:24.510 --> 00:05:26.920
It's just going to
be the k unit vector.

00:05:26.920 --> 00:05:28.230
It's going to go straight up.

00:05:28.230 --> 00:05:31.160
So what happens if we
take the curl of f dot k?

00:05:31.160 --> 00:05:34.030
If we just dot this with k.

00:05:34.030 --> 00:05:36.080
We're just dotting
this with this.

00:05:36.080 --> 00:05:39.730
Well, we're just going to end up
with this part right over here.

00:05:39.730 --> 00:05:43.930
So curl of f dot the unit
normal vector is just

00:05:43.930 --> 00:05:45.400
going to be equal
to this business.

00:05:45.400 --> 00:05:49.260
It's just going to be equal to
the partial of q with respect

00:05:49.260 --> 00:05:54.980
to x minus the partial
of p with respect to y.

00:05:54.980 --> 00:05:57.944
And this is neat because
using Stokes' theorem

00:05:57.944 --> 00:05:59.610
in the special case,
where we're dealing

00:05:59.610 --> 00:06:03.030
with a flattened-out
surface in the xy plane,

00:06:03.030 --> 00:06:07.960
in this situation, this just
boiled down to Green's theorem.

00:06:07.960 --> 00:06:12.030
This thing right over here just
boiled down to Green's theorem.

00:06:12.030 --> 00:06:15.920
So we see that Green's theorem
is really just a special case--

00:06:15.920 --> 00:06:17.840
let me write theorem
a little bit neater.

00:06:17.840 --> 00:06:20.390
We see that Green's
theorem is really

00:06:20.390 --> 00:06:22.800
just a special case
of Stokes' theorem,

00:06:22.800 --> 00:06:27.360
where our surface is flattened
out, and it's in the xy plane.

00:06:27.360 --> 00:06:30.140
So that should make us feel
pretty good, although we still

00:06:30.140 --> 00:06:32.240
have not proven Stokes' theorem.

00:06:32.240 --> 00:06:34.530
But the one thing that I do
like about this is seeing

00:06:34.530 --> 00:06:36.780
that Green's theorem and
Stokes' theorem is consistent

00:06:36.780 --> 00:06:39.430
is now it starts to make
sense of this right over here.

00:06:39.430 --> 00:06:40.810
When we first learned Green's
theorem, we were like,

00:06:40.810 --> 00:06:41.380
what is this?

00:06:41.380 --> 00:06:42.565
what's going on over here?

00:06:42.565 --> 00:06:44.190
But now this is
telling us this is just

00:06:44.190 --> 00:06:47.920
taking the curl in this
region along this surface.

00:06:47.920 --> 00:06:50.840
And now starts to make a lot
of sense based on the intuition

00:06:50.840 --> 00:06:54.090
that we saw in the last video.


Stokes' theorem intuition

WEBVTT
Kind: captions
Language: en

00:00:00.650 --> 00:00:04.160
So I've drawn multiple versions
of the exact same surface

00:00:04.160 --> 00:00:07.790
S, five copies of that
exact same surface.

00:00:07.790 --> 00:00:11.060
And what I want to do is think
about the value of the line

00:00:11.060 --> 00:00:14.320
integral-- let me write this
down-- the value of the line

00:00:14.320 --> 00:00:22.840
integral of F dot dr, where F
is the vector field that I've

00:00:22.840 --> 00:00:24.920
drawn in magenta in
each of these diagrams.

00:00:24.920 --> 00:00:27.802
And obviously, it's different
in each of these diagrams.

00:00:27.802 --> 00:00:30.010
And the only part of the
vector field that I've drawn

00:00:30.010 --> 00:00:31.560
is a part that's
along the surface.

00:00:31.560 --> 00:00:33.410
I could have drawn the
part of the vector field

00:00:33.410 --> 00:00:34.870
that's off the
surface, but we're only

00:00:34.870 --> 00:00:37.390
going to be concerned with
what's going on on the surface.

00:00:37.390 --> 00:00:38.960
So the vector field
could be defined

00:00:38.960 --> 00:00:42.080
in this entire three-dimensional
space in here as well.

00:00:42.080 --> 00:00:44.080
And these are obviously
different vector fields,

00:00:44.080 --> 00:00:47.060
and we can see that visually
based on how we drew it.

00:00:47.060 --> 00:00:49.070
And the contour that we
care about-- remember,

00:00:49.070 --> 00:00:51.444
we're going to take a line
integral, so the path matters.

00:00:51.444 --> 00:00:55.530
The path that we care about is
the counterclockwise boundary

00:00:55.530 --> 00:00:58.110
of our surface, so it's going
to be this right over here.

00:00:58.110 --> 00:01:02.210
The counterclockwise
boundary of our surface

00:01:02.210 --> 00:01:06.965
is what we're going to
be taking F dot dr along.

00:01:06.965 --> 00:01:09.514
So this right over here, and
let me draw the orientation.

00:01:09.514 --> 00:01:10.930
It's going to be
counterclockwise.

00:01:10.930 --> 00:01:15.730
And we're going to do it in
every one of these situations,

00:01:15.730 --> 00:01:19.520
for every one of these surfaces
and every one of these F's.

00:01:19.520 --> 00:01:22.520
And what I really want to think
about is how the value of F

00:01:22.520 --> 00:01:26.110
dot dr over that
contour, how it might

00:01:26.110 --> 00:01:27.714
change from example to example.

00:01:27.714 --> 00:01:29.130
And obviously, the
only difference

00:01:29.130 --> 00:01:34.230
between each of these is what
the vector field F is doing.

00:01:34.230 --> 00:01:37.180
So first let's think about
this example right over here.

00:01:37.180 --> 00:01:39.540
At this part of the
contour, this bottom part

00:01:39.540 --> 00:01:41.450
right over here,
our vector field

00:01:41.450 --> 00:01:45.160
is going in the exact same
direction as our line,

00:01:45.160 --> 00:01:46.620
as our contour.

00:01:46.620 --> 00:01:50.474
So we're going to get positive
values of F dot dr down here,

00:01:50.474 --> 00:01:52.140
and we're going to
keep summing them up.

00:01:52.140 --> 00:01:53.550
We're taking an integral.

00:01:53.550 --> 00:01:55.910
Then as we go up
the curve or as we

00:01:55.910 --> 00:01:58.520
go kind of uphill
right over here,

00:01:58.520 --> 00:02:02.180
we see that our vector field is
going essentially orthogonal.

00:02:02.180 --> 00:02:03.580
It's going perpendicular.

00:02:03.580 --> 00:02:05.760
It's going perpendicular
to our contour.

00:02:05.760 --> 00:02:08.610
So our F dot dr, we're not going
to get any value from that.

00:02:08.610 --> 00:02:11.665
F dot dr and all of these,
this part of the contour,

00:02:11.665 --> 00:02:13.300
it's all going to be 0.

00:02:13.300 --> 00:02:15.770
So we're not going to get
anything right over there.

00:02:15.770 --> 00:02:17.640
Actually, let me
just write nothing.

00:02:17.640 --> 00:02:20.020
So we get nothing
right over there.

00:02:20.020 --> 00:02:22.060
Maybe I'll write 0.

00:02:22.060 --> 00:02:24.270
I'll write we'll get
0 right over there.

00:02:24.270 --> 00:02:27.320
And then up here, when we're
at this part of the contour,

00:02:27.320 --> 00:02:31.580
our vector field is going in
the exact opposite direction

00:02:31.580 --> 00:02:32.430
as our path.

00:02:32.430 --> 00:02:34.247
Up here our path
is going, I guess,

00:02:34.247 --> 00:02:36.330
from the right to the left,
while our vector field

00:02:36.330 --> 00:02:38.200
is going from the
left to the right.

00:02:38.200 --> 00:02:40.330
And so we're actually going
to get negative values.

00:02:40.330 --> 00:02:42.500
We are going to get
negative values up here,

00:02:42.500 --> 00:02:45.700
and they're going to sum up to
a reasonable negative value.

00:02:45.700 --> 00:02:48.030
And if the vector
field is constant,

00:02:48.030 --> 00:02:49.900
and I kind of drew
it like it is,

00:02:49.900 --> 00:02:52.590
and if this length is
equal to this length,

00:02:52.590 --> 00:02:54.870
then these two values
are going to cancel out.

00:02:54.870 --> 00:02:57.900
When you add this positive
sum to this negative sum,

00:02:57.900 --> 00:02:59.520
they're going to be 0.

00:02:59.520 --> 00:03:01.570
And then once again, when
you go downhill again,

00:03:01.570 --> 00:03:03.410
the vector field
is perpendicular

00:03:03.410 --> 00:03:06.550
to our actual path, so
we're going to get 0.

00:03:06.550 --> 00:03:09.640
So based on the way
I've described it,

00:03:09.640 --> 00:03:14.140
your line integral of F dot
dr for this version of F

00:03:14.140 --> 00:03:17.090
in this example right over
here, it might all cancel out.

00:03:17.090 --> 00:03:19.780
You will get something, if we
make the assumptions that I

00:03:19.780 --> 00:03:23.400
made, this might be equal to 0.

00:03:23.400 --> 00:03:31.280
So in this example, F dot
dr could be equal to 0.

00:03:31.280 --> 00:03:33.670
Now let's think about what's
going on in this situation.

00:03:33.670 --> 00:03:37.240
In this one, just like the last
one, as we go along the bottom,

00:03:37.240 --> 00:03:39.980
the vector field is going
in the exact same direction

00:03:39.980 --> 00:03:43.020
as our contour, so we're
going to get positive values.

00:03:43.020 --> 00:03:45.400
And once we go up the
hill, the vector field

00:03:45.400 --> 00:03:46.992
is going perpendicular
to our path,

00:03:46.992 --> 00:03:48.950
so it's not going to add
anything really to it.

00:03:48.950 --> 00:03:51.290
So we're just going to
get 0 along this part.

00:03:51.290 --> 00:03:54.250
But then up here, our vector
field has switched directions.

00:03:54.250 --> 00:03:57.200
And once again, it's going
in the exact same direction

00:03:57.200 --> 00:04:00.550
as our path, so we're going to
get more positive values right

00:04:00.550 --> 00:04:01.380
over there.

00:04:01.380 --> 00:04:03.220
And then as we go
down here, it's

00:04:03.220 --> 00:04:05.570
not going to add anything
because our vector field is

00:04:05.570 --> 00:04:06.695
perpendicular to our path.

00:04:06.695 --> 00:04:08.270
So we're going to get 0.

00:04:08.270 --> 00:04:11.994
But notice, now these
two ends don't cancel out

00:04:11.994 --> 00:04:12.660
with each other.

00:04:12.660 --> 00:04:15.100
We're going to get
a positive value.

00:04:15.100 --> 00:04:17.470
We are going to get
a positive value.

00:04:17.470 --> 00:04:19.660
And what was the difference
between this version

00:04:19.660 --> 00:04:22.310
of F, this vector field,
and this vector field

00:04:22.310 --> 00:04:23.360
right over here?

00:04:23.360 --> 00:04:25.530
Well, this vector field
switched directions

00:04:25.530 --> 00:04:27.130
so that the top part
didn't cancel out

00:04:27.130 --> 00:04:28.210
with the bottom part.

00:04:28.210 --> 00:04:31.680
Or another way to think about
it is, it had some curl.

00:04:31.680 --> 00:04:33.480
There's a little bit
of spinning going on.

00:04:33.480 --> 00:04:36.210
If this was describing
the velocity of a fluid,

00:04:36.210 --> 00:04:39.490
and if you were to put
a stick right over there

00:04:39.490 --> 00:04:41.810
on the surface, the
stick would spin.

00:04:41.810 --> 00:04:43.610
It has some spin or
some curl, however

00:04:43.610 --> 00:04:44.610
you want to describe it.

00:04:44.610 --> 00:04:46.542
This right over
here has no curl.

00:04:46.542 --> 00:04:48.000
If you put a stick
right over here,

00:04:48.000 --> 00:04:49.710
it would just flow
with the fluid,

00:04:49.710 --> 00:04:52.130
but the stick itself
would not spin.

00:04:52.130 --> 00:04:54.660
So we got a positive value
for the line integral

00:04:54.660 --> 00:04:55.530
in this situation.

00:04:55.530 --> 00:04:58.130
And we also have, it looks
like, a positive curl.

00:05:02.950 --> 00:05:04.710
Now let's think about this one.

00:05:04.710 --> 00:05:09.270
In this situation, as we go
along this part of our contour,

00:05:09.270 --> 00:05:12.000
our vector field F is going
in the exact same direction,

00:05:12.000 --> 00:05:13.820
so we're going to
get positive values.

00:05:13.820 --> 00:05:18.600
Now as we go uphill,
our vector field F,

00:05:18.600 --> 00:05:20.780
it's also kind of turned
in that direction,

00:05:20.780 --> 00:05:23.090
and so we'll get
more positive values.

00:05:23.090 --> 00:05:25.390
And now as we go
in that direction,

00:05:25.390 --> 00:05:26.797
our vector field
F is still going

00:05:26.797 --> 00:05:28.380
in the direction of
our contour, we're

00:05:28.380 --> 00:05:30.260
going to get more
positive values.

00:05:30.260 --> 00:05:32.557
And as we go down, once
again the vector field F

00:05:32.557 --> 00:05:34.265
is going in the
direction of our contour,

00:05:34.265 --> 00:05:36.660
so we'll get even
more positive values.

00:05:36.660 --> 00:05:40.860
So in this situation, the
value of our line integral of F

00:05:40.860 --> 00:05:43.590
dot dr is even more positive.

00:05:47.490 --> 00:05:50.797
And we see that the
actual vector field along

00:05:50.797 --> 00:05:52.630
the surface-- and
remember, the vector field

00:05:52.630 --> 00:05:55.005
might be doing all sorts of
crazy things off the surface.

00:05:55.005 --> 00:05:57.740
Actually, let me draw that
in that same magenta color.

00:05:57.740 --> 00:06:00.240
It might be doing all sorts of
crazy things off the surface,

00:06:00.240 --> 00:06:01.864
but what we really
care about is what's

00:06:01.864 --> 00:06:02.940
happening on the surface.

00:06:02.940 --> 00:06:05.037
And because this
vector field is,

00:06:05.037 --> 00:06:06.620
I guess you could
say curling, or it's

00:06:06.620 --> 00:06:09.050
spinning along the
surface, it allows

00:06:09.050 --> 00:06:11.540
it to go with the boundary
along all the points,

00:06:11.540 --> 00:06:14.590
and we get a very positive
value for this line integral.

00:06:14.590 --> 00:06:16.170
So we have a higher curl.

00:06:16.170 --> 00:06:22.750
So more curl, it looks like, is
leading to a more positive line

00:06:22.750 --> 00:06:23.507
integral.

00:06:23.507 --> 00:06:25.090
Now let's think about
what's happening

00:06:25.090 --> 00:06:27.530
in this situation
right over here.

00:06:27.530 --> 00:06:30.870
This situation down
here, our vector field

00:06:30.870 --> 00:06:33.130
is going in the same
direction as our path,

00:06:33.130 --> 00:06:35.260
so we're going to
get positive values.

00:06:35.260 --> 00:06:38.330
Just like the first situation,
as we go up the hill like this

00:06:38.330 --> 00:06:40.435
or up the surface like
that, our vector field

00:06:40.435 --> 00:06:42.560
is perpendicular to our
surface, so it's really not

00:06:42.560 --> 00:06:44.770
going to add anything
to our line integral.

00:06:44.770 --> 00:06:48.909
And then as we go along this
top part, this first part

00:06:48.909 --> 00:06:50.950
of the top part right over
here, the vector field

00:06:50.950 --> 00:06:51.800
is going against us.

00:06:51.800 --> 00:06:53.620
So it's negative
right over here.

00:06:53.620 --> 00:06:56.360
We're going in the exact
opposite direction of our path.

00:06:56.360 --> 00:06:57.980
And then, right as
we get to the end,

00:06:57.980 --> 00:06:59.720
the vector field
flips direction,

00:06:59.720 --> 00:07:01.460
and we get a little
bit of positive right

00:07:01.460 --> 00:07:03.168
over here because it's
a little bit of it

00:07:03.168 --> 00:07:05.100
going in the same direction.

00:07:05.100 --> 00:07:06.920
And then we go back downhill.

00:07:06.920 --> 00:07:09.090
When we go back
downhill, it adds nothing

00:07:09.090 --> 00:07:12.225
because our vector field
is going perpendicular

00:07:12.225 --> 00:07:13.420
to our path.

00:07:13.420 --> 00:07:15.930
So the big difference between
this case and the case

00:07:15.930 --> 00:07:19.940
up here is the case
up here-- well,

00:07:19.940 --> 00:07:22.740
actually, I could compare
between these two or these two.

00:07:22.740 --> 00:07:24.740
But the difference between
this one and this one

00:07:24.740 --> 00:07:27.200
is that at least this
part of the vector field

00:07:27.200 --> 00:07:28.500
has switched directions.

00:07:28.500 --> 00:07:30.655
So we get a little
bit of positive value.

00:07:30.655 --> 00:07:32.280
And one way to think
about it, this one

00:07:32.280 --> 00:07:34.738
is going to be less positive
than that, if we take the line

00:07:34.738 --> 00:07:36.662
integral, but more
positive than that.

00:07:36.662 --> 00:07:38.120
And one other way
to think about it

00:07:38.120 --> 00:07:41.420
is, we have a little bit of
curl going on right over here.

00:07:41.420 --> 00:07:44.954
Our vector field switched
directions right around there,

00:07:44.954 --> 00:07:46.370
or I guess you
could say that it's

00:07:46.370 --> 00:07:47.536
spinning right around there.

00:07:47.536 --> 00:07:49.580
So if you put a stick,
if that was in the water,

00:07:49.580 --> 00:07:50.640
it would start spinning.

00:07:50.640 --> 00:07:52.840
But everywhere else,
there isn't a lot of curl.

00:07:52.840 --> 00:07:55.650
So you have some curl, but
it's over a little small region

00:07:55.650 --> 00:07:56.640
of the surface.

00:07:56.640 --> 00:07:59.090
While over here, you
had curl going on

00:07:59.090 --> 00:08:04.320
over a larger portion
of our surface.

00:08:04.320 --> 00:08:06.930
And so up here, you had
a more positive curl,

00:08:06.930 --> 00:08:08.750
more positive line integral.

00:08:08.750 --> 00:08:11.530
Here you have a curl
over less of the surface,

00:08:11.530 --> 00:08:14.420
and you're going to have a
less positive line integral.

00:08:14.420 --> 00:08:16.550
Now let's think about
this one over here.

00:08:16.550 --> 00:08:19.680
This vector field, along the
surface there is some curl.

00:08:19.680 --> 00:08:22.270
There's some curl going
on right over there.

00:08:22.270 --> 00:08:23.980
If you put a stick
in the water, if you

00:08:23.980 --> 00:08:27.180
view that as the velocity of
water, the stick would spin.

00:08:27.180 --> 00:08:29.210
So you have some curl.

00:08:29.210 --> 00:08:31.480
But then it switches
direction again,

00:08:31.480 --> 00:08:33.690
so you have curl there as
well, and it's actually

00:08:33.690 --> 00:08:35.232
curl in the opposite direction.

00:08:35.232 --> 00:08:37.440
So to some degree, if you
were to sum all of this up,

00:08:37.440 --> 00:08:39.000
maybe it would cancel out.

00:08:39.000 --> 00:08:40.570
And it makes sense.

00:08:40.570 --> 00:08:42.684
It makes sense that
it would cancel out

00:08:42.684 --> 00:08:44.310
because when you take
the line integral

00:08:44.310 --> 00:08:47.340
around the whole thing, just
like this first situation,

00:08:47.340 --> 00:08:49.235
it looks like it
will add up to 0.

00:08:49.235 --> 00:08:50.860
Because even though
you have some curl,

00:08:50.860 --> 00:08:52.970
the curls cancel out each other.

00:08:52.970 --> 00:08:56.060
And so when you go to this
top part of the surface,

00:08:56.060 --> 00:08:58.290
the vector field is going
in the exact same direction

00:08:58.290 --> 00:09:00.830
as this bottom part
of the surface.

00:09:00.830 --> 00:09:05.490
So if you were to take your line
integral, the same one that we

00:09:05.490 --> 00:09:07.965
care about, just like
the first situation

00:09:07.965 --> 00:09:11.626
it would be positive down
here, 0 as we go up the curve.

00:09:11.626 --> 00:09:13.500
And then as we go down
here, the vector field

00:09:13.500 --> 00:09:15.780
has switched directions
twice, so it's still

00:09:15.780 --> 00:09:18.820
going against the
path of our contour,

00:09:18.820 --> 00:09:20.470
just like this first situation.

00:09:20.470 --> 00:09:22.310
So it would be
negative up there.

00:09:22.310 --> 00:09:24.340
And then as we go
down, it would be 0.

00:09:24.340 --> 00:09:28.250
So this thing right
over here also

00:09:28.250 --> 00:09:31.340
looks just like the first one
because the curls essentially

00:09:31.340 --> 00:09:32.220
cancel out.

00:09:32.220 --> 00:09:33.880
We switched direction twice.

00:09:33.880 --> 00:09:40.450
So over here, our line
integral might also be 0.

00:09:40.450 --> 00:09:43.040
Now, the whole reason why I
went through this exercise

00:09:43.040 --> 00:09:45.830
is to give you an intuition
of why it might make sense

00:09:45.830 --> 00:09:48.000
that if we have more
curl happening over

00:09:48.000 --> 00:09:50.750
more of this surface,
why that might make

00:09:50.750 --> 00:09:54.120
the value of this line
integral be larger.

00:09:54.120 --> 00:09:55.990
And so, hopefully,
it starts to give you

00:09:55.990 --> 00:09:59.610
an intuition that maybe, just
maybe, the value of this line

00:09:59.610 --> 00:10:10.150
integral, the value of F
dot dr over that contour,

00:10:10.150 --> 00:10:12.706
over this contour that's
going in the counterclockwise

00:10:12.706 --> 00:10:14.580
direction-- we'll talk
more about orientation

00:10:14.580 --> 00:10:17.910
in future videos-- maybe
this is equal to the sum

00:10:17.910 --> 00:10:20.650
of the curls over the surface.

00:10:20.650 --> 00:10:21.920
And so let's think about this.

00:10:21.920 --> 00:10:23.720
It could be a surface integral.

00:10:23.720 --> 00:10:26.950
So we're going to
go over the surface,

00:10:26.950 --> 00:10:29.700
and what we care about
is the curl of F.

00:10:29.700 --> 00:10:33.700
We care about the curl
of F. But we don't just

00:10:33.700 --> 00:10:36.090
care about the curl of F
generally because F might

00:10:36.090 --> 00:10:38.960
be spinning in a direction, in
a way, that's off the surface.

00:10:38.960 --> 00:10:42.120
We care about how much it's
curling on the surface.

00:10:42.120 --> 00:10:47.650
So what we would want to do is
we'd want to take the curl of F

00:10:47.650 --> 00:10:50.920
and dot it with the
normal vector at any point

00:10:50.920 --> 00:10:55.790
to the surface and then multiply
that times the surface itself.

00:10:59.955 --> 00:11:02.080
And this is just to say
that the more surface where

00:11:02.080 --> 00:11:03.869
we have more curling
going on, the more

00:11:03.869 --> 00:11:06.160
that the line integral, the
value of the line integral,

00:11:06.160 --> 00:11:06.659
might be.

00:11:06.659 --> 00:11:09.587
And we saw that when we
compared these three examples.

00:11:09.587 --> 00:11:11.170
And another way of
writing all of this

00:11:11.170 --> 00:11:13.770
is the surface
integral-- let me write

00:11:13.770 --> 00:11:16.160
the surface in that
same brown color--

00:11:16.160 --> 00:11:21.220
the surface integral of the
curl of F, which would just

00:11:21.220 --> 00:11:24.356
be another vector that tells
us how much we are spinning

00:11:24.356 --> 00:11:26.230
generally, but we want
to care how much we're

00:11:26.230 --> 00:11:27.500
spinning along the surface.

00:11:27.500 --> 00:11:29.710
So we're dotting it
with the normal vector.

00:11:29.710 --> 00:11:36.080
Or another way to write this
whole thing is to say dot ds.

00:11:36.080 --> 00:11:40.342
So if you take essentially the
sum across the entire surface

00:11:40.342 --> 00:11:42.050
of how much we're
curling, how much we're

00:11:42.050 --> 00:11:45.740
spinning along that surface,
then maybe, just maybe,

00:11:45.740 --> 00:11:48.500
this will be equal to the
value of the line integral

00:11:48.500 --> 00:11:51.180
as we go around the
boundary of the surface.

00:11:51.180 --> 00:11:54.100
And it actually turns out
that this is the case.

00:11:54.100 --> 00:11:56.420
And obviously, I haven't
proven it to you here,

00:11:56.420 --> 00:11:58.960
but hopefully you have some
intuition why this makes sense.

00:11:58.960 --> 00:12:02.370
And this idea that
this is equal to this

00:12:02.370 --> 00:12:08.970
is called Stokes'
theorem, and we'll

00:12:08.970 --> 00:12:12.130
explore it more in
the next few videos.


MIT 2012 Commencement Address

WEBVTT
Kind: captions
Language: en

00:00:01.460 --> 00:00:06.076
It's now my honor to introduce our Commencement Day speaker,

00:00:06.076 --> 00:00:09.776
the founder of the online education phenomenon

00:00:09.776 --> 00:00:14.614
known as the Khan Academy, Sal Khan.

00:00:14.614 --> 00:00:17.958
Sal earned three degrees from MIT,

00:00:17.958 --> 00:00:21.578
a Bachelor's in mathematics, as well as

00:00:21.578 --> 00:00:23.150
a Bachelor's and a Master's degree

00:00:23.150 --> 00:00:25.993
in electrical engineering and computer science

00:00:25.993 --> 00:00:32.232
rounding out his education with a MBA from Harvard.

00:00:32.232 --> 00:00:36.533
Sal began his career as a hedge fund analyst.

00:00:36.533 --> 00:00:39.414
Within a few years, however, he discovered

00:00:39.414 --> 00:00:44.013
quite by accident what has become his calling.

00:00:44.013 --> 00:00:45.980
Through preparing video tutorials

00:00:45.980 --> 00:00:49.149
to help his young cousin learn math,

00:00:49.149 --> 00:00:53.477
he came to understand his own gifts as a teacher

00:00:53.477 --> 00:00:56.846
and a great gift that he can give to the world

00:00:56.846 --> 00:01:00.852
by creating an online library of 1000s of

00:01:00.852 --> 00:01:04.663
free bite-size videos that would shed light

00:01:04.663 --> 00:01:10.761
on any topic he chose from geometry to the GDP.

00:01:10.761 --> 00:01:14.125
Today, the Khan Academy channel on YouTube

00:01:14.125 --> 00:01:17.342
has more than 330,000 subscribers

00:01:17.342 --> 00:01:21.915
and has delivered more than 156,000,000 lessons

00:01:21.915 --> 00:01:24.518
and has for millions of students,

00:01:24.518 --> 00:01:28.422
replaced bafflement with understanding.

00:01:28.422 --> 00:01:31.959
For launching this revolutionary experiment

00:01:31.959 --> 00:01:35.396
in online learning, in the year 2012

00:01:35.396 --> 00:01:38.766
Time Magazine named Sal Khan as

00:01:38.766 --> 00:01:44.038
one of the 100 most influential people on the planet.

00:01:44.038 --> 00:01:47.274
This is not the first time that Sal has spoken

00:01:47.274 --> 00:01:51.759
from the commencement stage here in Killian Court

00:01:51.759 --> 00:01:55.910
because he also served as president of his class.

00:01:55.910 --> 00:02:02.657
In 1998, addressing his fellow graduates, Sal said,

00:02:02.657 --> 00:02:08.571
"It is no exaggeration to say that we will change the world."

00:02:08.571 --> 00:02:12.731
Having checked this assignment off his to-do list,

00:02:12.731 --> 00:02:14.794
he joins us here today.

00:02:14.794 --> 00:02:18.307
Ladies and gentleman, Sal Khan.

00:02:18.307 --> 00:02:26.976
[Applause]

00:02:26.976 --> 00:02:32.047
Now this is a surreal, a deep honor to be here.

00:02:32.047 --> 00:02:34.863
For a whole set of reasons and as was introduced

00:02:34.863 --> 00:02:38.366
obviously MIT has played quite a big role in my own life.

00:02:38.366 --> 00:02:39.626
But I think a much deeper role than

00:02:39.626 --> 00:02:42.644
many of you all might appreciate.

00:02:42.644 --> 00:02:44.498
Some of you might remember the late nineties,

00:02:44.498 --> 00:02:47.673
kind of when the first internet boom was happening,

00:02:47.673 --> 00:02:52.199
there was a lot of talk about online education and

00:02:52.199 --> 00:02:54.944
most of the conversation back then about online education

00:02:54.944 --> 00:02:58.278
not too different than now, was around how to profit off it,

00:02:58.278 --> 00:02:59.469
how to make money off it.

00:02:59.469 --> 00:03:03.114
Or some institutions were thinking about how to defend against it,

00:03:03.114 --> 00:03:08.529
or at least sit on the sidelines and see how things played out

00:03:08.529 --> 00:03:11.258
Then all of a sudden MIT jumped into the mix in 2001,

00:03:11.258 --> 00:03:14.027
and announced MIT OpenCourseWare.

00:03:14.027 --> 00:03:16.466
That it was going to take knowledge and resources

00:03:16.466 --> 00:03:20.762
that used to be behind the walls of elite institutions and

00:03:20.762 --> 00:03:24.226
not charge for them but give them away for free to the world

00:03:24.226 --> 00:03:28.557
And instead of saying how can we profit off of this, MIT said

00:03:28.557 --> 00:03:30.934
"Well there are some things that are higher than that."

00:03:30.934 --> 00:03:34.740
That if we could empower an unlimited number of people

00:03:34.740 --> 00:03:37.244
for all of time maybe, that is something

00:03:37.244 --> 00:03:39.720
we would be willing to spend resources on.

00:03:39.720 --> 00:03:43.857
When that happened I was just a couple years out of college

00:03:43.857 --> 00:03:45.491
working at a tech company in San Francisco

00:03:45.491 --> 00:03:48.417
I had no idea that my own career adventure

00:03:48.417 --> 00:03:50.597
would lead to what I am doing now.

00:03:50.597 --> 00:03:53.339
But when I read that press release

00:03:53.339 --> 00:03:56.003
I had never been so inspired.

00:03:56.003 --> 00:04:00.415
I had never felt so proud to come from this community.

00:04:00.415 --> 00:04:02.023
And frankly a couple of years later,

00:04:02.023 --> 00:04:04.445
when the videos that I had made for my cousins

00:04:04.445 --> 00:04:05.785
and it became clear that

00:04:05.785 --> 00:04:09.918
people who were not my cousins started watching.

00:04:09.918 --> 00:04:12.553
There was talk of, oh well this could be a business.

00:04:12.553 --> 00:04:15.155
I was in Silicon Valley, this is what it was all about.

00:04:15.155 --> 00:04:19.044
I worked for a hedge fund, a very for profit organization.

00:04:19.044 --> 00:04:21.962
But it was the memory of how I felt the first time

00:04:21.962 --> 00:04:25.158
I read that press release about OpenCourseWare

00:04:25.158 --> 00:04:28.421
and what it had become, that really gave me the clarity

00:04:28.421 --> 00:04:31.072
to understand what Khan Academy could be.

00:04:31.072 --> 00:04:33.967
That it could be this institution that could reach everyone

00:04:33.967 --> 00:04:37.385
and should transcend ideas of profits and businesses.

00:04:37.385 --> 00:04:38.990
And I say this not just to show

00:04:38.990 --> 00:04:40.987
the connection this had for Khan Academy,

00:04:40.987 --> 00:04:43.283
but there is a meta level idea here

00:04:43.283 --> 00:04:47.142
that these days you hear a lot of universities and institutions

00:04:47.142 --> 00:04:49.798
talk about teaching ethics, teching morality

00:04:49.798 --> 00:04:52.643
and instilling that in classes and

00:04:52.643 --> 00:04:55.963
they tell you to read about it, but MIT actually did it.

00:04:55.963 --> 00:04:57.798
MIT actually lived by its actions.

00:04:57.798 --> 00:05:01.676
It actually put principle over profit

00:05:01.676 --> 00:05:03.103
and it is continuing to do it.

00:05:03.103 --> 00:05:06.073
Now with MITx now with EdX in concert with Harvard

00:05:06.073 --> 00:05:07.875
it is continuing to push the envelope.

00:05:07.875 --> 00:05:09.849
And I am just in awe, it feels like

00:05:09.849 --> 00:05:11.474
we are living in a science fiction book of

00:05:11.474 --> 00:05:14.818
what might happen in education in the next few years.

00:05:14.818 --> 00:05:18.819
But my connections to MIT go even deeper.

00:05:18.819 --> 00:05:22.016
Obviously I went here, but my wife also graduated

00:05:22.016 --> 00:05:24.558
from MIT class of 2001.

00:05:24.558 --> 00:05:27.194
The president of Khan Academy and chief operating officer,

00:05:27.194 --> 00:05:31.002
he was my roommate freshman year at Next House.

00:05:31.002 --> 00:05:33.696
Room 343 for the two or three of you

00:05:33.696 --> 00:05:36.136
who might have shared that room at some point.

00:05:36.136 --> 00:05:39.206
One of our board members went to MIT

00:05:39.206 --> 00:05:41.275
and his wife went to MIT.

00:05:41.275 --> 00:05:43.611
I am just starting. That is just a small sample

00:05:43.611 --> 00:05:45.812
of all of the people we know at MIT.

00:05:45.812 --> 00:05:47.280
But maybe even more surprisingly

00:05:47.280 --> 00:05:49.583
the fact that of the people we know from MIT now

00:05:49.583 --> 00:05:55.441
90% of them are married to each other

00:05:55.441 --> 00:05:58.725
and any time you have this type of love

00:05:58.725 --> 00:06:00.961
come from one place

00:06:00.961 --> 00:06:03.363
I think one should introspect

00:06:03.363 --> 00:06:07.482
as romantic as the infinite corridor may be

00:06:07.482 --> 00:06:10.170
I have seen such extreme coupling here that

00:06:10.170 --> 00:06:13.273
I have often suspected that this whole place might just be

00:06:13.273 --> 00:06:21.640
a front for some type of DARPA funded breeding project.

00:06:21.640 --> 00:06:24.585
Someone knows what is going on up here.

00:06:24.585 --> 00:06:27.659
There are simpler explanations,

00:06:27.659 --> 00:06:29.890
and I think the most obvious one

00:06:29.890 --> 00:06:31.725
or at least the most clear to me

00:06:31.725 --> 00:06:33.464
is that the admissions office here

00:06:33.464 --> 00:06:35.829
at both the undergraduate and graduate level

00:06:35.829 --> 00:06:38.398
seems to have a somewhat unhealthy bias

00:06:38.398 --> 00:06:42.369
for only admitting extremely attractive people.

00:06:42.369 --> 00:06:47.280
You're welcome.

00:06:47.280 --> 00:06:51.299
I thought that would go over well.

00:06:51.299 --> 00:06:52.946
I think it goes even deeper than that.

00:06:52.946 --> 00:06:54.648
As long as I can remember,

00:06:54.648 --> 00:06:56.336
to any one who would listen to me,

00:06:56.336 --> 00:06:57.961
I have told people that MIT is

00:06:57.961 --> 00:07:02.280
the closest thing on this planet Earth to Hogwarts,

00:07:02.280 --> 00:07:06.388
to Harry Potter's wizarding school.

00:07:06.388 --> 00:07:10.149
That the ideas and the research and the science that

00:07:10.159 --> 00:07:12.232
percolates behind these walls,

00:07:12.232 --> 00:07:14.334
that's the closest thing to magic in the real world,

00:07:14.334 --> 00:07:17.214
and frankly to people outside of this campus

00:07:17.214 --> 00:07:20.797
looks like magic. The faculty we have here,

00:07:20.797 --> 00:07:24.075
these are the leading wizards of out time!

00:07:24.075 --> 00:07:31.618
The Dumbledore's and Mcgonagall's

00:07:31.618 --> 00:07:33.395
and I guess president Hawkfield

00:07:33.395 --> 00:07:36.932
you would be a Mcgonagall.

00:07:36.932 --> 00:07:42.084
The halls here they have secret passages and tunnels

00:07:42.084 --> 00:07:44.900
and around every corner there are strange and bizzare

00:07:44.900 --> 00:07:46.466
magical objects and creatures,

00:07:46.466 --> 00:07:51.952
some of whom may finish their thesis this decade.

00:07:51.952 --> 00:07:55.548
Maybe a few in the audience.

00:07:55.548 --> 00:07:57.810
When we're in Killian Court, it is almost a shrine.

00:07:57.810 --> 00:08:00.128
When you look at the names around us, you see

00:08:00.128 --> 00:08:05.624
Newton and Darwin and Galileo and Archimedes

00:08:05.624 --> 00:08:08.457
and these are the great wizards of history.

00:08:08.457 --> 00:08:09.850
And they remind us that

00:08:09.850 --> 00:08:13.238
we have inherited an ancient tradition, an ancient art,

00:08:13.238 --> 00:08:16.427
one that for much of history and even today to some degree

00:08:16.427 --> 00:08:19.132
has always been sometimes vilified or suppressed

00:08:19.132 --> 00:08:21.715
by forces of ignorance, but despite that,

00:08:21.715 --> 00:08:23.536
has always shone through and has been

00:08:23.536 --> 00:08:27.774
at least in my mind the prime cause of human advancement.

00:08:27.774 --> 00:08:30.530
The people who come here, the students who come here

00:08:30.530 --> 00:08:32.079
you!

00:08:32.079 --> 00:08:36.018
There are these young people from all over the world

00:08:36.018 --> 00:08:37.751
who are all off the charts in some special way

00:08:37.751 --> 00:08:39.953
for this kind of magic that goes on here.

00:08:39.953 --> 00:08:42.756
Some come from affluent rich educated families.

00:08:42.756 --> 00:08:44.401
Some of you come from poor families

00:08:44.401 --> 00:08:46.193
where you are the first to get a college degree.

00:08:46.193 --> 00:08:47.360
Some come from environments

00:08:47.360 --> 00:08:49.830
where their gifts were really celebrated.

00:08:49.830 --> 00:08:52.632
Some come from environments where most of their lives

00:08:52.632 --> 00:08:55.245
they have had to hide their passions, their gifts

00:08:55.245 --> 00:08:57.237
for fear of looking different.

00:08:57.237 --> 00:09:01.110
But they come here suspecting that this might be a place

00:09:01.110 --> 00:09:02.810
where they can spread their wings,

00:09:02.810 --> 00:09:05.322
where they can explore the world.

00:09:05.322 --> 00:09:07.647
They can finally look with clarity at

00:09:07.647 --> 00:09:09.107
the mysteries of the universe,

00:09:09.107 --> 00:09:11.290
the magic of everything that surrounds us

00:09:11.290 --> 00:09:14.819
and MIT delivers on that.

00:09:14.819 --> 00:09:16.790
It opens our minds to what is possible

00:09:16.790 --> 00:09:19.765
and even more uniquely it pushes us,

00:09:19.765 --> 00:09:24.064
and I do believe that MIT pushes us harder than

00:09:24.064 --> 00:09:27.734
probably any other institution in the world.

00:09:27.734 --> 00:09:30.598
But when you do that you take them to another level

00:09:30.598 --> 00:09:33.373
you take them to truly recognize what they are capable of

00:09:33.373 --> 00:09:35.075
and I think there is another side affect going to why

00:09:35.075 --> 00:09:37.711
there is so much love here.

00:09:37.711 --> 00:09:39.713
When you have people regardless of

00:09:39.713 --> 00:09:41.290
what they look like superficially,

00:09:41.290 --> 00:09:42.358
or where they come from,

00:09:42.358 --> 00:09:44.918
or what type of background they come from,

00:09:44.918 --> 00:09:46.720
but they all have that same core desire

00:09:46.720 --> 00:09:48.117
to understand the universe.

00:09:48.117 --> 00:09:49.789
They all have that same core desire

00:09:49.789 --> 00:09:51.491
to push humanity forward.

00:09:51.491 --> 00:09:53.663
And you bring them all together into a community like this

00:09:53.663 --> 00:09:55.328
and then you push them

00:09:55.328 --> 00:09:57.164
in a frankly very intense environment

00:09:57.164 --> 00:09:58.899
you cry together, you laugh together,

00:09:58.899 --> 00:10:00.600
you procrastinate together,

00:10:00.600 --> 00:10:03.744
you have sleepless nights together,

00:10:03.744 --> 00:10:06.306
you wander the halls together.

00:10:06.306 --> 00:10:09.476
It creates the deepest possible bond.

00:10:09.476 --> 00:10:12.112
It is like people who have fought in wars together.

00:10:12.112 --> 00:10:14.750
They have a shared experience that

00:10:14.750 --> 00:10:18.418
other people might not understand or even comprehend,

00:10:18.418 --> 00:10:21.159
and because of that, the rest of your life

00:10:21.159 --> 00:10:25.058
you will feel connected to other MIT people.

00:10:25.058 --> 00:10:27.290
You will want to be around them,

00:10:27.290 --> 00:10:28.814
and you will seek them out

00:10:28.814 --> 00:10:31.631
and if you are in kind of a mixed group of people

00:10:31.631 --> 00:10:33.867
and you hear someone who is not from MIT

00:10:33.867 --> 00:10:35.869
start talking about how impossible something is,

00:10:35.869 --> 00:10:39.301
how hard something is, or how difficult something is,

00:10:39.301 --> 00:10:41.308
you will seek out the other MIT person in the room

00:10:41.308 --> 00:10:45.749
and catch a glance and you will both share a little smirk.

00:10:45.749 --> 00:10:48.215
And while you are doing that smirk,

00:10:48.215 --> 00:10:51.184
if they are your preferred gender,

00:10:51.184 --> 00:10:55.463
you might see a certain twinkle in their eye

00:10:55.463 --> 00:11:00.512
and realize that you are irresistibly attracted to them.

00:11:00.512 --> 00:11:04.197
So coming here it really does feel like I have come to a family,

00:11:04.197 --> 00:11:06.298
a family that I am deeply connected to and,

00:11:06.298 --> 00:11:08.295
a family that I truly love.

00:11:08.295 --> 00:11:13.393
I hope to just give you some appreciation for the potential

00:11:13.393 --> 00:11:15.208
that you guys are about to leave with.

00:11:15.208 --> 00:11:17.978
And when I said that 14 years ago at my commencement

00:11:17.978 --> 00:11:21.229
it was kind of an idea on paper. But now I have seen what

00:11:21.229 --> 00:11:23.969
my peers in my class and above/below me have done.

00:11:23.969 --> 00:11:27.312
And it has really been nothing short of amazing.

00:11:27.312 --> 00:11:31.101
And with that I want to give you a sense of

00:11:31.101 --> 00:11:32.932
internal strength and happiness

00:11:32.932 --> 00:11:34.327
and not just because I care about you

00:11:34.327 --> 00:11:35.765
and I want you to be a happy person,

00:11:35.765 --> 00:11:37.436
but because I really do believe that

00:11:37.436 --> 00:11:38.597
for you to really reach your potential,

00:11:38.597 --> 00:11:41.198
you have to be centered and have a place to go.

00:11:41.198 --> 00:11:43.637
When things get a little bit tough

00:11:43.637 --> 00:11:45.639
and you should take all of this with a grain of salt.

00:11:45.639 --> 00:11:48.823
I am not a lot older than most of you.

00:11:48.823 --> 00:11:51.444
View me as your older brother or cousin or whatever

00:11:51.444 --> 00:11:53.864
and what I am about to say.

00:11:53.864 --> 00:11:56.311
These are things that I still try to live by,

00:11:56.311 --> 00:11:57.382
but I don't do.

00:11:57.382 --> 00:11:59.853
I am at least as imperfect as any of you guys.

00:11:59.853 --> 00:12:01.388
But there are things that when I am able to do

00:12:01.388 --> 00:12:04.057
I have found work quite well.

00:12:04.057 --> 00:12:07.494
The first is to just be as incredibly and

00:12:07.494 --> 00:12:10.269
maybe delusionally positive as possible.

00:12:10.269 --> 00:12:12.777
It is a very cynical place out there sometimes.

00:12:12.777 --> 00:12:15.235
And that cynicism will eat at your energy

00:12:15.235 --> 00:12:18.271
and will eat at your potential. To fight it,

00:12:18.271 --> 00:12:21.363
you should smile with every atom in your body.

00:12:21.363 --> 00:12:23.476
You should smile first thing in the morning.

00:12:23.476 --> 00:12:24.875
You should even force yourself.

00:12:24.875 --> 00:12:26.175
This is something I acutally do:

00:12:26.175 --> 00:12:28.683
If I am in a bad mood, I actually force a smile.

00:12:28.683 --> 00:12:31.516
It releases things in your brain.

00:12:31.516 --> 00:12:34.791
You should smile with your mouth,

00:12:34.791 --> 00:12:37.023
your eyes, your face, your body and

00:12:37.023 --> 00:12:40.760
evey living and non living thing that you see.

00:12:40.760 --> 00:12:42.495
You should recognize that

00:12:42.495 --> 00:12:44.542
the grass is greener on your side of the fence

00:12:44.542 --> 00:12:46.599
and even in the 1% chance that it is not.

00:12:46.599 --> 00:12:48.969
Just convincing yourself that it is greener

00:12:48.969 --> 00:12:52.027
will be a self fulfilling prophecy.

00:12:52.027 --> 00:12:56.476
If you find yourself in your life in an argument

00:12:56.476 --> 00:12:58.745
with someone whom you respect,

00:12:58.745 --> 00:13:01.201
whom you care deeply about,

00:13:01.201 --> 00:13:04.143
that is a time, and it is not an easy thing to do,

00:13:04.143 --> 00:13:08.319
but try to surrender your ego to the shared identity of

00:13:08.319 --> 00:13:11.558
the relationship that you have with that person.

00:13:11.558 --> 00:13:13.760
If possible, do the exact opposite of

00:13:13.760 --> 00:13:17.197
what your pride and ego tell you in the heat of a battle.

00:13:17.197 --> 00:13:21.034
If you have the strength to do it,

00:13:21.034 --> 00:13:24.550
right when you are about to try to get the last word

00:13:24.550 --> 00:13:27.540
and you are about to say that kind of vindictive

00:13:27.540 --> 00:13:30.944
or below the belt thing, just pause, don't do anything

00:13:30.944 --> 00:13:35.749
and try to just give them a super-mega-hug.

00:13:35.749 --> 00:13:39.759
Recognize that material loss or gain, we'll all have it.

00:13:39.759 --> 00:13:41.997
But when it happens, be upset a little bit

00:13:41.997 --> 00:13:44.057
or be happy a little bit but keep it all in perspective.

00:13:44.057 --> 00:13:48.165
They are all silly relative to the things that matter,

00:13:48.165 --> 00:13:50.296
your health, and your relationships.

00:13:50.296 --> 00:13:52.032
As much as possible try to make people feel that

00:13:52.032 --> 00:13:55.535
you are listening to them. And I have a secret here.

00:13:55.535 --> 00:13:57.404
The best way to make people feel that

00:13:57.404 --> 00:14:02.770
you are listening to them, is to listen to them.

00:14:02.770 --> 00:14:04.442
When you are stressed and

00:14:04.442 --> 00:14:06.780
there will be times of stress in your life,

00:14:06.780 --> 00:14:09.637
just look up at the night sky and imagine

00:14:09.637 --> 00:14:12.828
the scale of the universe, the age of the universe,

00:14:12.828 --> 00:14:14.554
the distance to the next star,

00:14:14.554 --> 00:14:18.560
the other sentient creatures on other M-class planets,

00:14:18.560 --> 00:14:21.027
who are also looking at the night sky,

00:14:21.027 --> 00:14:22.855
and think about that shared experience

00:14:22.855 --> 00:14:24.388
the civilizations that have come by,

00:14:24.388 --> 00:14:27.499
it will put your problems into a little bit of perspective.

00:14:27.499 --> 00:14:29.202
If you can take a walk through the woods,

00:14:29.202 --> 00:14:32.422
forget your name, your identity, your ambitions,

00:14:32.422 --> 00:14:33.873
and just for a little bit, for a moment

00:14:33.873 --> 00:14:34.741
realize what you are.

00:14:34.741 --> 00:14:36.342
Just another animal walking through the wood,

00:14:36.342 --> 00:14:37.577
just another mammal.

00:14:37.577 --> 00:14:40.713
Recognize that you are not quite sure why you are here,

00:14:40.713 --> 00:14:43.817
but you enjoy the ride and you love the mystery of everything

00:14:43.817 --> 00:14:45.588
that surrounds you, and you want to explore it more.

00:14:45.588 --> 00:14:47.260
As much as possible

00:14:47.260 --> 00:14:48.755
and this is something that is difficult and

00:14:48.755 --> 00:14:50.723
not something that I do anywhere near perfectly

00:14:50.723 --> 00:14:55.456
but I try, is try to build true empathy for everyone.

00:14:55.456 --> 00:14:57.337
And a thought experiment that I often do

00:14:57.337 --> 00:14:59.009
to get me into that frame of mind,

00:14:59.009 --> 00:15:00.681
and it is just a though experiment

00:15:00.681 --> 00:15:02.735
regardless of your actual spiritual beliefs.

00:15:02.735 --> 00:15:06.473
Imagine that time isn't a one way street.

00:15:06.473 --> 00:15:09.109
That you can go back and etc. etc.

00:15:09.109 --> 00:15:12.278
In your next life you could be reincarnated

00:15:12.278 --> 00:15:14.844
you could go back in time and be reincarnated as anyone.

00:15:14.844 --> 00:15:18.988
You literally, because of that, could be everybody in this room.

00:15:18.988 --> 00:15:21.754
And with that mental framework in place,

00:15:21.754 --> 00:15:25.277
imagine that in your next life you could be that person

00:15:25.277 --> 00:15:27.477
you could go back in time and be that person that

00:15:27.477 --> 00:15:29.799
you are having the conversation with right now.

00:15:29.799 --> 00:15:31.935
Maybe that you are arguing with right now or maybe

00:15:31.935 --> 00:15:34.768
the person that you are passing judgement on right now.

00:15:34.768 --> 00:15:37.275
And if that is the case, then in your next life

00:15:37.275 --> 00:15:38.771
you will have to put up with

00:15:38.771 --> 00:15:42.976
the current self righteous version of yourself.

00:15:42.976 --> 00:15:45.478
So I want to just bring it all together

00:15:45.478 --> 00:15:49.559
with another thought experiment that I like to do,

00:15:49.559 --> 00:15:53.019
that helps me focus where I want to put my energies.

00:15:53.019 --> 00:15:58.941
Imagine yourself in 50 years. You are in your early 70's.

00:15:58.941 --> 00:16:03.373
You are near the end of your career.

00:16:03.373 --> 00:16:05.441
Hey! Oh! Oh! Very good! [Laughter]

00:16:05.441 --> 00:16:07.200
We have a few models here

00:16:07.200 --> 00:16:10.410
if you have trouble visualizing that.

00:16:10.410 --> 00:16:15.472
But imagine at that stage you are on your couch in 2062

00:16:15.472 --> 00:16:17.516
and you have just finished watching

00:16:17.516 --> 00:16:22.953
the state of the union holograph by president Kardashian

00:16:22.953 --> 00:16:25.712
and you start to reflect on your life.

00:16:25.712 --> 00:16:28.452
You start to think of all the success you have had

00:16:28.452 --> 00:16:30.938
all the career sucesses, family sucesses,

00:16:30.938 --> 00:16:32.458
the great memories that you have had.

00:16:32.458 --> 00:16:33.927
But then you also start to think about the things

00:16:33.927 --> 00:16:36.533
you wish you did just a little bit different.

00:16:36.533 --> 00:16:40.201
Your regrets, and I can imagine what they might be.

00:16:40.201 --> 00:16:42.709
You'll wish you spent more time with your children.

00:16:42.709 --> 00:16:45.238
You'll wish you had told your spouse

00:16:45.238 --> 00:16:47.051
that you loved them more frequently.

00:16:47.051 --> 00:16:48.607
You'll wish that you had spend more time and

00:16:48.607 --> 00:16:50.543
told your parents how much you appreciated them,

00:16:50.543 --> 00:16:52.812
before they passed away.

00:16:52.812 --> 00:16:56.571
And just while that is happening, a genie appears.

00:16:56.571 --> 00:16:58.313
And the genie says,

00:16:58.313 --> 00:17:00.520
"Well I have been listening in to your regrets."

00:17:00.520 --> 00:17:02.214
"You seem like a good person."

00:17:02.214 --> 00:17:05.792
"I am willing to give you a 2nd chance, If you are open to it."

00:17:05.792 --> 00:17:07.660
And so you say "sure".

00:17:07.660 --> 00:17:10.330
And the genie snaps his fingers and you blink your eyes

00:17:10.330 --> 00:17:12.570
and when you open your eyes you find yourself right there

00:17:12.570 --> 00:17:16.668
right where you are right now June 8th 2012, in Killian Court.

00:17:16.668 --> 00:17:19.439
Some crazy guy is giving a commencement speech.

00:17:19.439 --> 00:17:20.836
And you say, Oh my God,

00:17:20.836 --> 00:17:25.295
I am in my 20-something fit and pain free body again,

00:17:25.295 --> 00:17:29.299
I am around my peers again.

00:17:29.299 --> 00:17:31.451
The genie was serious. I have a second chance.

00:17:31.451 --> 00:17:33.213
I can have all the successes,

00:17:33.213 --> 00:17:36.394
all of the adventures I had the first time around,

00:17:36.394 --> 00:17:38.391
but now I can optimize things.

00:17:38.391 --> 00:17:39.659
Now when I see my classmates and

00:17:39.659 --> 00:17:41.327
I give them that hug at commencement,

00:17:41.327 --> 00:17:43.197
I can hug them a little bit harder.

00:17:43.197 --> 00:17:45.398
I can show them how much I care about them.

00:17:45.398 --> 00:17:47.400
Now that my parents are back,

00:17:47.400 --> 00:17:49.836
I can finally tell them how much I appreciate them.

00:17:49.836 --> 00:17:52.973
I can finally give them more hugs, more time.

00:17:52.973 --> 00:17:54.087
I can do everything that

00:17:54.087 --> 00:17:56.876
I can laugh more, I can sing more, I can dance more.

00:17:56.876 --> 00:17:58.511
I can be more of a source of positivity

00:17:58.511 --> 00:18:01.518
for people around me and empower more people.

00:18:01.518 --> 00:18:03.507
And so here I am

00:18:03.507 --> 00:18:06.986
truly honored to be your commencement speaker.

00:18:06.986 --> 00:18:09.900
Just in awe of the potential that is here,

00:18:09.900 --> 00:18:12.325
the potential in this time that we are in,

00:18:12.325 --> 00:18:15.461
a time when the revolutions, the positive revolutions,

00:18:15.461 --> 00:18:18.728
are not going to be caused by generals and politicians.

00:18:18.728 --> 00:18:20.733
They are going to be caused by innovators like you

00:18:20.733 --> 00:18:24.370
and in this time to see you the wizards of tomorrow.

00:18:24.370 --> 00:18:25.272
I am just excited by what

00:18:25.272 --> 00:18:27.340
you are going to do with your second pass.

00:18:27.340 --> 00:18:29.212
Thank You.

00:18:29.242 --> 00:18:41.329
[Applause]


Vector representation of a surface integral

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:01.950
In the last video,
we figured out

00:00:01.950 --> 00:00:05.590
how to construct a unit
normal vector to a surface.

00:00:05.590 --> 00:00:08.800
And so now we can use that
back in our original surface

00:00:08.800 --> 00:00:10.750
integral to try to
simplify a little bit,

00:00:10.750 --> 00:00:13.320
or at least give us a clue how
we can calculate these things.

00:00:13.320 --> 00:00:14.820
And also, think
about different ways

00:00:14.820 --> 00:00:17.950
to represent this type
of a surface integral.

00:00:17.950 --> 00:00:20.710
So if we just substitute what
we came up as our normal vector,

00:00:20.710 --> 00:00:22.190
our unit normal
vector right here,

00:00:22.190 --> 00:00:26.120
we will get-- so
once again, it's

00:00:26.120 --> 00:00:31.540
the surface integral of F dot.

00:00:31.540 --> 00:00:33.499
And F dot all of this
business right over here.

00:00:33.499 --> 00:00:35.123
And I'm going to
write it all in white,

00:00:35.123 --> 00:00:36.900
just so it doesn't
take me too much time.

00:00:36.900 --> 00:00:38.930
So the partial of
r with respect to u

00:00:38.930 --> 00:00:41.940
crossed with the partial
of r with respect

00:00:41.940 --> 00:00:46.280
to v over the magnitude
of the same thing, partial

00:00:46.280 --> 00:00:49.560
of r with respect to u
crossed with the partial of r

00:00:49.560 --> 00:00:51.770
with respect to v.

00:00:51.770 --> 00:00:53.709
And now, we've
played with ds a lot.

00:00:53.709 --> 00:00:55.375
We know that the other
way to write ds--

00:00:55.375 --> 00:00:56.874
and I gave the
intuition, hopefully,

00:00:56.874 --> 00:00:58.840
for that several videos
ago when we first

00:00:58.840 --> 00:01:01.120
explored what a surface
integral was all about.

00:01:01.120 --> 00:01:05.770
We know that ds is--
it can be represented

00:01:05.770 --> 00:01:08.420
as the magnitude of the
partial of r with respect

00:01:08.420 --> 00:01:13.501
to u crossed with the partial
of r with respect to v du dv.

00:01:13.501 --> 00:01:17.020
And Obviously, the du dv, it
could be written as dv du.

00:01:17.020 --> 00:01:20.410
You could write it as da,
a little chunk of area

00:01:20.410 --> 00:01:23.310
and the uv plane or
in the uv domain.

00:01:23.310 --> 00:01:26.010
And actually, since now this
integral's in terms of uv,

00:01:26.010 --> 00:01:27.760
we're no longer taking
a surface integral.

00:01:27.760 --> 00:01:30.880
We're now taking a double
integral over the uv domain.

00:01:30.880 --> 00:01:33.150
So you could say kind
of a region in uv.

00:01:33.150 --> 00:01:38.100
So I'll say R to say that's
it's a region in the uv plane

00:01:38.100 --> 00:01:39.550
that we're now thinking about.

00:01:39.550 --> 00:01:41.770
But there's probably a
huge-- or there should be,

00:01:41.770 --> 00:01:43.720
or I'm guessing there's a
huge simplification that's

00:01:43.720 --> 00:01:44.928
popping out at you right now.

00:01:44.928 --> 00:01:47.130
We're dividing by the
magnitude of the cross product

00:01:47.130 --> 00:01:48.588
of these two vectors
and then we're

00:01:48.588 --> 00:01:52.040
multiplying by the magnitude of
the cross product of these two

00:01:52.040 --> 00:01:52.695
vectors.

00:01:52.695 --> 00:01:54.070
Those are just
scalar quantities.

00:01:54.070 --> 00:01:56.140
You divide by something
and multiply by something.

00:01:56.140 --> 00:01:58.598
Well, that's just the same
thing as multiplying or dividing

00:01:58.598 --> 00:01:59.390
by 1.

00:01:59.390 --> 00:02:01.420
So these two
characters cancel out,

00:02:01.420 --> 00:02:05.930
and our integral simplifies
to the double integral

00:02:05.930 --> 00:02:09.250
over that region, the
corresponding region in the uv

00:02:09.250 --> 00:02:14.200
plane, of F-- of
our vector field F

00:02:14.200 --> 00:02:16.880
dotted with this cross product.

00:02:16.880 --> 00:02:19.130
This is going to give us
a vector right over here.

00:02:19.130 --> 00:02:20.740
That's going to
give us a vector.

00:02:20.740 --> 00:02:22.282
It gives us actually
a normal vector.

00:02:22.282 --> 00:02:24.031
And then when you
divide by its magnitude,

00:02:24.031 --> 00:02:25.520
it gives you a
unit normal vector.

00:02:25.520 --> 00:02:28.780
So this, you're going to
take the dot product of F

00:02:28.780 --> 00:02:32.700
with r, the partial
of r with respect

00:02:32.700 --> 00:02:41.170
to u crossed with the partial
of r with respect to v du dv.

00:02:41.170 --> 00:02:45.550
Let me scroll over to the
right a little bit, du dv.

00:02:45.550 --> 00:02:47.590
And we'll see in the
few videos from now

00:02:47.590 --> 00:02:50.270
that this is essentially how we
go about actually calculating

00:02:50.270 --> 00:02:50.860
these things.

00:02:50.860 --> 00:02:52.390
If you have a
parameterization, you

00:02:52.390 --> 00:02:54.640
can then get everything in
terms of a double integral,

00:02:54.640 --> 00:02:56.509
in terms of uv this way.

00:02:56.509 --> 00:02:58.800
Now, the last thing I want
to do is explore another way

00:02:58.800 --> 00:03:01.050
that you'll see a surface
integral like this written.

00:03:01.050 --> 00:03:03.379
It all comes from,
really, writing this part

00:03:03.379 --> 00:03:04.170
in a different way.

00:03:04.170 --> 00:03:05.630
But it hopefully gives you
a little bit more intuition

00:03:05.630 --> 00:03:07.700
of what this thing
is even saying.

00:03:07.700 --> 00:03:09.240
So I'm just going to rewrite.

00:03:09.240 --> 00:03:12.570
I'm going to rewrite this
chunk right over here.

00:03:12.570 --> 00:03:14.122
I'm just going to
rewrite that chunk.

00:03:14.122 --> 00:03:16.580
And I'm going to use slightly
different notation because it

00:03:16.580 --> 00:03:19.030
will hopefully help make
a little bit more sense.

00:03:19.030 --> 00:03:21.100
So the partial of r
with respect to u I

00:03:21.100 --> 00:03:25.930
can write as the partial
of r with respect to u.

00:03:25.930 --> 00:03:27.600
And we're taking
the cross product.

00:03:27.600 --> 00:03:30.300
Let me make my u's a little bit
more u-like so we confuse them

00:03:30.300 --> 00:03:31.190
with v's.

00:03:31.190 --> 00:03:32.940
And we're taking the
cross product of that

00:03:32.940 --> 00:03:36.400
with the partial
of r with respect

00:03:36.400 --> 00:03:40.130
to v. So very small
changes in our vector--

00:03:40.130 --> 00:03:41.930
in our parameterization
right here,

00:03:41.930 --> 00:03:44.100
our position vector given
a small change in v. Very

00:03:44.100 --> 00:03:47.850
small changes in the vector
given a small change in u.

00:03:47.850 --> 00:03:50.670
And then we're multiplying
that times du dv.

00:03:50.670 --> 00:03:56.240
We're multiplying
that times du dv.

00:03:56.240 --> 00:03:59.750
Now, du and dv are
just scalar quantities.

00:03:59.750 --> 00:04:01.290
They're infinitesimally small.

00:04:01.290 --> 00:04:03.084
But for the sake
of this argument,

00:04:03.084 --> 00:04:04.750
you can just view--
they're not vectors,

00:04:04.750 --> 00:04:06.500
they're just scalar quantities.

00:04:06.500 --> 00:04:09.730
And so you can
essentially include them--

00:04:09.730 --> 00:04:11.530
if you have the cross product.

00:04:11.530 --> 00:04:18.470
If you have a cross b times some
scalar value-- I don't know, x,

00:04:18.470 --> 00:04:23.610
you could rewrite this
as x times a cross b,

00:04:23.610 --> 00:04:28.530
or you could write
this as a cross x times

00:04:28.530 --> 00:04:30.700
b, because x is
just a scalar value.

00:04:30.700 --> 00:04:31.540
It's just a number.

00:04:31.540 --> 00:04:33.206
So we could do the
same thing over here.

00:04:33.206 --> 00:04:35.650
We can rewrite all
of this business as--

00:04:35.650 --> 00:04:39.630
and I'm going to group the du
where we have the partial--

00:04:39.630 --> 00:04:41.300
or with respect to u
in the denominator.

00:04:41.300 --> 00:04:42.966
And I'll do the same
thing with the v's.

00:04:42.966 --> 00:04:48.430
And so you will get the
partial of r with respect

00:04:48.430 --> 00:04:52.440
to u times du,
times that scalar.

00:04:52.440 --> 00:04:54.240
So that'll give us a vector.

00:04:54.240 --> 00:04:55.750
And we're going to cross that.

00:04:55.750 --> 00:04:58.530
We're going to cross that with
the partial of r with respect

00:04:58.530 --> 00:05:06.650
to v dv.

00:05:06.650 --> 00:05:08.814
Now, these might
look notationally

00:05:08.814 --> 00:05:10.480
like two different
things, but that just

00:05:10.480 --> 00:05:12.470
comes from the
necessity of when we

00:05:12.470 --> 00:05:16.780
take partial derivatives to say,
oh, no, this vector function

00:05:16.780 --> 00:05:18.887
is defined-- it's a function
of multiple variables

00:05:18.887 --> 00:05:20.720
and this is taking a
derivative with respect

00:05:20.720 --> 00:05:22.010
to only one of them.

00:05:22.010 --> 00:05:23.510
So this is, how
much does our vector

00:05:23.510 --> 00:05:26.330
change when you have a
very small change in u?

00:05:26.330 --> 00:05:28.880
But this is also an
infinitesimally small change

00:05:28.880 --> 00:05:31.950
in u over here, we're just using
slightly different notation.

00:05:31.950 --> 00:05:33.280
So for the sake of--
and this is a little bit

00:05:33.280 --> 00:05:35.280
loosey-goosey mathematics,
but it will hopefully

00:05:35.280 --> 00:05:37.860
give you the intuition
for why this thing could

00:05:37.860 --> 00:05:39.790
be written in a different way.

00:05:39.790 --> 00:05:42.290
These are essentially
the same quantity.

00:05:42.290 --> 00:05:44.770
So if you divide by something
and multiply by something,

00:05:44.770 --> 00:05:46.072
you can cancel them out.

00:05:46.072 --> 00:05:48.280
If you divide by something
and multiply by something,

00:05:48.280 --> 00:05:49.840
you can cancel them out.

00:05:49.840 --> 00:05:52.070
And all you're left
with then-- all

00:05:52.070 --> 00:05:55.670
you're left with is
the differential of r.

00:05:55.670 --> 00:05:57.080
And since we lost
the information

00:05:57.080 --> 00:05:58.496
that it's in the
u-direction, I'll

00:05:58.496 --> 00:06:00.830
write here, the differential
of r in the u-direction.

00:06:00.830 --> 00:06:02.891
I don't want to get
the notation confused.

00:06:02.891 --> 00:06:04.140
This is just the differential.

00:06:04.140 --> 00:06:06.000
This is just how much r changed.

00:06:06.000 --> 00:06:08.820
This is not the partial
derivative of r with respect

00:06:08.820 --> 00:06:09.455
to u.

00:06:09.455 --> 00:06:11.350
This right over here
is, how much does r

00:06:11.350 --> 00:06:15.320
change given per unit change,
per small change in u?

00:06:15.320 --> 00:06:19.190
This just says a differential
in the direction of-- I

00:06:19.190 --> 00:06:22.530
guess as u changes, this is
how much that infinitely small

00:06:22.530 --> 00:06:24.060
change that just r changes.

00:06:24.060 --> 00:06:27.460
This isn't change in r with
respect to change in u.

00:06:27.460 --> 00:06:28.710
And we're going to cross that.

00:06:28.710 --> 00:06:31.775
Now, we're going to cross
that with the partial of r,

00:06:31.775 --> 00:06:34.900
the partial of r
in the v-direction.

00:06:34.900 --> 00:06:37.280
Now, this right over here,
let's just conceptualize this.

00:06:37.280 --> 00:06:39.920
And this goes back to
our original visions

00:06:39.920 --> 00:06:42.330
of what a surface
integral was all about.

00:06:42.330 --> 00:06:44.314
If we're on a surface--
and I'll draw surface.

00:06:44.314 --> 00:06:45.480
Let me draw another surface.

00:06:45.480 --> 00:06:47.970
I won't use the one that
I had already drawn on.

00:06:47.970 --> 00:06:51.077
If we draw a surface, and for
a very small change in u--

00:06:51.077 --> 00:06:52.910
and we're not going to
think about the rate.

00:06:52.910 --> 00:06:56.220
We're just thinking about
kind of the change in r.

00:06:56.220 --> 00:06:57.790
You're going in that direction.

00:06:57.790 --> 00:07:01.220
So if that thing
looks like this,

00:07:01.220 --> 00:07:05.310
this is actually kind of a
distance moved on the surface.

00:07:05.310 --> 00:07:08.060
Because remember, this
isn't the derivative.

00:07:08.060 --> 00:07:09.350
This is the differential.

00:07:09.350 --> 00:07:11.840
So it's just a small
change along the surface,

00:07:11.840 --> 00:07:13.510
that's that over there.

00:07:13.510 --> 00:07:16.710
And that this is a small
change when you change v. So

00:07:16.710 --> 00:07:19.050
it's also a change
along the surface.

00:07:19.050 --> 00:07:21.310
When you take the cross
product of these two things,

00:07:21.310 --> 00:07:23.430
you get a vector
that is orthogonal.

00:07:23.430 --> 00:07:26.570
You get a vector that is
normal to the surface.

00:07:26.570 --> 00:07:33.740
So it is normal to the
surface and its magnitude--

00:07:33.740 --> 00:07:36.340
and we saw this when we first
learned about cross products.

00:07:36.340 --> 00:07:40.080
Its magnitude is
equal to the area that

00:07:40.080 --> 00:07:42.680
is defined by these two vectors.

00:07:42.680 --> 00:07:49.302
So its magnitude
is equal to area.

00:07:49.302 --> 00:07:50.760
So in a lot of
ways, you can really

00:07:50.760 --> 00:07:52.420
think of it-- you
really could think

00:07:52.420 --> 00:07:57.020
of it as a unit normal
vector times ds.

00:07:57.020 --> 00:07:59.960
And so the way that we would,
I guess notationally do this,

00:07:59.960 --> 00:08:02.470
is we can call this--
because this is kind of a ds,

00:08:02.470 --> 00:08:04.045
but it's a vector
version of the ds.

00:08:04.045 --> 00:08:06.760
Over here, this is just
an area right over here.

00:08:06.760 --> 00:08:08.830
This is just a scalar value.

00:08:08.830 --> 00:08:12.250
But now, we have a vector
that points normally

00:08:12.250 --> 00:08:14.930
from the surface,
but its magnitude

00:08:14.930 --> 00:08:17.300
is the same thing as
that ds that we were just

00:08:17.300 --> 00:08:18.340
talking about.

00:08:18.340 --> 00:08:22.070
So we can call this thing right
over here, we can call this ds.

00:08:22.070 --> 00:08:25.140
And the key difference here
is this is a vector now.

00:08:25.140 --> 00:08:28.240
So we'll call it ds with
a little vector over it

00:08:28.240 --> 00:08:29.570
to know that this thing.

00:08:29.570 --> 00:08:33.200
This isn't the scalar ds that
is just concerned with the area.

00:08:33.200 --> 00:08:34.970
But when you view
things this way,

00:08:34.970 --> 00:08:39.289
we just saw that this entire
thing simplifies to ds.

00:08:39.289 --> 00:08:43.419
Then our whole surface
integral can be rewritten.

00:08:43.419 --> 00:08:45.200
Instead of writing
it like this, we

00:08:45.200 --> 00:08:50.480
can write it as the
integral or the surface

00:08:50.480 --> 00:08:54.050
integral-- those integral
signs were too fancy.

00:08:54.050 --> 00:08:57.794
The surface integral of F dot.

00:08:57.794 --> 00:09:00.750
And instead of saying a
normal vector times the scalar

00:09:00.750 --> 00:09:03.040
quantity, that little chunk
of area on the surface,

00:09:03.040 --> 00:09:07.731
we can now just call that
the vector differential ds.

00:09:07.731 --> 00:09:10.230
And I want to make it clear,
these are two different things.

00:09:10.230 --> 00:09:12.280
This is a vector.

00:09:12.280 --> 00:09:14.130
This is essentially
what we're calling it.

00:09:14.130 --> 00:09:16.655
This right over here is a
scalar times a normal vector.

00:09:16.655 --> 00:09:18.030
So these are three
different ways

00:09:18.030 --> 00:09:19.800
of really representing
the same thing.

00:09:19.800 --> 00:09:22.175
And in different contexts,
you will see different things,

00:09:22.175 --> 00:09:25.110
depending on what the author of
whoever's trying to communicate

00:09:25.110 --> 00:09:26.820
is trying to communicate.

00:09:26.820 --> 00:09:29.990
This right over here is the one
that we'll use most frequently

00:09:29.990 --> 00:09:34.192
as we actually try to calculate
these surface integrals.