3D divergence theorem intuition

WEBVTT
Kind: captions
Language: en

00:00:00.499 --> 00:00:02.750
We've already explored a
two-dimensional version

00:00:02.750 --> 00:00:03.940
of the divergence theorem.

00:00:03.940 --> 00:00:09.880
If I have some region-- so this
is my region right over here.

00:00:09.880 --> 00:00:13.550
We'll call it R. And let's
call the boundary of my region,

00:00:13.550 --> 00:00:16.860
let's call that C. And
if I have some vector

00:00:16.860 --> 00:00:20.710
field in this region, so let me
draw a vector field like this.

00:00:20.710 --> 00:00:26.130
If I draw a vector
field just like that,

00:00:26.130 --> 00:00:28.320
our two-dimensional
divergence theorem,

00:00:28.320 --> 00:00:31.220
which we really derived
from Green's theorem,

00:00:31.220 --> 00:00:37.570
told us that the flux across
our boundary of this region--

00:00:37.570 --> 00:00:38.810
so let me write that out.

00:00:38.810 --> 00:00:41.090
The flux across the
boundary, so the flux

00:00:41.090 --> 00:00:43.980
is essentially going
to be the vector field.

00:00:43.980 --> 00:00:46.970
It's going to be
our vector field F

00:00:46.970 --> 00:00:50.040
dotted with the normal
outward-facing vector.

00:00:50.040 --> 00:00:52.080
So the normal
vector at any point

00:00:52.080 --> 00:00:55.230
is this outward-facing
vector, So our vector field,

00:00:55.230 --> 00:00:59.745
dotted with the normal-facing
vector at our boundary times

00:00:59.745 --> 00:01:01.120
our little chunk
of the boundary.

00:01:06.630 --> 00:01:10.880
If we were to sum them all
up over the entire boundary--

00:01:10.880 --> 00:01:15.960
let me write that a
little bit neater--

00:01:15.960 --> 00:01:19.300
that's the same thing as summing
up over the entire region.

00:01:19.300 --> 00:01:21.740
So let's sum up over
the entire region.

00:01:21.740 --> 00:01:25.490
So summing up over this entire
region, each little chunk

00:01:25.490 --> 00:01:31.730
of area, dA-- we could call
that dx dy if we wanted to

00:01:31.730 --> 00:01:33.410
if we're dealing
in the xy domain

00:01:33.410 --> 00:01:36.667
right over here, but each
little chunk of area times

00:01:36.667 --> 00:01:38.500
the divergence of F,
which is really saying,

00:01:38.500 --> 00:01:42.080
how much is that vector
field pulling apart?

00:01:42.080 --> 00:01:47.910
So it's times the
divergence of F.

00:01:47.910 --> 00:01:49.540
And hopefully, it
made intuitive sense.

00:01:49.540 --> 00:01:51.780
The way that I drew this
vector field right over here,

00:01:51.780 --> 00:01:53.350
you see everything's
kind of coming out.

00:01:53.350 --> 00:01:55.308
You could almost call
this a source right here,

00:01:55.308 --> 00:01:57.990
where the vector field seems
like it's popping out of there.

00:01:57.990 --> 00:02:00.770
This has positive
divergence right over here.

00:02:00.770 --> 00:02:02.670
And so because of
this, you actually

00:02:02.670 --> 00:02:05.177
see that the vector
field at the boundary

00:02:05.177 --> 00:02:07.510
is actually going in the
direction of the normal vector,

00:02:07.510 --> 00:02:09.635
pretty close to the direction
of the normal vector,

00:02:09.635 --> 00:02:10.419
so it makes sense.

00:02:10.419 --> 00:02:12.860
You have positive
divergence, and this

00:02:12.860 --> 00:02:14.310
is going to be a positive value.

00:02:14.310 --> 00:02:15.950
The vector field is
going, for the most part,

00:02:15.950 --> 00:02:17.533
in the direction of
the normal vector.

00:02:17.533 --> 00:02:20.130
So the larger this is,
the larger that is.

00:02:20.130 --> 00:02:21.770
So hopefully, some
intuitive sense.

00:02:21.770 --> 00:02:26.520
If you had another
vector field-- so

00:02:26.520 --> 00:02:29.700
let me draw another region--
that looked like this, so

00:02:29.700 --> 00:02:31.270
I could draw a
couple situations.

00:02:31.270 --> 00:02:33.560
So one where there's
very limited divergence,

00:02:33.560 --> 00:02:35.070
maybe it's just a constant.

00:02:35.070 --> 00:02:37.500
The vector field
doesn't really change

00:02:37.500 --> 00:02:41.860
as you go in any
given direction.

00:02:41.860 --> 00:02:45.150
Over here you'll
get positive fluxes.

00:02:45.150 --> 00:02:47.060
I don't know what the
plural of flux is.

00:02:47.060 --> 00:02:50.040
You'll get positive fluxes,
because the vector field seems

00:02:50.040 --> 00:02:53.440
to be going in roughly the same
direction as our normal vector.

00:02:53.440 --> 00:02:55.650
But here, you'll
get a negative flux.

00:02:55.650 --> 00:02:57.230
So stuff is coming in here.

00:02:57.230 --> 00:02:59.370
If you imagine your vector
field is essentially

00:02:59.370 --> 00:03:01.500
some type of mass
density times volume,

00:03:01.500 --> 00:03:03.480
and we've thought
about that before,

00:03:03.480 --> 00:03:05.650
this is showing how
much stuff is coming in,

00:03:05.650 --> 00:03:07.330
and then stuff is coming out.

00:03:07.330 --> 00:03:11.690
So your net flux will
be close to zero.

00:03:11.690 --> 00:03:13.791
Stuff is coming in, and
stuff is coming out.

00:03:13.791 --> 00:03:15.290
Here, you're just
saying, hey, stuff

00:03:15.290 --> 00:03:18.281
is constantly coming
out of this surface.

00:03:18.281 --> 00:03:19.780
So hopefully, this
gives you a sense

00:03:19.780 --> 00:03:21.670
that here you have
very low divergence,

00:03:21.670 --> 00:03:26.010
and you would have a low
flux, total aggregate flux,

00:03:26.010 --> 00:03:27.200
going through your boundary.

00:03:27.200 --> 00:03:28.866
Here you have a high
divergence, and you

00:03:28.866 --> 00:03:30.820
would have a high
aggregate flux.

00:03:30.820 --> 00:03:33.230
I could draw another situation.

00:03:33.230 --> 00:03:36.550
So this is my region
R. And let's say

00:03:36.550 --> 00:03:38.110
that we have
negative divergence,

00:03:38.110 --> 00:03:39.809
or we could even
call it convergence.

00:03:39.809 --> 00:03:41.600
Convergence isn't an
actual technical term,

00:03:41.600 --> 00:03:46.140
but you could imagine if the
vector field is converging

00:03:46.140 --> 00:03:49.570
within R, well,
the divergence is

00:03:49.570 --> 00:03:51.349
going to be negative
in this situation.

00:03:51.349 --> 00:03:53.890
It's actually converging, which
is the opposite of diverging.

00:03:53.890 --> 00:03:57.650
So the divergence is
negative in this situation.

00:03:57.650 --> 00:03:59.790
And also the flux
across the boundary

00:03:59.790 --> 00:04:00.790
is going to be negative.

00:04:00.790 --> 00:04:02.200
Because as we see
here, the way I

00:04:02.200 --> 00:04:04.100
drew it, across most
of this boundary,

00:04:04.100 --> 00:04:06.840
the vector field is going
in the opposite direction.

00:04:06.840 --> 00:04:10.590
It's going in the opposite
direction as our normal vector

00:04:10.590 --> 00:04:11.280
at any point.

00:04:11.280 --> 00:04:12.779
So hopefully, this
gives you a sense

00:04:12.779 --> 00:04:15.390
of why there's this connection
between the divergence

00:04:15.390 --> 00:04:19.381
over the region and the
flux across the boundary.

00:04:19.381 --> 00:04:21.089
Well, now we're just
going to extend this

00:04:21.089 --> 00:04:24.560
to three dimensions, and it's
the exact same reasoning.

00:04:24.560 --> 00:04:26.900
If we have a-- and I'll
define it a little bit more

00:04:26.900 --> 00:04:31.440
precisely in future videos--
a simple, solid region.

00:04:31.440 --> 00:04:32.530
So let me just draw it.

00:04:32.530 --> 00:04:35.500
And I'm going to try to
draw it in three dimensions.

00:04:35.500 --> 00:04:39.370
So let's say it looks
something like that.

00:04:39.370 --> 00:04:40.986
And one way to think
about it is this

00:04:40.986 --> 00:04:43.360
is going to be a region that
doesn't bend back on itself.

00:04:43.360 --> 00:04:46.010
And if you have a region that
bends back on itself-- well,

00:04:46.010 --> 00:04:47.593
we'll think about
it in multiple ways.

00:04:47.593 --> 00:04:49.794
But out of all the volumes
of three dimensions

00:04:49.794 --> 00:04:51.460
that you can imagine,
these are the ones

00:04:51.460 --> 00:04:53.539
that don't bend
back on themselves.

00:04:53.539 --> 00:04:55.080
And there are some
that you might not

00:04:55.080 --> 00:04:57.490
be able to imagine that
would also not make the case.

00:04:57.490 --> 00:05:02.050
But even if you had ones
that bent back on themselves,

00:05:02.050 --> 00:05:05.520
you could separate them out
into other ones that don't.

00:05:05.520 --> 00:05:08.820
So here is just a simple
solid region over here.

00:05:08.820 --> 00:05:10.320
I'll make it look
three dimensional.

00:05:10.320 --> 00:05:13.830
So maybe if it was transparent,
you would see it like that.

00:05:13.830 --> 00:05:16.950
And then you see the
front of it like that.

00:05:16.950 --> 00:05:20.990
So it's this kind of elliptical,
circular, blob-looking thing.

00:05:20.990 --> 00:05:22.610
So that could be the back of it.

00:05:22.610 --> 00:05:24.390
And then if you go
to the front, it

00:05:24.390 --> 00:05:26.350
could look something like that.

00:05:26.350 --> 00:05:30.760
So this is our
simple solid region.

00:05:30.760 --> 00:05:33.940
I'll call it-- well,
I'll call it R still.

00:05:33.940 --> 00:05:35.870
But we're dealing with
a three dimensional.

00:05:35.870 --> 00:05:40.260
We are now dealing with a
three-dimensional region.

00:05:40.260 --> 00:05:43.240
And now the boundary of
this is no longer a line.

00:05:43.240 --> 00:05:44.490
We're now in three dimensions.

00:05:44.490 --> 00:05:46.640
The boundary is a surface.

00:05:46.640 --> 00:05:57.020
So I'll call that S. S
is the boundary of R.

00:05:57.020 --> 00:05:59.250
And now let's throw on
a vector field here.

00:05:59.250 --> 00:06:01.330
Now, this is a vector
field in three dimensions.

00:06:01.330 --> 00:06:02.913
And now let's imagine
that we actually

00:06:02.913 --> 00:06:06.160
have positive divergence
of our vector field

00:06:06.160 --> 00:06:08.490
within this region
right over here.

00:06:08.490 --> 00:06:09.990
So we have positive divergence.

00:06:09.990 --> 00:06:12.450
So you can imagine that
it's kind of-- the vector

00:06:12.450 --> 00:06:16.250
field within the region, it's
a source of the vector field,

00:06:16.250 --> 00:06:18.080
or the vector field
is diverging out.

00:06:21.410 --> 00:06:23.260
That's just the case I
drew right over here.

00:06:23.260 --> 00:06:25.468
And the other thing we want
to say about vector field

00:06:25.468 --> 00:06:28.360
S, it's oriented in a way that
its normal vector is outward

00:06:28.360 --> 00:06:33.600
facing, so outward
normal vector.

00:06:33.600 --> 00:06:36.130
So the normal, it's oriented
so that the surface--

00:06:36.130 --> 00:06:38.170
the normal vector is like that.

00:06:38.170 --> 00:06:39.660
The other option
is that you have

00:06:39.660 --> 00:06:41.150
an inward-facing normal vector.

00:06:41.150 --> 00:06:43.820
But we're assuming it's
an outward-facing N. Well,

00:06:43.820 --> 00:06:46.600
then we just extrapolate
this to three dimensions.

00:06:46.600 --> 00:06:49.750
We essentially say the
flux across the surface.

00:06:49.750 --> 00:06:52.790
So the flux across the surface,
you would take your vector

00:06:52.790 --> 00:07:01.220
field, dot it with the
normal vector at the surface,

00:07:01.220 --> 00:07:04.330
and then multiply that times
a little chunk of surface,

00:07:04.330 --> 00:07:07.860
so multiply that times a
little chunk of surface,

00:07:07.860 --> 00:07:11.305
and then sum it up along the
whole surface, so sum it up.

00:07:11.305 --> 00:07:12.930
So it's going to be
a surface integral.

00:07:12.930 --> 00:07:14.620
So this is flux
across the surface.

00:07:21.450 --> 00:07:26.490
It's going to be equal
to-- if we were to sum up

00:07:26.490 --> 00:07:31.045
the divergence, if we were to
sum up across the whole volume,

00:07:31.045 --> 00:07:32.420
so now if we're
summing up things

00:07:32.420 --> 00:07:36.232
on every little chunk of volume
over here in three dimensions,

00:07:36.232 --> 00:07:38.690
we're going to have to take
integrals along each dimension.

00:07:38.690 --> 00:07:43.060
So it's going to be
a triple integral

00:07:43.060 --> 00:07:47.870
over the region of
the divergence of F.

00:07:47.870 --> 00:07:49.730
So we're going to
say, how much is F?

00:07:49.730 --> 00:07:51.480
What is the divergence
at F at each point?

00:07:51.480 --> 00:07:56.800
And then multiply it times the
volume of that little chunk

00:07:56.800 --> 00:07:59.120
to sense of how much
is it totally diverging

00:07:59.120 --> 00:08:01.130
in that volume.

00:08:01.130 --> 00:08:02.220
And then you sum it up.

00:08:02.220 --> 00:08:03.970
That should be
equal to the flux.

00:08:03.970 --> 00:08:05.720
It's completely
analogous to what's here.

00:08:05.720 --> 00:08:08.690
Here we had a flux
across the line.

00:08:08.690 --> 00:08:10.290
We had essentially
a two-dimensional--

00:08:10.290 --> 00:08:12.623
or I guess we could say it's
a one-dimensional boundary,

00:08:12.623 --> 00:08:18.090
so flux across the curve.

00:08:18.090 --> 00:08:20.680
And here we have the
flux across a surface.

00:08:20.680 --> 00:08:23.210
Here we were summing the
divergence in the region.

00:08:23.210 --> 00:08:24.710
Here we're summing
it in the volume.

00:08:24.710 --> 00:08:26.412
But it's the exact same logic.

00:08:26.412 --> 00:08:27.870
If you had a vector
field like this

00:08:27.870 --> 00:08:30.330
that was fairly constant
going through the surface,

00:08:30.330 --> 00:08:32.230
on one side you would
have a negative flux.

00:08:32.230 --> 00:08:33.799
On the other side, you
would have a positive flux,

00:08:33.799 --> 00:08:35.252
and they would
roughly cancel out.

00:08:35.252 --> 00:08:36.710
And that makes
sense, because there

00:08:36.710 --> 00:08:38.380
would be no diverging going on.

00:08:38.380 --> 00:08:40.854
If you had a converging vector
field, where it's coming in,

00:08:40.854 --> 00:08:42.520
the flux would be
negative, because it's

00:08:42.520 --> 00:08:45.010
going in the opposite
direction of the normal vector.

00:08:45.010 --> 00:08:48.370
And so the divergence
would be negative as well,

00:08:48.370 --> 00:08:51.250
because essentially the vector
field would be converging.

00:08:51.250 --> 00:08:53.500
So hopefully this
gives you an intuition

00:08:53.500 --> 00:08:57.400
of what the divergence theorem
is actually saying something

00:08:57.400 --> 00:09:01.060
very, very, very, very-- almost
common sense or intuitive.

00:09:01.060 --> 00:09:02.584
And now in the
next few videos, we

00:09:02.584 --> 00:09:04.250
can do some worked
examples, just so you

00:09:04.250 --> 00:09:07.830
feel comfortable computing or
manipulating these integrals.

00:09:07.830 --> 00:09:10.600
And then we'll do a couple of
proof videos, where we actually

00:09:10.600 --> 00:09:13.132
prove the divergence theorem.


Undefined and indeterminate

WEBVTT
Kind: captions
Language: en

00:00:00.201 --> 00:00:03.381
So, once again let's think of yourself as some type of

00:00:03.381 --> 00:00:07.885
ancient philosopher/mathematician, who is trying to extend

00:00:07.885 --> 00:00:12.317
mathematics as much as possible and try to make sure that you're not being lazy

00:00:12.317 --> 00:00:15.537
and leaving things undefined, when you might be able to define them.

00:00:15.537 --> 00:00:19.916
Whenever you start extending mathematics, especially in the realm of

00:00:19.916 --> 00:00:23.700
multiplication and division, there are few things that you hold dear to.

00:00:23.700 --> 00:00:27.569
You feel that if you define some type of division operation,

00:00:27.569 --> 00:00:31.083
that needs to be undone by multiplication; this is close to your heart.

00:00:31.083 --> 00:00:34.369
So you assume... You want to assume...

00:00:34.369 --> 00:00:39.067
You would like to assume that any type of division operation, if you start

00:00:39.067 --> 00:00:42.672
with some number and if you divide with a number

00:00:42.672 --> 00:00:45.851
over which... - division by that number is defined - so when

00:00:45.851 --> 00:00:47.903
I divide by some number and then multiply

00:00:47.903 --> 00:00:51.896
by that same number that this should get me this original number

00:00:51.896 --> 00:00:55.189
right over here, this should give me x right over here.

00:00:55.189 --> 00:00:59.315
And this happens when we just multiply and divide with regular numbers.

00:00:59.315 --> 00:01:04.619
If I get 3 divided by 2 times 2, that's gonna get me 3. If I say

00:01:04.619 --> 00:01:08.271
10 divided by 5 times 5,

00:01:08.271 --> 00:01:13.439
that's going to get me 10. The other things that I want to assume...

00:01:13.439 --> 00:01:17.364
- and this is very close to my heart - I feel that any type of definitions I make

00:01:17.364 --> 00:01:35.067
have to be constant with the idea x*0 has to be 0 or any x.

00:01:35.067 --> 00:01:37.368
So these are close to my heart. I wanna extend

00:01:37.368 --> 00:01:41.299
mathematics. These two things are things that cannot be contradicted,

00:01:41.299 --> 00:01:42.719
cannot be untrue.

00:01:42.719 --> 00:01:47.980
Now, that out of the way. You wanna start exploring the divide-by-0 question.

00:01:47.980 --> 00:01:52.353
So the first thing that you say: "Well, let me just try to define it." So let's

00:01:52.353 --> 00:01:53.850
start, let's assume

00:01:53.850 --> 00:01:57.586
that I have, so this is... So let's make a further assumption...

00:01:57.586 --> 00:02:02.283
that x is some non-zero number.

00:02:02.283 --> 00:02:05.465
Let's just say, well,maybe the best way of finding out what

00:02:05.465 --> 00:02:09.303
x divided by 0 should be, how I should divide it, let's just assume

00:02:09.303 --> 00:02:12.140
there is define, and then come up with any results

00:02:12.140 --> 00:02:14.741
that there might be, there might be a resolve for.

00:02:14.741 --> 00:02:17.639
So let's say that x divided by 0

00:02:25.673 --> 00:02:29.351
is equal to k.

00:02:29.351 --> 00:02:31.278
Well, if this is true and if we are defining what it means

00:02:31.278 --> 00:02:37.656
to divide by zero, then we are assuming that if we multiply by zero,

00:02:37.656 --> 00:02:41.846
we'll get our original number right over here. This is something that we are not willing to

00:02:41.846 --> 00:02:46.880
contradict. So let's see what happens: x divided by 0 is equal to k.

00:02:46.880 --> 00:02:49.764
On the left hand side we have a divide by zero

00:02:49.764 --> 00:02:53.279
and than multiplied by zero. Well then if two things are equal,

00:02:53.279 --> 00:02:57.197
if I do something to one thing inorder for them to stay equal, I have to do it

00:02:57.197 --> 00:02:59.949
to the other thing. This has to be equal to that.

00:02:59.949 --> 00:03:03.096
I have to multiply the left hand AND the right hand side by zero.

00:03:03.096 --> 00:03:07.575
Well, then by this assumption that I am never willing to give up, this left hand side

00:03:07.575 --> 00:03:13.713
right over here, must be equal to x.

00:03:13.713 --> 00:03:17.163
And by this assumption right over here, that I am not willing to give up,

00:03:17.163 --> 00:03:19.733
This right hand side right over here

00:03:19.733 --> 00:03:25.285
must be equal to 0.

00:03:25.285 --> 00:03:27.697
But I just hit a contradiction!

00:03:27.697 --> 00:03:31.513
I assume that x does not equal to 0, and now I am being forced to say

00:03:31.513 --> 00:03:36.798
that x=0. And I am not willing to give up the idea,

00:03:36.798 --> 00:03:39.551
I am not willing to give up either of these ideas. I am defining

00:03:39.551 --> 00:03:43.630
what it means to divide by zero. Or if I am defining what it means to divide by anything...

00:03:43.630 --> 00:03:47.101
...that if I then multiply by that something, that I should get my original number.

00:03:47.101 --> 00:03:50.500
And I am not willing to give up the idea that anything times 0 is 0.

00:03:50.500 --> 00:03:55.578
So all of these things... The only thing that I can give up

00:03:55.578 --> 00:03:59.396
is this right over here. And I'll say, well, I guess k will have to

00:03:59.396 --> 00:04:02.763
stay undefined.

00:04:02.763 --> 00:04:09.121
This whole contradiction happened because I attempted to define what x/0 is.

00:04:09.121 --> 00:04:10.771
Now that out of the way... OK...

00:04:10.771 --> 00:04:15.058
This was a situation when x does not equal zero. But what about

00:04:15.058 --> 00:04:20.548
when x DOES equal zero. So let's think about that a little bit.

00:04:20.548 --> 00:04:23.008
And once again, I will try to

00:04:23.008 --> 00:04:27.266
define it. So I will assume...

00:04:27.266 --> 00:04:32.280
that 0 divided by 0 is equal to some number.

00:04:32.280 --> 00:04:39.148
Well once again, so let's say it is equal to k again. And so, once again...

00:04:39.148 --> 00:04:43.213
we are trying to do the same logic, so we'll write 0/0

00:04:43.213 --> 00:04:46.141
is equal to k. Actually, let me colourcode these zeros.

00:04:46.141 --> 00:04:50.193
This will be a magenta zero and this is a blue zero

00:04:50.193 --> 00:04:54.090
right over here. And once again, I am not willing to give up the idea that

00:04:54.090 --> 00:04:57.915
if I start with a number x, I divide it by something over which division is

00:04:57.915 --> 00:05:02.280
defined, and then I multiply by that something, I should get my original x again.

00:05:02.280 --> 00:05:05.751
I can't give this up. Otherwise it doesn't seem like a good definition

00:05:05.751 --> 00:05:10.024
for the division. So what I am gonna do - I am gonna multiply the left-hand side

00:05:10.024 --> 00:05:13.996
times 0 and by this property that I am

00:05:13.996 --> 00:05:18.299
not willing to give up, the left-hand side should simplify to this magenta

00:05:18.299 --> 00:05:21.650
zero. It should simplify to this over here. But once again,

00:05:21.650 --> 00:05:25.629
anything I do to once side of the equation, inorder for the equation to hold

00:05:25.629 --> 00:05:29.042
true, I need to the other side of the equation. And these two were equal beforehand.

00:05:29.042 --> 00:05:33.064
Any operation I do to this inorder for it to still be equal , I need to do to that.

00:05:33.064 --> 00:05:38.664
So let me multiply the right-hand side by zero.

00:05:38.664 --> 00:05:40.142
So the left I get

00:05:40.142 --> 00:05:44.396
0, I just get this magenta 0, and on the right

00:05:44.396 --> 00:05:48.267
I could just write the zero here, but I won't simplify

00:05:48.267 --> 00:05:54.037
it. I get k times 0.

00:05:54.037 --> 00:05:55.792
Well, this I see right over here...

00:05:55.792 --> 00:06:00.200
This actually is not a contradiction. This actually is true for any k,

00:06:00.200 --> 00:06:03.906
This is one of the core assumptions that I've made in my mathematics

00:06:03.906 --> 00:06:09.765
that I am not willing to give up. So this is true

00:06:09.765 --> 00:06:18.762
True for any k. It's not a contradiction. But the problem here is I wanna come up with

00:06:18.762 --> 00:06:22.693
the k, I'd like a resolve for a k. It would not be nice if this turned out to be 0.

00:06:22.693 --> 00:06:25.338
if this turned out to be one, or if this turned out to be negative one

00:06:25.348 --> 00:06:28.071
But now I see, given the assumptions right here

00:06:28.077 --> 00:06:33.031
this could be ANY... this could be absolutely any k

00:06:33.054 --> 00:06:36.500
I cannot determine what k this should be

00:06:36.515 --> 00:06:40.932
This could be a hundred thousand, this could be 75, it could be anything

00:06:40.932 --> 00:06:42.633
true for any k

00:06:42.633 --> 00:06:48.993
I cannot determine

00:06:49.016 --> 00:06:51.236
what k this should be

00:06:51.236 --> 00:06:54.229
and that's why when you get a little bit more nuance in early math

00:06:54.229 --> 00:06:57.402
people will say, well 0 divided by 0, well we don't know what that's gonna be

00:06:57.402 --> 00:06:59.515
there's no consistent answer there

00:06:59.515 --> 00:07:01.496
so we're just going to call it undefined

00:07:01.496 --> 00:07:04.585
there's no good answer that seems better than any other answer

00:07:04.585 --> 00:07:07.255
but now we see a little bit nuance here

00:07:07.255 --> 00:07:09.667
one divided by zero... you just couldn't define it

00:07:09.667 --> 00:07:11.470
it led to direct contradictions

00:07:11.470 --> 00:07:14.800
zero divided by zero... it could be anything

00:07:14.800 --> 00:07:16.214
you can't determine it

00:07:16.214 --> 00:07:18.934
and so that's why, when you do higher level math

00:07:18.934 --> 00:07:21.677
and you'll often hear this when you take a calculus course

00:07:21.677 --> 00:07:31.567
we see that zero divided by zero is indeterminate


Why zero divided by zero is undefined/indeterminate

WEBVTT
Kind: captions
Language: en

00:00:00.095 --> 00:00:04.042
In the last video we saw why when we take any non-zero number

00:00:04.042 --> 00:00:08.686
divided by zero why mathematicians have left that as being undefined.

00:00:08.686 --> 00:00:12.355
But it might have raised a question in your brain. What about

00:00:12.355 --> 00:00:16.534
zero divided by zero? Isn't there an argument

00:00:16.534 --> 00:00:20.343
why that could be defined? So we're gonna think about zero

00:00:20.343 --> 00:00:24.615
divided by zero. Well there's a couple of lines of reasoning here.

00:00:24.615 --> 00:00:28.033
One, you could start taking numbers closer and closer to zero

00:00:28.033 --> 00:00:30.002
and dividing them by themselves.

00:00:30.002 --> 00:00:34.600
So for example, you take 0.1 divided by 0.1.

00:00:34.600 --> 00:00:37.200
Well that's gonna be one. Let's get even closer to zero:

00:00:37.200 --> 00:00:44.863
0.001 divided by 0.001. Well, that also equals one.

00:00:44.863 --> 00:00:54.197
Let's get super close to zero: 0.000001 divided by 0.000001.

00:00:54.197 --> 00:00:58.005
Well once again, that also equals one. And it didn't even matter whether

00:00:58.005 --> 00:01:02.700
these were positive or negative. I could make these negative and I'd still get the same result.

00:01:02.700 --> 00:01:07.340
Negative this thing divided by negative this thing still gets me to one.

00:01:07.340 --> 00:01:13.005
So based on this logic you might say, "Hey, well this seems like a pretty reasonable argument for

00:01:13.005 --> 00:01:18.532
zero divided by zero to be defined as being equal to one. "

00:01:18.532 --> 00:01:24.337
But someone could come along and say, "Well what happens if we divide zero by numbers closer and closer to zero;

00:01:24.337 --> 00:01:29.367
not a number by itself, but zero by smaller and smaller numbers, or numbers closer and closer to zero."

00:01:29.367 --> 00:01:41.194
And so they say, "For example, zero divided by 0.1, well that's just going to be zero. Zero divided by 0.001,

00:01:41.194 --> 00:01:50.668
well that's also going to be to zero. 0 divided by 0.000001 is also going to be equal to zero."

00:01:50.668 --> 00:01:54.424
And it didn't matter whether we were dividing by a positive or negative number.

00:01:54.424 --> 00:01:57.489
Make all of these negatives, you still get the same answer.

00:01:57.489 --> 00:02:03.015
So this line of reasoning tells you that it's completely legitimate, to think at least that maybe

00:02:03.015 --> 00:02:09.842
zero divided by zero could be equal to zero. And these are equally valid arguments.

00:02:09.842 --> 00:02:15.415
And because they're equally valid, and frankly neither of them is consistent with the rest of mathematics,

00:02:15.415 --> 99:59:59.999
once again mathematicians have left zero divided by zero as undefined.


Why dividing by zero is undefined

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:02.540
Comedian Steven
Wright-- and I guess

00:00:02.540 --> 00:00:05.510
we can credit him with being
a bit of a philosopher--

00:00:05.510 --> 00:00:09.954
once commented that "Black holes
are where God divided by zero."

00:00:09.954 --> 00:00:11.620
And I won't get in
to the physics of it,

00:00:11.620 --> 00:00:14.540
and obviously the metaphor
breaks down in certain ways

00:00:14.540 --> 00:00:19.080
But it is strangely appropriate,
because black holes are where

00:00:19.080 --> 00:00:22.600
our current understanding of
physics seems to break down

00:00:22.600 --> 00:00:26.710
and dividing by zero, as simple
of idea as that seems to be,

00:00:26.710 --> 00:00:29.820
is where our mathematics
also breaks down.

00:00:29.820 --> 00:00:33.370
This is "undefined."

00:00:33.370 --> 00:00:35.512
Sometimes when you see
"undefined" in math class

00:00:35.512 --> 00:00:36.970
it seems like a
very strange thing.

00:00:36.970 --> 00:00:38.700
It seems like a
very bizarre idea.

00:00:38.700 --> 00:00:41.140
But it really means exactly
what the word means.

00:00:41.140 --> 00:00:43.570
Mathematicians
have never defined

00:00:43.570 --> 00:00:45.970
what it must mean
to divide by zero.

00:00:45.970 --> 00:00:46.932
What is that value?

00:00:46.932 --> 00:00:48.390
And the reason they
haven't done it

00:00:48.390 --> 00:00:51.000
is because they couldn't
come up with a good answer.

00:00:51.000 --> 00:00:56.230
There's no good answer
here, no good definition.

00:00:56.230 --> 00:00:59.090
And because of that,
any non-zero number,

00:00:59.090 --> 00:01:01.950
divided by zero, is
left just "undefined."

00:01:01.950 --> 00:01:04.160
7 divided by 0.

00:01:04.160 --> 00:01:06.130
8 divided by 0.

00:01:06.130 --> 00:01:08.120
Negative 1 divided by 0.

00:01:08.120 --> 00:01:11.940
We say all of these things
are just "undefined."

00:01:11.940 --> 00:01:14.340
You might say, well if
we can just define it,

00:01:14.340 --> 00:01:17.220
let's at least try to come
up with a definition of what

00:01:17.220 --> 00:01:20.590
it means to take a non-zero
number divided by zero.

00:01:20.590 --> 00:01:22.220
So let's do that right now.

00:01:22.220 --> 00:01:25.417
We could just take the simplest
of all non-zero numbers.

00:01:25.417 --> 00:01:26.500
We'll just do it with one.

00:01:26.500 --> 00:01:28.810
But we could have done this
with any non-zero number.

00:01:28.810 --> 00:01:30.852
Let's take the example of one.

00:01:30.852 --> 00:01:32.685
Since we don't know
what it means-- or we're

00:01:32.685 --> 00:01:35.060
trying to figure out what it
means to divide by zero Lets

00:01:35.060 --> 00:01:37.572
just try out really, really,
small positive numbers.

00:01:37.572 --> 00:01:39.780
Let's divide by really,
really small positive numbers

00:01:39.780 --> 00:01:42.670
and see what happens as
we get close to zero.

00:01:42.670 --> 00:01:49.010
So lets divide by 0.1 Well,
this will get us to 10.

00:01:49.010 --> 00:01:55.530
If we divide 1 by 0.01
that gets us to 100.

00:01:55.530 --> 00:01:58.310
If I go really,
really close to zero.

00:01:58.310 --> 00:02:05.598
If I divide 1 by
0.000001 this gets us--

00:02:05.598 --> 00:02:08.922
so this is not a tenth,
hundredth, thousandth,

00:02:08.922 --> 00:02:10.380
ten thousandth,
hundred thousandth.

00:02:10.380 --> 00:02:11.213
This is a millionth.

00:02:11.213 --> 00:02:16.720
1 divided by a millionth, that's
going to give us 1 million.

00:02:16.720 --> 00:02:18.090
So we see a pattern here.

00:02:18.090 --> 00:02:21.190
As we divide one by smaller
and smaller and smaller

00:02:21.190 --> 00:02:25.020
positive numbers, we get a
larger and larger and larger

00:02:25.020 --> 00:02:25.520
value.

00:02:25.520 --> 00:02:27.600
Based on just this you
might say, well, hey,

00:02:27.600 --> 00:02:31.310
I've got somewhat of a
definition for 1 divided by 0.

00:02:31.310 --> 00:02:35.994
Maybe we can say that 1 divided
by 0 is positive infinity.

00:02:35.994 --> 00:02:38.160
As we get smaller and smaller
positive numbers here,

00:02:38.160 --> 00:02:41.240
we get super super large
numbers right over here.

00:02:41.240 --> 00:02:44.310
But then, your friend
might say, well,

00:02:44.310 --> 00:02:48.210
that worked when we divided by
positive numbers close to zero

00:02:48.210 --> 00:02:50.380
but what happens when we
divide by negative numbers

00:02:50.380 --> 00:02:51.590
close to zero?

00:02:51.590 --> 00:02:53.180
So lets try those out.

00:02:53.180 --> 00:03:00.420
Well, 1 divided by negative 0.1,
that's going to be negative 10.

00:03:00.420 --> 00:03:04.790
1 divided by
negative 0.01, that's

00:03:04.790 --> 00:03:07.170
going to be negative 100.

00:03:07.170 --> 00:03:13.800
And, if we go all the way to 1
divided by negative 0.000001--

00:03:13.800 --> 00:03:16.250
yup, I drew the same
number of zeros--

00:03:16.250 --> 00:03:19.910
that gets us to
negative 1 million.

00:03:19.910 --> 00:03:24.270
So you when we keep dividing
1 by negative numbers that

00:03:24.270 --> 00:03:27.010
are closer and closer and
closer and closer to zero,

00:03:27.010 --> 00:03:28.470
we get a very different answer.

00:03:28.470 --> 00:03:31.660
We actually start approaching
negative infinity.

00:03:31.660 --> 00:03:34.060
So over here we said maybe it
would be positive infinity,

00:03:34.060 --> 00:03:36.600
but you can make an
equally strong argument

00:03:36.600 --> 00:03:38.700
that it could be a
very different number.

00:03:38.700 --> 00:03:41.710
Negative infinity is going
the exact opposite direction.

00:03:41.710 --> 00:03:44.110
So you could make an
equally strong argument

00:03:44.110 --> 00:03:46.030
that it should be
negative infinity.

00:03:46.030 --> 00:03:47.910
And this is why
mathematicians say

00:03:47.910 --> 00:03:49.730
there's just no
good answer here.

00:03:49.730 --> 00:03:52.720
Especially one that's consistent
with the rest of mathematics.

00:03:52.720 --> 00:03:55.580
They could have just said
it's equal to 42 or something

00:03:55.580 --> 00:03:56.180
like that.

00:03:56.180 --> 00:03:58.550
But that would make no sense.

00:03:58.550 --> 00:04:00.230
It's neither one
of these values,

00:04:00.230 --> 00:04:02.810
and it wouldn't be consistent
with everything else we know.

00:04:02.810 --> 00:04:06.560
So they just left the
whole thing "undefined."


Subtracting a negative = adding a positive

WEBVTT
Kind: captions
Language: en

00:00:00.050 --> 00:00:05.325
This is Steve and this is Steve's uncle
and Steve is in a bit of a bind.

00:00:05.545 --> 00:00:09.809
Not only does he not have any money
but he also owes Michael,

00:00:09.809 --> 00:00:11.353
who is not depicted here.

00:00:11.353 --> 00:00:16.088
He owes Michael $3. So he actually
has a negative net worth.

00:00:16.134 --> 00:00:18.724
Steve has a net worth of -3.

00:00:18.724 --> 00:00:23.664
Steve's uncle cares about Steve
and feels bad that he doesn't even have

00:00:23.664 --> 00:00:25.894
a 0 net worth;
he has a negative net worth!

00:00:25.894 --> 00:00:29.413
And so Steve's uncle wants to at least
take away some of this pain.

00:00:29.413 --> 00:00:34.812
He at least wants to get Steve back
to a neutral net worth, a 0 net worth.

00:00:34.862 --> 00:00:42.600
So he decides to take away
Steve's negative net worth.

00:00:42.600 --> 00:00:45.568
He wants to take away that -3.

00:00:45.648 --> 00:00:51.703
What happens if you take away that -3?
Well, that should get you back to 0.

00:00:51.703 --> 00:00:54.200
If you take away anything...
if you have something

00:00:54.200 --> 00:00:56.707
and you take it away,
it should go back to 0.

00:00:56.707 --> 00:01:01.218
Similarly if you owe people things
and that owing people things

00:01:01.218 --> 00:01:03.969
is taken away, it also gets back to 0.

00:01:03.969 --> 00:01:07.713
And another way to think about it is
how Steve's uncle would actually

00:01:07.713 --> 00:01:11.767
take that liability,
take that debt away from Steve.

00:01:12.077 --> 00:01:16.493
Well, the easiest way he could do it
if Steve is starting at -3 net worth

00:01:16.493 --> 00:01:22.684
is for his uncle to give Steve $3,
so that he(Steve) can get back

00:01:22.684 --> 00:01:24.675
to a net worth of 0.

00:01:24.675 --> 00:01:27.716
So hopefully, that makes it
a little bit clearer,

00:01:27.719 --> 00:01:32.059
that this and this
are intuitively equivalent.


Stokes' theorem proof part 7

WEBVTT
Kind: captions
Language: en

00:00:00.570 --> 00:00:02.220
Where we left off
in the last video,

00:00:02.220 --> 00:00:04.800
we'd expressed our line
integral around the boundary

00:00:04.800 --> 00:00:06.120
of our surface.

00:00:06.120 --> 00:00:10.090
So this is f.dr where our
path is this boundary right

00:00:10.090 --> 00:00:11.750
over here-- this path c.

00:00:11.750 --> 00:00:13.780
We had expressed it
in terms of a line

00:00:13.780 --> 00:00:17.360
integral around this boundary--
around path c1-- which

00:00:17.360 --> 00:00:20.160
is the boundary of
region R. And the reason

00:00:20.160 --> 00:00:21.990
why this is going
to be valuable to us

00:00:21.990 --> 00:00:25.080
is now we can directly apply
Green's Theorem to this

00:00:25.080 --> 00:00:28.760
to turn this into a double
integral over this region

00:00:28.760 --> 00:00:32.990
right over here-- the region
that it is actually bounding.

00:00:32.990 --> 00:00:34.230
So let's actually do that.

00:00:34.230 --> 00:00:36.510
We're just applying
Green's Theorem here.

00:00:36.510 --> 00:00:43.947
So Green's Theorem.

00:00:43.947 --> 00:00:46.030
And actually, let me box
this off right over here.

00:00:46.030 --> 00:00:47.571
This was just
something that we wrote

00:00:47.571 --> 00:00:50.010
to remind ourselves
in the last video.

00:00:50.010 --> 00:00:52.170
So let me just box it
off right over here.

00:00:52.170 --> 00:00:54.260
But when you apply
Green's Theorem,

00:00:54.260 --> 00:00:56.760
you get that this is going
to be the exact same thing

00:00:56.760 --> 00:00:59.310
as the double integral
over the region

00:00:59.310 --> 00:01:01.440
that c1 bounds-- that
region, R, that's

00:01:01.440 --> 00:01:05.319
in our xy plane-- of the
partial of this with respect

00:01:05.319 --> 00:01:16.620
to x of this business.

00:01:16.620 --> 00:01:18.790
So I'll do that in
that same green color.

00:01:18.790 --> 00:01:24.800
Q plus R times the partial
of z with respect to y

00:01:24.800 --> 00:01:30.330
minus the partial
with respect to y

00:01:30.330 --> 00:01:34.772
of P plus R times the partial
of z with respect to x.

00:01:42.730 --> 00:01:46.280
And then, dA, little
differential of our region.

00:01:46.280 --> 00:01:46.780
dA.

00:01:49.590 --> 00:01:52.910
And so let's actually
calculate this right here.

00:01:52.910 --> 00:01:55.370
You take the partials of
each of these expressions.

00:01:55.370 --> 00:01:58.000
And we'll see that if we
expand them and then simplify,

00:01:58.000 --> 00:01:59.958
we'll get something very
similar-- or actually,

00:01:59.958 --> 00:02:02.040
hopefully, identical--
to this right over here.

00:02:02.040 --> 00:02:05.500
And that'll show that this line
integral for this special case

00:02:05.500 --> 00:02:08.210
is the same thing as this
surface integral, which

00:02:08.210 --> 00:02:11.330
will prove Stokes' theorem
for this special case.

00:02:11.330 --> 00:02:12.260
So let's do it.

00:02:12.260 --> 00:02:14.780
So let's just apply the
partial derivative operator.

00:02:14.780 --> 00:02:17.280
So first we want to take the
partial derivative with respect

00:02:17.280 --> 00:02:18.070
to Q.

00:02:18.070 --> 00:02:20.570
And we need to remind
ourselves-- and we did this way

00:02:20.570 --> 00:02:25.640
up here where we first thought
about it-- we saw that P, Q,

00:02:25.640 --> 00:02:28.230
and R, they're each
functions of x, y, and z.

00:02:28.230 --> 00:02:30.550
So we're assuming that's
how they're represented.

00:02:30.550 --> 00:02:33.801
And if z was not a
function of x, then,

00:02:33.801 --> 00:02:36.050
if we were to take the partial
of Q with respect to x,

00:02:36.050 --> 00:02:38.980
we would just write it as a
partial of Q with respect to x.

00:02:38.980 --> 00:02:42.270
But we know that we've
assumed that z is itself

00:02:42.270 --> 00:02:44.980
a function of x and y.

00:02:44.980 --> 00:02:48.250
So if we're taking the partial
with respect to x over here,

00:02:48.250 --> 00:02:50.550
we have to think
first-- Well, how

00:02:50.550 --> 00:02:52.970
can Q directly change
with respect to x?

00:02:52.970 --> 00:02:55.550
And then, how can it change
due to something else changing

00:02:55.550 --> 00:02:56.100
due to x?

00:02:56.100 --> 00:02:59.590
And so that other thing that
could change due to x is z. y

00:02:59.590 --> 00:03:02.980
is independent of x, but
z is a function of x.

00:03:02.980 --> 00:03:05.234
So let's keep that in mind.

00:03:05.234 --> 00:03:07.650
So we're really going to do
the multi-variable chain rule.

00:03:07.650 --> 00:03:12.570
So when we try to take the
derivative of this part,

00:03:12.570 --> 00:03:14.330
of this whole function
with respect to x,

00:03:14.330 --> 00:03:18.000
we have to think about--
How will Q change directly

00:03:18.000 --> 00:03:19.410
with respect to x?

00:03:19.410 --> 00:03:22.820
And to that, we have to add how
Q could change due to changes

00:03:22.820 --> 00:03:24.740
in other variables
due to changes in x.

00:03:24.740 --> 00:03:26.197
And the only other
variable that Q

00:03:26.197 --> 00:03:28.280
is a function of that could
change due to a change

00:03:28.280 --> 00:03:30.010
in x is z.

00:03:30.010 --> 00:03:35.420
So Q could also change to z
because z has changed due to x.

00:03:35.420 --> 00:03:38.400
So the operator here is
the partial with respect

00:03:38.400 --> 00:03:40.860
to x plus the partial
of Q with respect

00:03:40.860 --> 00:03:45.220
to z times the partial
of z with respect to x.

00:03:45.220 --> 00:03:48.620
If we rewrote Q so that
z was substituted with

00:03:48.620 --> 00:03:51.465
x's and y's-- because it is a
function of x and y's-- then we

00:03:51.465 --> 00:03:53.840
would just have to write this
first term right over here.

00:03:53.840 --> 00:03:55.500
But we're assuming
this is expressed

00:03:55.500 --> 00:03:57.170
as a function of x, y, and z.

00:03:57.170 --> 00:03:59.480
And z itself is a function of x.

00:03:59.480 --> 00:04:03.850
And so that's why we had to use
the multi-variable chain rule.

00:04:03.850 --> 00:04:05.380
Now, let's move
to the next part.

00:04:05.380 --> 00:04:09.710
And both of these might
have some x's in them.

00:04:09.710 --> 00:04:12.170
So we have to use the
product rule right over here.

00:04:12.170 --> 00:04:19.050
So first, we can take the
derivative of r with respect

00:04:19.050 --> 00:04:19.890
to x.

00:04:19.890 --> 00:04:21.822
And then, multiply
that times z sub y.

00:04:21.822 --> 00:04:23.780
Then, we have to take
the derivative of z sub y

00:04:23.780 --> 00:04:26.770
with respect to x and
multiply that times R.

00:04:26.770 --> 00:04:30.660
So this is going to be plus.

00:04:30.660 --> 00:04:32.830
If we take the derivative
of this with respect

00:04:32.830 --> 00:04:37.230
to x, same exact logic-- R
can change directly due to x,

00:04:37.230 --> 00:04:40.080
and it can change due to y.

00:04:40.080 --> 00:04:43.470
And that could change due
to z and multiply that times

00:04:43.470 --> 00:04:46.160
how z could change due to x.

00:04:46.160 --> 00:04:48.750
Once again, you could view this
as the multi-variable chain

00:04:48.750 --> 00:04:49.969
rule in action here.

00:04:49.969 --> 00:04:51.510
But of course, we
take the derivative

00:04:51.510 --> 00:04:53.980
of the first term
times the second term.

00:04:53.980 --> 00:04:55.665
And I'll do the second
term in magenta.

00:04:58.510 --> 00:05:02.670
So the partial of z with
respect to y plus the derivative

00:05:02.670 --> 00:05:04.624
of the second term--
which is the partial of z

00:05:04.624 --> 00:05:06.540
with respect to y, and
then taking the partial

00:05:06.540 --> 00:05:08.500
of that with respect to
x, which we could just

00:05:08.500 --> 00:05:10.760
write as that--
times the first term.

00:05:14.700 --> 00:05:16.300
So that's the
partial with respect

00:05:16.300 --> 00:05:18.960
to x of all of this
business right over here.

00:05:18.960 --> 00:05:21.070
And then, we need to
subtract the partial

00:05:21.070 --> 00:05:22.320
of this with respect to y.

00:05:22.320 --> 00:05:24.720
And we're going to use
the exact same logic.

00:05:24.720 --> 00:05:26.510
So then, we're
going to subtract--

00:05:26.510 --> 00:05:29.100
and I'll put it in
parentheses like that.

00:05:29.100 --> 00:05:32.100
So P could change
directly due to y.

00:05:32.100 --> 00:05:34.720
I will circle P. Let
me do it with a color

00:05:34.720 --> 00:05:36.180
that I haven't used yet.

00:05:36.180 --> 00:05:39.770
P could change
directly due to y.

00:05:39.770 --> 00:05:43.750
So we could say the partial
of P with respect to y,

00:05:43.750 --> 00:05:47.150
but it could also change due
to z changing because of y.

00:05:47.150 --> 00:05:50.570
So plus the partial
with respect to z times

00:05:50.570 --> 00:05:55.976
the partial of z
with respect to y--

00:05:55.976 --> 00:05:58.670
and I'll do R maybe
in that same color--

00:05:58.670 --> 00:06:00.454
plus the derivative of R.

00:06:00.454 --> 00:06:02.620
Well, we already figured
that, but actually now it's

00:06:02.620 --> 00:06:04.700
with respect to x,
not with respect to y.

00:06:04.700 --> 00:06:05.749
You have to be careful.

00:06:05.749 --> 00:06:07.790
So it's going to be the
partial of R with respect

00:06:07.790 --> 00:06:11.060
to y plus the partial
of R with respect to z

00:06:11.060 --> 00:06:14.820
times the partial of z
with respect to y times

00:06:14.820 --> 00:06:23.304
z sub x plus-- now,
we take the derivative

00:06:23.304 --> 00:06:24.970
of the second term
times the first term.

00:06:24.970 --> 00:06:27.800
The derivative of the partial
of z with respect to x, then

00:06:27.800 --> 00:06:30.015
with respect to y,
is going to be z.

00:06:35.060 --> 00:06:42.360
And then we're going to
multiply that times R.

00:06:42.360 --> 00:06:45.010
And now, let's see if
we can expand this out.

00:06:45.010 --> 00:06:46.340
And hopefully, things simplify.

00:06:46.340 --> 00:06:47.650
And just a reminder,
I'm just working

00:06:47.650 --> 00:06:49.233
on the inside of
this double integral.

00:06:49.233 --> 00:06:53.630
I'll rewrite the double
integral and the dA

00:06:53.630 --> 00:06:56.044
once I get this all
cleaned up a little bit.

00:06:56.044 --> 00:06:57.460
So let me rewrite
it a little bit.

00:06:57.460 --> 00:07:02.540
So this is equal to
the partial of Q.

00:07:02.540 --> 00:07:04.660
I'll try to color-code
it the same way.

00:07:04.660 --> 00:07:07.070
And this is really just
algebra at this point.

00:07:07.070 --> 00:07:09.670
The partial of Q with
respect to x plus the partial

00:07:09.670 --> 00:07:12.260
of Q with respect to z times
the partial of z with respect

00:07:12.260 --> 00:07:18.210
to x plus the partial
of R with respect

00:07:18.210 --> 00:07:22.850
to x times the partial
of z with respect to y.

00:07:22.850 --> 00:07:26.640
And then plus the partial
of R with respect to z

00:07:26.640 --> 00:07:28.190
times the partial
of z with respect

00:07:28.190 --> 00:07:31.680
to x times the partial
of z with respect to y.

00:07:31.680 --> 00:07:34.030
And then we have this
term right over here--

00:07:34.030 --> 00:07:40.380
which I'll just do in
purple-- plus the partial of z

00:07:40.380 --> 00:07:44.170
with respect to y, and
then respect to x, times R.

00:07:44.170 --> 00:07:46.680
And now, we're going to subtract
all of this business right

00:07:46.680 --> 00:07:47.320
over here.

00:07:50.140 --> 00:07:51.890
I'll do it in the blue.

00:07:51.890 --> 00:07:54.420
Minus the partial
of P with respect

00:07:54.420 --> 00:08:00.280
to y minus the partial of
P with respect to z times

00:08:00.280 --> 00:08:03.090
the partial of z
with respect to y.

00:08:03.090 --> 00:08:05.910
And then we're going
to subtract from that.

00:08:05.910 --> 00:08:09.380
Minus the partial
of r with respect

00:08:09.380 --> 00:08:13.400
to y times-- we want to
distribute this-- times

00:08:13.400 --> 00:08:17.439
the partial of z with respect
to x minus the partial of R.

00:08:17.439 --> 00:08:18.730
This gets a little bit tedious.

00:08:18.730 --> 00:08:20.890
But hopefully, it'll get
us to where we need to go.

00:08:20.890 --> 00:08:24.150
The partial of R with respect
to z times the partial of z

00:08:24.150 --> 00:08:29.710
with respect to y times the
partial of z with respect to x.

00:08:29.710 --> 00:08:32.141
And then, finally, this
term right over here

00:08:32.141 --> 00:08:34.390
minus this-- because we have
that negative out there--

00:08:34.390 --> 00:08:37.280
minus the partial
of z with respect

00:08:37.280 --> 00:08:43.030
to y, then with respect to x,
and then with respect to yR.

00:08:43.030 --> 00:08:45.010
Now, let's see if we
can simplify things.

00:08:45.010 --> 00:08:48.090
So the first thing-- this
and this look to be the same.

00:08:48.090 --> 00:08:51.840
We just can commute the order
in which we actually multiply.

00:08:51.840 --> 00:08:53.490
But these are the
exact same term.

00:08:53.490 --> 00:08:56.320
So that is going to
cancel out with that.

00:08:56.320 --> 00:09:03.150
And because we
assumed way up here

00:09:03.150 --> 00:09:05.845
that we have continuous second
derivatives of the function

00:09:05.845 --> 00:09:08.030
z-- z is a function
of x and y, that that

00:09:08.030 --> 00:09:13.340
is equal to that-- we can
now say that these two

00:09:13.340 --> 00:09:15.970
right over here are going to
be the negatives of each other

00:09:15.970 --> 00:09:18.410
or that they are
going to cancel out.

00:09:18.410 --> 00:09:20.550
And so this simplified
things a good bit.

00:09:20.550 --> 00:09:23.344
And let's see if
I can group terms

00:09:23.344 --> 00:09:25.010
in a way that might
start to make sense.

00:09:25.010 --> 00:09:26.551
And actually, I'm
going to try to see

00:09:26.551 --> 00:09:28.420
if I can make them
similar to this.

00:09:28.420 --> 00:09:32.277
So I have all the terms that
have a z sub x and the z sub y

00:09:32.277 --> 00:09:33.360
and then the rest of them.

00:09:33.360 --> 00:09:35.860
So z sub x I'll do in blue.

00:09:35.860 --> 00:09:37.970
So you have the terms
that have a z sub x in it.

00:09:37.970 --> 00:09:40.610
So you have this
term right over here

00:09:40.610 --> 00:09:41.900
and this term right over here.

00:09:41.900 --> 00:09:44.270
And we can factor
out the z sub x.

00:09:44.270 --> 00:09:46.770
And we get the partial
of z with respect

00:09:46.770 --> 00:09:50.330
to x times the partial
of Q with respect

00:09:50.330 --> 00:09:55.400
to z minus the partial
of R with respect to y.

00:09:55.400 --> 00:09:57.582
And then, let's do
the-- and I want

00:09:57.582 --> 00:09:59.470
to get the colors
the same way, too.

00:09:59.470 --> 00:10:00.910
I did yellow next.

00:10:00.910 --> 00:10:03.150
Plus, I have all the
terms of the partial of z

00:10:03.150 --> 00:10:04.150
with respect to y.

00:10:04.150 --> 00:10:07.470
So it's that term and
that term right over here.

00:10:07.470 --> 00:10:11.700
So it becomes plus the
partial of z with respect to y

00:10:11.700 --> 00:10:14.110
times the partial
of R with respect

00:10:14.110 --> 00:10:18.250
to x minus the partial
of p with respect to z.

00:10:18.250 --> 00:10:20.410
And then we have
these last two terms.

00:10:20.410 --> 00:10:22.700
And I used the color
green up there,

00:10:22.700 --> 00:10:24.730
so I'll use the
color green again.

00:10:24.730 --> 00:10:26.440
So for these two
terms, I'll just

00:10:26.440 --> 00:10:29.750
write plus the partial
of Q with respect

00:10:29.750 --> 00:10:35.850
to x minus the partial
of P with respect to y.

00:10:35.850 --> 00:10:38.750
So our double integrals
have kind of-- I

00:10:38.750 --> 00:10:40.390
can't really say
simplified-- but we

00:10:40.390 --> 00:10:42.270
can rewrite it like this.

00:10:42.270 --> 00:10:47.370
And we don't want to forget
this was all a simplification

00:10:47.370 --> 00:10:50.526
of our double integral
over the region dA.

00:10:50.526 --> 00:10:52.150
This is what we have
been able to using

00:10:52.150 --> 00:10:54.270
Green's theorem and the
multi-variable chain rule

00:10:54.270 --> 00:10:55.460
and whatever else.

00:10:55.460 --> 00:10:57.070
We've been able to
say that that line

00:10:57.070 --> 00:10:58.861
integral around the
boundary of our surface

00:10:58.861 --> 00:11:00.430
is the same thing as this.

00:11:00.430 --> 00:11:04.010
And now, we can compare that to
what our surface integral was.

00:11:04.010 --> 00:11:05.570
So let's see if I have space.

00:11:05.570 --> 00:11:07.580
So copy.

00:11:07.580 --> 00:11:11.052
And then, let me see if I have
space up here to paste it.

00:11:11.052 --> 00:11:13.260
Well, it doesn't look like
I actually have much space

00:11:13.260 --> 00:11:15.730
to paste it, although
I'll try anyway.

00:11:15.730 --> 00:11:20.890
So if I paste it, you see
that they are identical.

00:11:20.890 --> 00:11:22.080
They are identical.

00:11:22.080 --> 00:11:25.100
Our line integral is
identical to this.

00:11:25.100 --> 00:11:27.070
We get the exact same thing.

00:11:27.070 --> 00:11:32.100
So our line integral,
f.dr, around this path c

00:11:32.100 --> 00:11:34.940
simplified to this and our
surface integral simplified

00:11:34.940 --> 00:11:35.440
to this.

00:11:35.440 --> 00:11:37.710
So using the
assumptions we had, they

00:11:37.710 --> 00:11:39.710
both simplified
to the same thing.

00:11:39.710 --> 00:11:41.740
So now we know, for
this special case,

00:11:41.740 --> 00:11:45.660
our line integral is equal
to our surface integral.

00:11:45.660 --> 00:11:47.801
And we are done.


Stokes' theorem proof part 6

WEBVTT
Kind: captions
Language: en

00:00:00.590 --> 00:00:03.550
Where we had left off, we had
expressed our line integral

00:00:03.550 --> 00:00:06.939
over the boundary of our
surface in terms of dt,

00:00:06.939 --> 00:00:08.210
in the dt domain.

00:00:08.210 --> 00:00:11.634
We expressed what f dot dr
is going to be equal to.

00:00:11.634 --> 00:00:13.050
What I'm going to
do in this video

00:00:13.050 --> 00:00:14.924
is do a little bit of
algebraic manipulation,

00:00:14.924 --> 00:00:17.104
and then we will actually
apply Green's theorem.

00:00:17.104 --> 00:00:19.020
And whether or not we
have time in this video,

00:00:19.020 --> 00:00:20.436
we'll then manipulate
that to show

00:00:20.436 --> 00:00:22.840
that it's the exact same thing
that we saw in the earlier

00:00:22.840 --> 00:00:25.450
part, where we evaluated
the surface integral.

00:00:25.450 --> 00:00:26.427
And so let's do that.

00:00:26.427 --> 00:00:28.510
So this integral right
over here is the same thing

00:00:28.510 --> 00:00:30.660
as the integral from a to b.

00:00:30.660 --> 00:00:32.159
And then what I'm
going to do is I'm

00:00:32.159 --> 00:00:35.830
going to group the things that
are being multiplied by dx dt.

00:00:35.830 --> 00:00:39.050
So if I group them and
then factor out a dx dt,

00:00:39.050 --> 00:00:42.864
so if I take this part right
here and this part right

00:00:42.864 --> 00:00:44.780
over here, I'm essentially
going to distribute

00:00:44.780 --> 00:00:46.260
the R. Let me make it clear.

00:00:46.260 --> 00:00:48.750
I'm going to distribute
the R. And so I'm

00:00:48.750 --> 00:00:51.260
going to group that and
that right over there,

00:00:51.260 --> 00:00:54.760
I will be left with-- and
I'll factor out the dx dt.

00:00:54.760 --> 00:01:01.080
I will be left with P plus
R times the partial of z

00:01:01.080 --> 00:01:04.940
with respect to x times dx dt.

00:01:08.450 --> 00:01:11.430
And I'll do the same
thing for the dy dt's.

00:01:11.430 --> 00:01:13.270
So that's that part
right over there.

00:01:13.270 --> 00:01:15.020
And then the R is going
to get distributed

00:01:15.020 --> 00:01:16.850
in this thing right over here.

00:01:16.850 --> 00:01:26.436
So it's going to be plus
Q plus R-- the R is going

00:01:26.436 --> 00:01:30.310
to get distributed-- times the
partial of z with respect to y.

00:01:30.310 --> 00:01:31.935
And I'm going to
factor out a dy dt.

00:01:35.380 --> 00:01:38.240
And then we can't
forget all of that

00:01:38.240 --> 00:01:44.190
is going to be multiplied by dt.

00:01:44.190 --> 00:01:46.690
Now, this right over
here is interesting,

00:01:46.690 --> 00:01:51.730
because it's starting to
look very similar to what

00:01:51.730 --> 00:01:54.560
we had up here, where we just
had our theoretical vector

00:01:54.560 --> 00:01:55.210
field.

00:01:55.210 --> 00:01:59.630
In fact, let me
copy and paste it.

00:01:59.630 --> 00:02:03.390
So actually, I don't know if I'm
on the right a layer of my work

00:02:03.390 --> 00:02:04.660
right over here.

00:02:04.660 --> 00:02:08.490
So let me copy and paste.

00:02:08.490 --> 00:02:09.750
No, that didn't work.

00:02:09.750 --> 00:02:11.960
Let me try it one more time.

00:02:11.960 --> 00:02:18.104
So if I try to copy,
I'll go a layer down.

00:02:18.104 --> 00:02:19.520
I'm using an art
program for this.

00:02:19.520 --> 00:02:23.940
Copy-- and then I think
this will work-- and paste.

00:02:23.940 --> 00:02:25.010
There we go.

00:02:25.010 --> 00:02:26.710
So this is a result
that we had before.

00:02:26.710 --> 00:02:29.200
This is kind of a
template to look at.

00:02:29.200 --> 00:02:31.260
But what is going on
over here if we just

00:02:31.260 --> 00:02:32.750
look at this template?

00:02:32.750 --> 00:02:37.970
We see that we're
in the t domain.

00:02:37.970 --> 00:02:40.600
We're integrating over
t right over here.

00:02:40.600 --> 00:02:41.850
But then we have these things.

00:02:41.850 --> 00:02:45.770
We have some function that's
a function of x and y times dx

00:02:45.770 --> 00:02:48.480
dt, and then some function,
the function of x and y times

00:02:48.480 --> 00:02:51.180
dy dt, and we're integrating
with respect to dt.

00:02:51.180 --> 00:02:53.430
Well, that's exactly what
we're doing right over here.

00:02:53.430 --> 00:02:55.650
We can distribute
the dt, and we have

00:02:55.650 --> 00:02:58.460
something that looks exactly
like this right over here,

00:02:58.460 --> 00:03:02.590
where M is analogous to
this piece right over here.

00:03:02.590 --> 00:03:03.760
Let me make it clear.

00:03:03.760 --> 00:03:09.700
M, this piece right over
here, looks a lot like M

00:03:09.700 --> 00:03:11.260
in our example right over here.

00:03:11.260 --> 00:03:14.580
It's being multiplied by
dx dt, and then this dt,

00:03:14.580 --> 00:03:18.110
which you can distribute, and
this piece right over here,

00:03:18.110 --> 00:03:20.680
looks a lot like N.

00:03:20.680 --> 00:03:22.490
And so, we can
say that, well, we

00:03:22.490 --> 00:03:23.990
have something that
looks like this.

00:03:23.990 --> 00:03:27.140
We can rewrite it
like this and go back

00:03:27.140 --> 00:03:30.820
into the-- kind of
deparametrize it.

00:03:30.820 --> 00:03:35.400
So this thing is going to
be equal to now the line

00:03:35.400 --> 00:03:37.430
integral of C1.

00:03:37.430 --> 00:03:39.160
We are in the xy plane.

00:03:39.160 --> 00:03:43.320
We started with the curve C,
but now we're going to go to C1.

00:03:43.320 --> 00:03:44.850
It's completely analogous.

00:03:44.850 --> 00:03:47.650
These are only functions
of x's and y's.

00:03:47.650 --> 00:03:49.010
Everything here is.

00:03:49.010 --> 00:03:55.250
So now, this is going to be
the line integral over C1--

00:03:55.250 --> 00:04:00.126
and I could even draw it as
that, if I like-- of M dx,

00:04:00.126 --> 00:04:01.000
and that makes sense.

00:04:01.000 --> 00:04:04.925
Because if you multiply dt times
dx dt, the dt's cancel out,

00:04:04.925 --> 00:04:06.470
and you're just left with dx.

00:04:06.470 --> 00:04:09.620
So M times dx, so let
me write it that way.

00:04:09.620 --> 00:04:15.320
So it's going to be P plus
R times the partial of z

00:04:15.320 --> 00:04:22.089
with respect to x dx plus n.

00:04:22.089 --> 00:04:23.910
Let me scroll to the
right a little bit.

00:04:23.910 --> 00:04:28.030
Plus n, which is Q plus
R times the partial

00:04:28.030 --> 00:04:32.500
of z with respect to y dy.

00:04:32.500 --> 00:04:34.130
And then this is
really interesting,

00:04:34.130 --> 00:04:36.887
because this path that we
are now concerned with,

00:04:36.887 --> 00:04:37.970
it's completely analogous.

00:04:37.970 --> 00:04:40.053
I hope you don't think I'm
doing some voodoo here.

00:04:40.053 --> 00:04:42.570
This statement is completely
analogous to this statement,

00:04:42.570 --> 00:04:45.480
where M could be this
and N could be this.

00:04:45.480 --> 00:04:47.450
And so, we can revert
it back to now the path

00:04:47.450 --> 00:04:50.190
C1 that sits in the xy plane.

00:04:50.190 --> 00:04:52.290
Not our original
boundary C, but now we're

00:04:52.290 --> 00:04:55.740
just dealing with a
boundary in the xy plane.

00:04:55.740 --> 00:04:57.000
So it reverts to this.

00:04:57.000 --> 00:04:59.420
But what's powerful about
getting it to this point,

00:04:59.420 --> 00:05:06.730
is we can now apply Green's
theorem to this to essentially

00:05:06.730 --> 00:05:09.580
turn this into a double
integral over the region

00:05:09.580 --> 00:05:13.240
that this path surrounds,
the original region

00:05:13.240 --> 00:05:15.850
over this region R.
And when we change

00:05:15.850 --> 00:05:18.080
how we manipulate that,
when we play around with it,

00:05:18.080 --> 00:05:21.030
we're going to see that we get
the exact same result that we

00:05:21.030 --> 00:05:22.544
got in earlier videos.

00:05:22.544 --> 00:05:23.960
And I'll leave you
there, and I'll

00:05:23.960 --> 00:05:26.950
see if I can do that
in the next video.


Stokes' theorem proof part 5

WEBVTT
Kind: captions
Language: en

00:00:00.600 --> 00:00:02.340
Now that we have
a parameterization

00:00:02.340 --> 00:00:05.090
for the boundary of our
surface right up here,

00:00:05.090 --> 00:00:08.039
let's think a little bit
about what the line integral--

00:00:08.039 --> 00:00:10.700
and this was the left side of
our original Stokes' theorem

00:00:10.700 --> 00:00:12.820
statement-- what the line
integral over the path

00:00:12.820 --> 00:00:19.320
C of F, our original vector
field F, dot dr is going to be.

00:00:19.320 --> 00:00:20.940
And now, once
again, this R we're

00:00:20.940 --> 00:00:24.400
talking about the path of
this boundary right over here,

00:00:24.400 --> 00:00:26.480
C. Not C1 that we
did to kind of get

00:00:26.480 --> 00:00:28.830
some of our foundational
material going.

00:00:28.830 --> 00:00:38.250
So F dot dr-- well,
we remember what F is.

00:00:38.250 --> 00:00:40.010
F was all the way up here.

00:00:40.010 --> 00:00:43.340
Its i, j, and k components
were just P, Q, and R.

00:00:43.340 --> 00:00:45.050
So that's kind of
easy to remember.

00:00:45.050 --> 00:00:48.730
But let's think about
what dr is equal to.

00:00:48.730 --> 00:00:50.230
And we're going to
have to break out

00:00:50.230 --> 00:00:51.771
a little bit of our
three-dimensional

00:00:51.771 --> 00:00:54.310
or our multivariable chain
rule right over here.

00:00:54.310 --> 00:01:03.701
So dr is the same thing
as dr dt times dt.

00:01:03.701 --> 00:01:05.200
So we really just
have to figure out

00:01:05.200 --> 00:01:07.540
what the derivative of
R with respect to t.

00:01:07.540 --> 00:01:08.320
And this one we're going
to have to break out

00:01:08.320 --> 00:01:09.611
a little bit of the chain rule.

00:01:09.611 --> 00:01:10.960
So let me write this down.

00:01:10.960 --> 00:01:17.450
So dr dt is going to be
equal to the derivative of x

00:01:17.450 --> 00:01:24.462
with respect to t times i
plus the derivative of y

00:01:24.462 --> 00:01:25.920
with respect to
t-- I'm just taking

00:01:25.920 --> 00:01:29.590
the derivative with
respect to t-- j.

00:01:29.590 --> 00:01:33.980
Because z is a function of
x, which is a function of t,

00:01:33.980 --> 00:01:36.460
and z is also a function of
y, which is a function of t,

00:01:36.460 --> 00:01:40.032
we're going to have to break out
our multivariable chain rule.

00:01:40.032 --> 00:01:41.740
So if we want to take
the derivative of z

00:01:41.740 --> 00:01:43.698
with respect to t-- I'll
do it separately here,

00:01:43.698 --> 00:01:45.370
and then I'll write
it down down here.

00:01:45.370 --> 00:01:49.950
The derivative of z with respect
to t-- the way I conceptualize

00:01:49.950 --> 00:01:52.420
is, what's all the
different ways that z

00:01:52.420 --> 00:01:54.800
can change from a change in t?

00:01:54.800 --> 00:01:56.670
Well, it could
change because x is

00:01:56.670 --> 00:01:59.230
changing due to a change in t.

00:01:59.230 --> 00:02:02.490
So z could change due
to x, the partial of z

00:02:02.490 --> 00:02:06.555
with respect to x, when x
changes with respect to t.

00:02:06.555 --> 00:02:08.680
But then that's not the
only way that z can change.

00:02:08.680 --> 00:02:11.920
We have to add to that how z
can change with respect to y,

00:02:11.920 --> 00:02:18.130
partial of z with respect
to y times how fast

00:02:18.130 --> 00:02:20.920
or how y is changing
with respect to t.

00:02:20.920 --> 00:02:23.390
And this is just our
multivariable chain rule.

00:02:23.390 --> 00:02:26.430
And so this is dz dt.

00:02:26.430 --> 00:02:28.170
So I'll just write
it right over here.

00:02:28.170 --> 00:02:29.610
And I'll use slightly
different notation

00:02:29.610 --> 00:02:31.651
that's consistent with
what we were doing before,

00:02:31.651 --> 00:02:34.140
and it'll help make things
a little bit clearer.

00:02:34.140 --> 00:02:41.650
So it's going to be the partial
of z with respect to x, dx dt,

00:02:41.650 --> 00:02:45.010
plus-- actually, let
me write it this way--

00:02:45.010 --> 00:02:51.490
plus the partial of z
with respect to y dy dt.

00:02:51.490 --> 00:02:54.830
And then we're going to
multiply everything times our k.

00:03:00.140 --> 00:03:04.340
So with this out of the way,
if we wanted what dr is,

00:03:04.340 --> 00:03:09.410
dr is just going to be
this whole thing times dt.

00:03:09.410 --> 00:03:10.780
So let's do that.

00:03:10.780 --> 00:03:14.130
So now we can rewrite our
line integral right over here.

00:03:14.130 --> 00:03:16.260
And we're going to now
go into the t domain.

00:03:16.260 --> 00:03:20.820
And so t is going to
go between a and b.

00:03:20.820 --> 00:03:23.820
And F dot dr--
remember, F's components

00:03:23.820 --> 00:03:28.760
were just the functions P,
Q, and R. And each of those

00:03:28.760 --> 00:03:30.540
were functions of x, y, and z.

00:03:30.540 --> 00:03:32.490
And z is a function
of x and y, so we'll

00:03:32.490 --> 00:03:34.490
have to think about all
of that in a little bit.

00:03:34.490 --> 00:03:38.010
Use a little more of our
multivariable chain rule.

00:03:38.010 --> 00:03:39.700
But when we take
the dot products,

00:03:39.700 --> 00:03:41.460
we're just going to take
the corresponding components

00:03:41.460 --> 00:03:42.290
and multiply them.

00:03:42.290 --> 00:03:47.050
So it is going to be-- actually,
let me just copy and paste it.

00:03:47.050 --> 00:03:50.439
Let me rewrite it down here.

00:03:50.439 --> 00:03:51.855
Our vector field
F-- and I'm going

00:03:51.855 --> 00:03:53.230
to write it a
little bit shorter.

00:03:53.230 --> 00:04:00.970
Our vector field F is P times i
plus Q times j plus R times k.

00:04:00.970 --> 00:04:04.570
So when we take our dot
product of F dot dr,

00:04:04.570 --> 00:04:07.120
we're essentially taking the
dot product of this and this.

00:04:07.120 --> 00:04:09.590
And we have to throw
a dt at the end.

00:04:09.590 --> 00:04:24.260
So we're going to get P
times dx dt plus Q times dy

00:04:24.260 --> 00:04:35.160
dt plus R times all of
this business over here,

00:04:35.160 --> 00:04:37.370
which is the partial
of z with respect

00:04:37.370 --> 00:04:46.130
to x dx dt plus the partial
of z with respect to y dy dt.

00:04:46.130 --> 00:04:49.240
And then we have to multiply
all of this times dt.

00:04:49.240 --> 00:04:51.060
We can't forget that
part right over there.

00:04:51.060 --> 00:04:57.002
So we're going to multiply all
of this right here times dt.

00:04:57.002 --> 00:04:58.960
Now, I'm going to leave
you there in this video

00:04:58.960 --> 00:05:01.770
just because I'm afraid of
making careless mistakes.

00:05:01.770 --> 00:05:03.400
What we're going
to do now is we're

00:05:03.400 --> 00:05:05.780
going to rearrange
this whole thing,

00:05:05.780 --> 00:05:09.860
recognize that this is the same
thing as this right over here.

00:05:09.860 --> 00:05:13.370
Then that gives it a form that
we can apply Green's theorem

00:05:13.370 --> 00:05:15.870
using this boundary
right over here.

00:05:15.870 --> 00:05:18.660
And then when we do a little
bit more algebraic manipulation,

00:05:18.660 --> 00:05:22.590
we're going to see that this
thing simplifies to this thing

00:05:22.590 --> 00:05:24.740
right over here
and proves Stokes'

00:05:24.740 --> 00:05:27.830
theorem for our special case.


Stokes' theorem proof part 4

WEBVTT
Kind: captions
Language: en

00:00:00.500 --> 00:00:03.104
Let's continue with our
proof of Stokes' theorem.

00:00:03.104 --> 00:00:04.520
And this time we're
going to focus

00:00:04.520 --> 00:00:05.840
on the other side
of Stokes' theorem.

00:00:05.840 --> 00:00:07.410
We're going to try
to figure out what

00:00:07.410 --> 00:00:11.270
is the line integral over the
boundary C-- where this is C

00:00:11.270 --> 00:00:15.792
right over here, the boundary
of our surface of f dot dr.

00:00:15.792 --> 00:00:17.500
And what we're going
to see is that we're

00:00:17.500 --> 00:00:20.270
going to get the exact same
result as we have right up

00:00:20.270 --> 00:00:21.440
here.

00:00:21.440 --> 00:00:24.970
Before we do that, I'm going to
take a little bit of a detour

00:00:24.970 --> 00:00:26.270
to kind of build up to this.

00:00:26.270 --> 00:00:28.866
So let's just take this and
put it to the side for now.

00:00:28.866 --> 00:00:30.990
Actually, let me actually
just delete it right now.

00:00:30.990 --> 00:00:32.365
And what I'm
actually going to do

00:00:32.365 --> 00:00:34.980
is I'm going to focus
on this region down

00:00:34.980 --> 00:00:37.370
here-- this region
in the xy plane.

00:00:40.290 --> 00:00:42.970
This is path C, which is
the boundary of our surface.

00:00:42.970 --> 00:00:44.740
I'm going to focus
on the path that

00:00:44.740 --> 00:00:46.300
is the boundary of this region.

00:00:46.300 --> 00:00:48.720
This path that sits
in the xy plane.

00:00:51.310 --> 00:00:54.040
I will call that path C1.

00:00:54.040 --> 00:00:56.440
And so, one, we can think
about a parameterization

00:00:56.440 --> 00:00:58.630
of just that path
in the xy plane.

00:00:58.630 --> 00:01:03.620
We could say that C1
could be parameterized

00:01:03.620 --> 00:01:11.370
as x is equal to x of t and
y is also a function of t.

00:01:11.370 --> 00:01:13.340
And t is obviously
our parameter.

00:01:13.340 --> 00:01:17.590
And it can go between A and B.

00:01:17.590 --> 00:01:20.380
So maybe when t is equal
to A, it sits right here.

00:01:20.380 --> 00:01:22.710
And then, as t gets larger
and larger and larger,

00:01:22.710 --> 00:01:24.110
it goes all the way around.

00:01:24.110 --> 00:01:25.930
And eventually, when
t is equal to B,

00:01:25.930 --> 00:01:27.990
it gets to that
exact same point.

00:01:27.990 --> 00:01:30.057
So that's our
parameterization right there.

00:01:30.057 --> 00:01:32.640
And now, just to make the rest
of this proof a little bit more

00:01:32.640 --> 00:01:34.490
understandable, I'm
going to give you

00:01:34.490 --> 00:01:36.730
a little bit of a
review of something.

00:01:36.730 --> 00:01:41.270
Imagine that we have some vector
field G. And G, at minimum, is

00:01:41.270 --> 00:01:43.310
defined on the xy plane.

00:01:43.310 --> 00:01:44.960
And it could be
defined other places,

00:01:44.960 --> 00:01:57.549
but let's say that G is equal
to m of x, y i plus N of x, y j.

00:01:57.549 --> 00:01:58.590
And this is all a review.

00:01:58.590 --> 00:02:00.640
We've seen this a long time ago.

00:02:00.640 --> 00:02:03.280
What would be the line
integral over the path C1?

00:02:03.280 --> 00:02:06.270
Not C, but this path that
sits in the xy plane.

00:02:06.270 --> 00:02:10.100
What would be the line
integral over the path C1?

00:02:10.100 --> 00:02:12.810
I like to write that sometimes.

00:02:12.810 --> 00:02:15.230
And I'm using G so I don't
get confused with F--

00:02:15.230 --> 00:02:20.040
our original vector field--
of G, our vector field

00:02:20.040 --> 00:02:21.940
here, along that path.

00:02:21.940 --> 00:02:36.605
G dot dr. Well, dr is just going
to be equal to dxi plus dyj.

00:02:39.520 --> 00:02:42.070
So if you take the dot product
of these two things right

00:02:42.070 --> 00:02:46.440
over here, you're going to get
the line integral over our path

00:02:46.440 --> 00:02:46.940
C1.

00:02:46.940 --> 00:02:48.727
Remember, C1 is
this path down here.

00:02:48.727 --> 00:02:50.310
Let me do it in that
same color so you

00:02:50.310 --> 00:02:52.420
don't think I'm
changing colors on you.

00:02:52.420 --> 00:02:54.570
The line integral
over our path C1.

00:02:54.570 --> 00:02:56.130
But when you take
this dot product,

00:02:56.130 --> 00:02:58.740
you multiply the x
components and then add that

00:02:58.740 --> 00:03:00.300
to the product of
the y component.

00:03:00.300 --> 00:03:16.030
So you have m times
dx plus n times dy.

00:03:16.030 --> 00:03:20.880
I just took the dot product
of G and dr. n times dy.

00:03:20.880 --> 00:03:22.950
And when you evaluate
these things,

00:03:22.950 --> 00:03:28.700
one way to think about it
is that dx is the same thing

00:03:28.700 --> 00:03:32.140
as-- let me write it up here
in a different color-- dx

00:03:32.140 --> 00:03:35.250
is the same thing as the
derivative of x with respect

00:03:35.250 --> 00:03:36.690
to t, dt.

00:03:36.690 --> 00:03:39.530
And same logic for y. dy
is equal to the derivative

00:03:39.530 --> 00:03:42.130
of y with respect to t, dt.

00:03:42.130 --> 00:03:44.320
One way to think about
it, these dt's cancel out.

00:03:44.320 --> 00:03:45.956
And you are just left with dx.

00:03:45.956 --> 00:03:47.830
And this is an important
thing to think about

00:03:47.830 --> 00:03:50.020
because then this allows
us to take this line

00:03:50.020 --> 00:03:53.390
integral into the
domain of our parameter.

00:03:53.390 --> 00:03:59.770
So then, this will be
equal to the integral

00:03:59.770 --> 00:04:01.260
in the domain of our parameter.

00:04:01.260 --> 00:04:03.180
So now we are in the t domain.

00:04:03.180 --> 00:04:06.210
And t is going to
vary between a and b.

00:04:06.210 --> 00:04:09.200
We are in the t domain
between a and b.

00:04:09.200 --> 00:04:14.460
This is going to be equal to m
times-- instead of writing dx,

00:04:14.460 --> 00:04:16.690
I'm going to write dx/dt dt.

00:04:16.690 --> 00:04:20.810
So it's going to be dx--
let me write it this way.

00:04:20.810 --> 00:04:25.900
dx-- the derivative
of x with respect

00:04:25.900 --> 00:04:34.270
to t-- dt-- that's the
first expression-- plus n--

00:04:34.270 --> 00:04:38.080
and then, the exact same
thing-- times dy dt.

00:04:38.080 --> 00:04:40.470
n times dy dt.

00:04:48.600 --> 00:04:51.450
These are all
equivalent statements.

00:04:51.450 --> 00:04:54.120
Now, with all of that out
of the way-- and all of this

00:04:54.120 --> 00:04:57.190
is really just a reminder so
that the rest of this proof

00:04:57.190 --> 00:04:59.370
becomes a little bit intuitive.

00:04:59.370 --> 00:05:01.330
With that out of the
way, let's come up

00:05:01.330 --> 00:05:03.920
with the parameterization
for this path

00:05:03.920 --> 00:05:08.710
up here, for C.
Remember, we just

00:05:08.710 --> 00:05:10.290
did C1 down in the xy plane.

00:05:10.290 --> 00:05:11.860
Now we're going to
do C that sits up

00:05:11.860 --> 00:05:14.770
here that kind of rises
above the xy plane.

00:05:14.770 --> 00:05:19.340
Well, for C, the
parameterizations for x and y

00:05:19.340 --> 00:05:20.970
can still be the
exact same thing

00:05:20.970 --> 00:05:22.810
because the x and
y values are going

00:05:22.810 --> 00:05:24.820
to be the exact same thing.

00:05:24.820 --> 00:05:26.880
The x and y value there
is the exact same thing

00:05:26.880 --> 00:05:28.650
as the x and y value there.

00:05:28.650 --> 00:05:31.290
The only difference is we
now have a z component.

00:05:31.290 --> 00:05:32.980
We defined it way up here.

00:05:32.980 --> 00:05:37.720
Our z component is going to
be a function of x and y.

00:05:37.720 --> 00:05:42.200
It tells us how high to go.

00:05:42.200 --> 00:05:50.080
We can parameterize C as--
maybe I'll write it as a vector.

00:05:50.080 --> 00:05:52.730
So let me parameterize.

00:05:52.730 --> 00:05:55.980
So I'll write C as a vector.

00:05:55.980 --> 00:05:57.445
Actually, no.

00:05:57.445 --> 00:05:58.770
Let me write it this way.

00:05:58.770 --> 00:06:04.470
Let me write C. Let me
do that purple color.

00:06:04.470 --> 00:06:08.420
C, we can say is x
is x of t-- actually,

00:06:08.420 --> 00:06:10.450
let me write this as a vector.

00:06:10.450 --> 00:06:12.210
And I'll use a
vector r, not to be

00:06:12.210 --> 00:06:14.850
confused with this
r right over here.

00:06:14.850 --> 00:06:16.840
So these are two different
r's, but I'll just

00:06:16.840 --> 00:06:19.090
use r because that tends
to be the convention.

00:06:19.090 --> 00:06:21.450
So in order to
parameterize C, it's

00:06:21.450 --> 00:06:23.200
going to be the
position vector r, which

00:06:23.200 --> 00:06:24.930
is going to be a function of t.

00:06:24.930 --> 00:06:32.891
And x is still just going
to be x of t i plus y of tj.

00:06:32.891 --> 00:06:34.640
And now we're going
to have a z component.

00:06:34.640 --> 00:06:37.000
And z is going to be
a function of x and y,

00:06:37.000 --> 00:06:39.050
which are, in turn,
functions of t.

00:06:39.050 --> 00:06:41.890
So z is a function of x--
which is a function of t--

00:06:41.890 --> 00:06:46.190
and a function of y-- which is a
function of t k-- that tells us

00:06:46.190 --> 00:06:49.390
how high above to, essentially,
get each of those points.

00:06:49.390 --> 00:06:51.700
And then, once again, we
know that t is between a

00:06:51.700 --> 00:06:55.130
and b. t is greater than
or equal to a and less than

00:06:55.130 --> 00:06:56.160
or equal to b.

00:06:56.160 --> 00:06:58.530
So we have that
parameterization right there.

00:06:58.530 --> 00:07:02.110
And now we can start
to think about the line

00:07:02.110 --> 00:07:06.620
integral of f dot
dr along this path.

00:07:06.620 --> 00:07:08.340
Before, we did dr
along this path.

00:07:08.340 --> 00:07:11.510
Now we're going to do dr on
this path right over here.

00:07:11.510 --> 00:07:14.550
This is now our
parameterization r.

00:07:14.550 --> 00:07:18.860
So I'll leave you there, and
I'll see you in the next video.


Why a negative times a negative makes intuitive sense

WEBVTT
Kind: captions
Language: en

00:00:00.000 --> 00:00:03.181
So you, as the ancient philosopher in mathematics

00:00:03.181 --> 00:00:07.185
have concluded in order for the multiplication of positive and negative numbers to be

00:00:07.185 --> 00:00:10.177
consistent with everything you've been constructing so far

00:00:10.177 --> 00:00:13.502
with all the other properties of multiplication that you know so far

00:00:13.502 --> 00:00:17.178
that you need a negative number times a positive

00:00:17.178 --> 00:00:20.770
number or a positive times a negative to give you a negative number

00:00:20.770 --> 00:00:24.181
and a negative times a negative

00:00:24.181 --> 00:00:27.846
to give you a positive number and so you accept

00:00:27.846 --> 00:00:30.845
it's all consistent so far.. this deal does not make complete concrete

00:00:30.845 --> 00:00:35.770
sense to you, you want to have a slightly deeper institution than just having to accept its

00:00:35.770 --> 00:00:40.369
consistent with the distributive property and whatever else and so you try another

00:00:40.369 --> 00:00:45.377
thought experiment, you say "well what is just a basic multiplication way of doing it?"

00:00:45.377 --> 00:00:47.369
So if I say, two times

00:00:47.369 --> 00:00:51.183
three, one way to

00:00:51.183 --> 00:00:55.103
to conceptualize is basic multiplication is really repeating

00:00:55.103 --> 00:00:58.036
addition, so you could view this as two threes

00:00:58.036 --> 00:01:02.351
so let me write three plus three

00:01:02.351 --> 00:01:05.703
and notice there are two of them, there are two of these

00:01:05.703 --> 00:01:09.770
or you could view this as three twos, and so this is the same thing as

00:01:09.770 --> 00:01:13.434
two plus two plus two and there are

00:01:13.434 --> 00:01:17.100
three of them, and either way you can conceptualize

00:01:17.100 --> 00:01:20.602
as you get the same exact answer. This is going to be equal

00:01:20.602 --> 00:01:24.597
to six, fair enough!

00:01:24.597 --> 00:01:27.569
Now, you knew this before you even tried to tackle negative numbers.

00:01:27.569 --> 00:01:30.518
Now let's try to make one of these negatives and see what

00:01:30.518 --> 00:01:33.369
happens. Let's do two

00:01:33.369 --> 00:01:35.516
times negative

00:01:35.516 --> 00:01:42.103
three, I want to make the negative into a different color. Two times

00:01:42.103 --> 00:01:46.309
negative three.

00:01:46.309 --> 00:01:49.851
Well, one way you could view this is the same analogy

00:01:49.851 --> 00:01:52.853
here, it's negative three twice so it would be

00:01:52.853 --> 00:01:56.767
negative.. I'll try to color code it

00:01:56.767 --> 00:02:01.042
negative three and then another negative

00:02:01.042 --> 00:02:05.180
three or you could say negative three minus three

00:02:05.180 --> 00:02:08.568
or, and this is the interesting thing, instead of

00:02:08.568 --> 00:02:11.036
over here there's a two times positive three

00:02:11.036 --> 00:02:14.310
you added two, three times.

00:02:14.310 --> 00:02:16.262
But since here is two times negative three

00:02:16.262 --> 00:02:19.036
you could also imagine you are going to subtract two, three times

00:02:19.036 --> 00:02:21.700
So instead of up here, I could

00:02:21.700 --> 00:02:26.519
written two plus two plus two because this is a positive

00:02:26.519 --> 00:02:29.436
two right over here, but since we're doing this over negative three

00:02:29.436 --> 00:02:33.702
we could imagine subtracting two, three times, so this would be

00:02:33.702 --> 00:02:37.930
subtracting two (repeated)

00:02:37.930 --> 00:02:43.186
subtract another two right over here, subtract another two

00:02:43.186 --> 00:02:46.102
and then you subtract another two

00:02:46.102 --> 00:02:54.936
notice you did it, once again, you did it

00:02:54.936 --> 00:02:59.932
three times, so this is a negative three, so essentially you are subtracting

00:02:59.932 --> 00:03:03.770
two, three times. And either way, you can conceptualize

00:03:03.770 --> 00:03:07.264
right over here, you are going to get negative six

00:03:07.264 --> 00:03:10.186
negative six is the answer.

00:03:10.186 --> 00:03:16.264
Now, so you are already starting to feel better about this part right over here

00:03:16.264 --> 00:03:18.369
negative times a positive, or a positive times a negative

00:03:18.369 --> 00:03:21.600
is going to give you a negative. Now lets take to the really un-intuitive one

00:03:21.600 --> 00:03:24.685
and measure negative times a negative, and all of a sudden negatives kind of cancel

00:03:24.685 --> 00:03:28.037
to give you a positive. Now why is that the case? Well we can just build from

00:03:28.037 --> 00:03:30.969
this example right over here. Let's say we had

00:03:30.969 --> 00:03:35.934
a negative two, lets say we had

00:03:35.934 --> 00:03:38.103
negative two, let me do it a different color,

00:03:38.103 --> 00:03:42.851
let's say we had a negative two, I already used this color

00:03:42.851 --> 00:03:45.239
negative two times

00:03:45.239 --> 00:03:48.702
negative three.

00:03:48.702 --> 00:03:53.975
So now, we can d- actually I'll do this one first.

00:03:53.975 --> 00:03:57.775
Let's do multiplying something by negative three so we'll

00:03:57.775 --> 00:04:01.234
repeatedly subtract that thing three times whatever that thing is

00:04:01.234 --> 00:04:05.769
so now the thing isn't a positive two so the thing over

00:04:05.769 --> 00:04:08.636
here is a positive two but the thing we're going to subtract is a negative two

00:04:08.636 --> 00:04:10.902
So let me make it clear, this says we are going to subtract something

00:04:10.902 --> 00:04:13.969
three times, so we subtract something three times, so

00:04:13.969 --> 00:04:17.104
subtracting something (repeatedly) three times

00:04:17.104 --> 00:04:20.642
That's what this part right over here tells us

00:04:20.642 --> 00:04:24.302
and we'll do this, exactly three times

00:04:24.302 --> 00:04:28.369
Over here, it was a positive two we subtracted three times, now we're going to

00:04:28.369 --> 00:04:32.271
do a negative two, now we're going to do a negative two

00:04:32.271 --> 00:04:35.770
and we know from subtracting negative numbers, we already

00:04:35.770 --> 00:04:39.434
built this intuition that subtracting a negative is the same thing

00:04:40.368 --> 00:04:46.105
it's the same thing as adding a positive, and so this

00:04:46.105 --> 00:04:50.370
this is going to be the same thing as two plus two plus two and

00:04:50.370 --> 00:04:53.606
we're told once again, gives you a positive

00:04:53.606 --> 00:04:56.901
six, you can same use the same logic over here, now

00:04:56.901 --> 00:05:00.302
instead of adding negative three twice, really I could have written this as

00:05:00.302 --> 00:05:03.854
negative three as this example

00:05:03.854 --> 00:05:05.569
negative three

00:05:05.569 --> 00:05:11.523
negative three, and we added it

00:05:11.523 --> 00:05:15.299
we added it, now let me put a plus here to make it clear

00:05:15.299 --> 00:05:18.602
over here we added it twice, we added negative three

00:05:18.602 --> 00:05:23.349
two times, or here since we have a negative two, we're going to subtract

00:05:23.349 --> 00:05:26.184
to negative three twice, so we're going to subtract something

00:05:26.184 --> 00:05:30.102
and we're going to subtract something again, and that something is going to be

00:05:30.102 --> 00:05:33.440
our negative three, it's going to be our negative three, so

00:05:33.440 --> 00:05:37.036
negative, negative and put our three right over here

00:05:37.036 --> 00:05:41.241
and once again, subtracting negative three is like taking away

00:05:41.241 --> 00:05:43.182
someone's debt, which is essentially giving them money,

00:05:43.182 --> 00:05:48.274
this is the same thing as adding three plus three which is once again six. So now

00:05:48.274 --> 00:05:51.439
you, the ancient philosopher, feel pretty good. Not only this

00:05:51.439 --> 00:05:55.105
all consistent with all the mathematics you know

00:05:55.105 --> 00:05:58.369
the distributive property is also the property of multiplying something

00:05:58.369 --> 00:06:00.767
times something all these things you already know, and now

00:06:00.767 --> 00:06:04.772
this actually makes conceptual sense to you, this is actually very consistent with

00:06:04.772 --> 00:06:08.108
with your notion, your original notion, or one of the possible notions

00:06:08.108 --> 00:06:12.108
of multiplication which is as repeated addition


Why a negative times a negative is a positive

WEBVTT
Kind: captions
Language: en

00:00:00.300 --> 00:00:06.635
Lets say you are an Ancient Philosopher who was building up mathematics who was building mathematics from the ground up

00:00:06.635 --> 00:00:13.783
And you already have a reasonable of what a negative number could or should represent and you know how to add and subtract negative numbers

00:00:13.783 --> 00:00:15.767
But now you are faced with a conundrum

00:00:15.767 --> 00:00:18.501
What happens when you multiply negative numbers?

00:00:18.501 --> 00:00:21.233
Either when you multiply a positive number times a negative number

00:00:21.233 --> 00:00:23.900
Or when you multiply two negative numbers

00:00:23.900 --> 00:00:25.167
So, for example

00:00:25.167 --> 00:00:32.502
You aren't quite sure what should happen if you were to multiply (and im just picking two numbers where one is positive and one is negative)

00:00:32.502 --> 00:00:37.081
What would happen if you were to multiply 5 times negative 3

00:00:37.081 --> 00:00:39.773
You're not quite sure about this just yet

00:00:39.773 --> 00:00:43.702
You're also not quite sure what would happen if you multiply two negative numbers.

00:00:43.702 --> 00:00:48.669
So lets say negative two times negative 6

00:00:48.669 --> 00:00:50.835
This is also unclear to you

00:00:50.835 --> 00:00:56.569
What you do know, because you are a mathematician, is however you define this or whatever this should be

00:00:56.569 --> 00:01:02.410
It should hopefully be consistant with all of the other properties of mathematics that you already know

00:01:02.410 --> 00:01:05.233
And preferably all of the other properties of multiplication

00:01:05.233 --> 00:01:08.233
That would make you feel comfortable that you are getting this right.

00:01:08.233 --> 00:01:11.233
and later we can think about other ways to get the intuition

00:01:11.233 --> 00:01:14.608
for what these might be allowed you to actually make sense

00:01:14.608 --> 00:01:18.034
but to make this make consistent with the rest of mathematics

00:01:18.034 --> 00:01:21.300
that you know, you go into a little bit of a thought experiment

00:01:21.300 --> 00:01:24.700
you say, well, what should five times

00:01:24.700 --> 00:01:28.524
three plus negative three equal

00:01:28.524 --> 00:01:32.681
well you already have a philosophy of adding negative numbers

00:01:32.681 --> 00:01:35.551
or adding positive numbers to negative numbers, you know negative three is

00:01:35.551 --> 00:01:39.298
the opposite of three, but you add three to negative three

00:01:39.298 --> 00:01:42.423
you're going to get zero, so this is going to be equal to

00:01:42.423 --> 00:01:45.520
five times zero

00:01:45.520 --> 00:01:52.102
based on how you already thought about adding a negative number

00:01:52.102 --> 00:01:54.369
to a positive, and anything times zero is going to be

00:01:54.369 --> 00:01:57.900
zero, so this expression right over here should be zero

00:01:57.900 --> 00:02:02.283
but you see, I want to multiply positive and negative

00:02:02.283 --> 00:02:04.697
numbers to be consistent with this distributive property so

00:02:04.697 --> 00:02:07.367
I should should be able to distribute this five

00:02:07.367 --> 00:02:12.943
and for math to be consistent, and math should be consistent, I should

00:02:12.943 --> 00:02:16.548
get the exact same answer, so let's distribute this five so

00:02:16.548 --> 00:02:18.103
we get five times three

00:02:18.103 --> 00:02:23.417
is going to write out as five times three

00:02:23.417 --> 00:02:27.020
let me write this multiplication sign, not this dot

00:02:27.020 --> 00:02:31.434
five times three, so i distributed there

00:02:31.434 --> 00:02:35.925
plus five times negative three

00:02:35.925 --> 00:02:39.358
i'll do that in yellow, five times negative three

00:02:39.358 --> 00:02:45.843
and this whole thing we just said should be equal to zero

00:02:45.843 --> 00:02:50.700
it should be equal to zero, well five times three

00:02:50.700 --> 00:02:54.843
those are two positive numbers, we should know what should should be, that is going to be fifteen

00:02:54.843 --> 00:02:57.825
now we get this thing, fifteen plus

00:02:57.825 --> 00:03:01.506
times whatever five times negative three is

00:03:01.506 --> 00:03:09.433
needs to be equal to zero in order to be consistent with all the other mathematics

00:03:09.433 --> 00:03:12.806
that we know, well what plus fifteen is going to

00:03:12.806 --> 00:03:16.498
be equal to zero, well the opposite of fifteen in order for this

00:03:16.498 --> 00:03:20.000
to be true, in order for this to be consistent with all the other mathematics we know

00:03:20.000 --> 00:03:23.588
this right over here needs to be equal to

00:03:23.588 --> 00:03:27.690
negative fifteen until you say five times negative three

00:03:27.690 --> 00:03:31.170
in order to be consistent with all other mathematics we know, needs to

00:03:31.170 --> 00:03:34.433
be equal to negative fifteen. That's also consistent

00:03:34.433 --> 00:03:38.251
with the intuition of adding negative three repeatedly

00:03:38.251 --> 00:03:41.302
five times, now look above above us

00:03:41.302 --> 00:03:44.769
slightly higher so you can see ideas of multiplying

00:03:44.769 --> 00:03:48.272
two negatives, but we can do the exact same product experiment.

00:03:48.272 --> 00:03:52.033
We want whatever this answer to be consistent with the rest of mathematics

00:03:52.033 --> 00:03:53.769
that we know so we can

00:03:53.769 --> 00:03:58.347
do the same product experiment. What would negative two times

00:03:58.347 --> 00:04:02.035
six plus negative six to be equal to.

00:04:02.035 --> 00:04:05.012
Well, six plus negative six is going to be zero.

00:04:05.012 --> 00:04:09.145
Negative two times zero, anything times zero, needs to be equal to

00:04:09.145 --> 00:04:12.367
zero, but then once again, we can distribute

00:04:12.367 --> 00:04:16.300
negative two times six so we get

00:04:16.300 --> 00:04:23.621
negative two times six, then plus negative two times

00:04:23.621 --> 00:04:27.102
negative six plus negative two

00:04:27.102 --> 00:04:30.767
times negative six, then once again all of this is going to be

00:04:30.767 --> 00:04:34.634
equal to zero, now based on the five experiment we just

00:04:34.634 --> 00:04:37.569
did, we said "well this needs to be equal to negative twelve"

00:04:37.569 --> 00:04:41.415
or we can view this as going to the six twice

00:04:41.415 --> 00:04:44.501
left direction on the number line which gets us to negative twelve

00:04:44.501 --> 00:04:48.433
or you could say repeatedly adding negative twos

00:04:48.433 --> 00:04:52.003
times six would also get you to negative twelve and now

00:04:52.003 --> 00:04:55.147
we also saw over here we want to multiply a positive and a negative

00:04:55.147 --> 00:04:58.769
we got the negative so

00:04:58.769 --> 00:05:02.501
this could be, you know, going to be equal to negative twelve

00:05:02.501 --> 00:05:05.648
so we have negative twelve plus

00:05:05.648 --> 00:05:08.234
whatever this business is

00:05:08.234 --> 00:05:13.833
going to have to be equal to zero (repeated)

00:05:13.833 --> 00:05:17.369
in order to be consistent with all the other mathematics that we know

00:05:17.369 --> 00:05:21.621
and so what plus negative twelve is going to equal to be zero

00:05:21.621 --> 00:05:25.101
Well, positive twelve plus negative twelve is going to equal to zero

00:05:25.101 --> 00:05:27.835
so this needs to be equal to positive twelve in

00:05:27.835 --> 00:05:31.564
order to be consistent with all the other mathematics we know

00:05:31.564 --> 00:05:35.609
so there we get the idea that this is going positive

00:05:35.609 --> 00:05:39.153
twelve. I'll leave you there and I'll see

00:05:39.153 --> 00:05:42.569
if I can make a few other videos that can also give you a conceptual understanding

00:05:42.569 --> 00:05:46.569
of why these are true