0:00:00.000,0:00:00.500


0:00:00.500,0:00:01.950
In the last video,[br]we figured out

0:00:01.950,0:00:05.590
how to construct a unit[br]normal vector to a surface.

0:00:05.590,0:00:08.800
And so now we can use that[br]back in our original surface

0:00:08.800,0:00:10.750
integral to try to[br]simplify a little bit,

0:00:10.750,0:00:13.320
or at least give us a clue how[br]we can calculate these things.

0:00:13.320,0:00:14.820
And also, think[br]about different ways

0:00:14.820,0:00:17.950
to represent this type[br]of a surface integral.

0:00:17.950,0:00:20.710
So if we just substitute what[br]we came up as our normal vector,

0:00:20.710,0:00:22.190
our unit normal[br]vector right here,

0:00:22.190,0:00:26.120
we will get-- so[br]once again, it's

0:00:26.120,0:00:31.540
the surface integral of F dot.

0:00:31.540,0:00:33.499
And F dot all of this[br]business right over here.

0:00:33.499,0:00:35.123
And I'm going to[br]write it all in white,

0:00:35.123,0:00:36.900
just so it doesn't[br]take me too much time.

0:00:36.900,0:00:38.930
So the partial of[br]r with respect to u

0:00:38.930,0:00:41.940
crossed with the partial[br]of r with respect

0:00:41.940,0:00:46.280
to v over the magnitude[br]of the same thing, partial

0:00:46.280,0:00:49.560
of r with respect to u[br]crossed with the partial of r

0:00:49.560,0:00:51.770
with respect to v.

0:00:51.770,0:00:53.709
And now, we've[br]played with ds a lot.

0:00:53.709,0:00:55.375
We know that the other[br]way to write ds--

0:00:55.375,0:00:56.874
and I gave the[br]intuition, hopefully,

0:00:56.874,0:00:58.840
for that several videos[br]ago when we first

0:00:58.840,0:01:01.120
explored what a surface[br]integral was all about.

0:01:01.120,0:01:05.770
We know that ds is--[br]it can be represented

0:01:05.770,0:01:08.420
as the magnitude of the[br]partial of r with respect

0:01:08.420,0:01:13.501
to u crossed with the partial[br]of r with respect to v du dv.

0:01:13.501,0:01:17.020
And Obviously, the du dv, it[br]could be written as dv du.

0:01:17.020,0:01:20.410
You could write it as da,[br]a little chunk of area

0:01:20.410,0:01:23.310
and the uv plane or[br]in the uv domain.

0:01:23.310,0:01:26.010
And actually, since now this[br]integral's in terms of uv,

0:01:26.010,0:01:27.760
we're no longer taking[br]a surface integral.

0:01:27.760,0:01:30.880
We're now taking a double[br]integral over the uv domain.

0:01:30.880,0:01:33.150
So you could say kind[br]of a region in uv.

0:01:33.150,0:01:38.100
So I'll say R to say that's[br]it's a region in the uv plane

0:01:38.100,0:01:39.550
that we're now thinking about.

0:01:39.550,0:01:41.770
But there's probably a[br]huge-- or there should be,

0:01:41.770,0:01:43.720
or I'm guessing there's a[br]huge simplification that's

0:01:43.720,0:01:44.928
popping out at you right now.

0:01:44.928,0:01:47.130
We're dividing by the[br]magnitude of the cross product

0:01:47.130,0:01:48.588
of these two vectors[br]and then we're

0:01:48.588,0:01:52.040
multiplying by the magnitude of[br]the cross product of these two

0:01:52.040,0:01:52.695
vectors.

0:01:52.695,0:01:54.070
Those are just[br]scalar quantities.

0:01:54.070,0:01:56.140
You divide by something[br]and multiply by something.

0:01:56.140,0:01:58.598
Well, that's just the same[br]thing as multiplying or dividing

0:01:58.598,0:01:59.390
by 1.

0:01:59.390,0:02:01.420
So these two[br]characters cancel out,

0:02:01.420,0:02:05.930
and our integral simplifies[br]to the double integral

0:02:05.930,0:02:09.250
over that region, the[br]corresponding region in the uv

0:02:09.250,0:02:14.200
plane, of F-- of[br]our vector field F

0:02:14.200,0:02:16.880
dotted with this cross product.

0:02:16.880,0:02:19.130
This is going to give us[br]a vector right over here.

0:02:19.130,0:02:20.740
That's going to[br]give us a vector.

0:02:20.740,0:02:22.282
It gives us actually[br]a normal vector.

0:02:22.282,0:02:24.031
And then when you[br]divide by its magnitude,

0:02:24.031,0:02:25.520
it gives you a[br]unit normal vector.

0:02:25.520,0:02:28.780
So this, you're going to[br]take the dot product of F

0:02:28.780,0:02:32.700
with r, the partial[br]of r with respect

0:02:32.700,0:02:41.170
to u crossed with the partial[br]of r with respect to v du dv.

0:02:41.170,0:02:45.550
Let me scroll over to the[br]right a little bit, du dv.

0:02:45.550,0:02:47.590
And we'll see in the[br]few videos from now

0:02:47.590,0:02:50.270
that this is essentially how we[br]go about actually calculating

0:02:50.270,0:02:50.860
these things.

0:02:50.860,0:02:52.390
If you have a[br]parameterization, you

0:02:52.390,0:02:54.640
can then get everything in[br]terms of a double integral,

0:02:54.640,0:02:56.509
in terms of uv this way.

0:02:56.509,0:02:58.800
Now, the last thing I want[br]to do is explore another way

0:02:58.800,0:03:01.050
that you'll see a surface[br]integral like this written.

0:03:01.050,0:03:03.379
It all comes from,[br]really, writing this part

0:03:03.379,0:03:04.170
in a different way.

0:03:04.170,0:03:05.630
But it hopefully gives you[br]a little bit more intuition

0:03:05.630,0:03:07.700
of what this thing[br]is even saying.

0:03:07.700,0:03:09.240
So I'm just going to rewrite.

0:03:09.240,0:03:12.570
I'm going to rewrite this[br]chunk right over here.

0:03:12.570,0:03:14.122
I'm just going to[br]rewrite that chunk.

0:03:14.122,0:03:16.580
And I'm going to use slightly[br]different notation because it

0:03:16.580,0:03:19.030
will hopefully help make[br]a little bit more sense.

0:03:19.030,0:03:21.100
So the partial of r[br]with respect to u I

0:03:21.100,0:03:25.930
can write as the partial[br]of r with respect to u.

0:03:25.930,0:03:27.600
And we're taking[br]the cross product.

0:03:27.600,0:03:30.300
Let me make my u's a little bit[br]more u-like so we confuse them

0:03:30.300,0:03:31.190
with v's.

0:03:31.190,0:03:32.940
And we're taking the[br]cross product of that

0:03:32.940,0:03:36.400
with the partial[br]of r with respect

0:03:36.400,0:03:40.130
to v. So very small[br]changes in our vector--

0:03:40.130,0:03:41.930
in our parameterization[br]right here,

0:03:41.930,0:03:44.100
our position vector given[br]a small change in v. Very

0:03:44.100,0:03:47.850
small changes in the vector[br]given a small change in u.

0:03:47.850,0:03:50.670
And then we're multiplying[br]that times du dv.

0:03:50.670,0:03:56.240
We're multiplying[br]that times du dv.

0:03:56.240,0:03:59.750
Now, du and dv are[br]just scalar quantities.

0:03:59.750,0:04:01.290
They're infinitesimally small.

0:04:01.290,0:04:03.084
But for the sake[br]of this argument,

0:04:03.084,0:04:04.750
you can just view--[br]they're not vectors,

0:04:04.750,0:04:06.500
they're just scalar quantities.

0:04:06.500,0:04:09.730
And so you can[br]essentially include them--

0:04:09.730,0:04:11.530
if you have the cross product.

0:04:11.530,0:04:18.470
If you have a cross b times some[br]scalar value-- I don't know, x,

0:04:18.470,0:04:23.610
you could rewrite this[br]as x times a cross b,

0:04:23.610,0:04:28.530
or you could write[br]this as a cross x times

0:04:28.530,0:04:30.700
b, because x is[br]just a scalar value.

0:04:30.700,0:04:31.540
It's just a number.

0:04:31.540,0:04:33.206
So we could do the[br]same thing over here.

0:04:33.206,0:04:35.650
We can rewrite all[br]of this business as--

0:04:35.650,0:04:39.630
and I'm going to group the du[br]where we have the partial--

0:04:39.630,0:04:41.300
or with respect to u[br]in the denominator.

0:04:41.300,0:04:42.966
And I'll do the same[br]thing with the v's.

0:04:42.966,0:04:48.430
And so you will get the[br]partial of r with respect

0:04:48.430,0:04:52.440
to u times du,[br]times that scalar.

0:04:52.440,0:04:54.240
So that'll give us a vector.

0:04:54.240,0:04:55.750
And we're going to cross that.

0:04:55.750,0:04:58.530
We're going to cross that with[br]the partial of r with respect

0:04:58.530,0:05:06.650
to v dv.

0:05:06.650,0:05:08.814
Now, these might[br]look notationally

0:05:08.814,0:05:10.480
like two different[br]things, but that just

0:05:10.480,0:05:12.470
comes from the[br]necessity of when we

0:05:12.470,0:05:16.780
take partial derivatives to say,[br]oh, no, this vector function

0:05:16.780,0:05:18.887
is defined-- it's a function[br]of multiple variables

0:05:18.887,0:05:20.720
and this is taking a[br]derivative with respect

0:05:20.720,0:05:22.010
to only one of them.

0:05:22.010,0:05:23.510
So this is, how[br]much does our vector

0:05:23.510,0:05:26.330
change when you have a[br]very small change in u?

0:05:26.330,0:05:28.880
But this is also an[br]infinitesimally small change

0:05:28.880,0:05:31.950
in u over here, we're just using[br]slightly different notation.

0:05:31.950,0:05:33.280
So for the sake of--[br]and this is a little bit

0:05:33.280,0:05:35.280
loosey-goosey mathematics,[br]but it will hopefully

0:05:35.280,0:05:37.860
give you the intuition[br]for why this thing could

0:05:37.860,0:05:39.790
be written in a different way.

0:05:39.790,0:05:42.290
These are essentially[br]the same quantity.

0:05:42.290,0:05:44.770
So if you divide by something[br]and multiply by something,

0:05:44.770,0:05:46.072
you can cancel them out.

0:05:46.072,0:05:48.280
If you divide by something[br]and multiply by something,

0:05:48.280,0:05:49.840
you can cancel them out.

0:05:49.840,0:05:52.070
And all you're left[br]with then-- all

0:05:52.070,0:05:55.670
you're left with is[br]the differential of r.

0:05:55.670,0:05:57.080
And since we lost[br]the information

0:05:57.080,0:05:58.496
that it's in the[br]u-direction, I'll

0:05:58.496,0:06:00.830
write here, the differential[br]of r in the u-direction.

0:06:00.830,0:06:02.891
I don't want to get[br]the notation confused.

0:06:02.891,0:06:04.140
This is just the differential.

0:06:04.140,0:06:06.000
This is just how much r changed.

0:06:06.000,0:06:08.820
This is not the partial[br]derivative of r with respect

0:06:08.820,0:06:09.455
to u.

0:06:09.455,0:06:11.350
This right over here[br]is, how much does r

0:06:11.350,0:06:15.320
change given per unit change,[br]per small change in u?

0:06:15.320,0:06:19.190
This just says a differential[br]in the direction of-- I

0:06:19.190,0:06:22.530
guess as u changes, this is[br]how much that infinitely small

0:06:22.530,0:06:24.060
change that just r changes.

0:06:24.060,0:06:27.460
This isn't change in r with[br]respect to change in u.

0:06:27.460,0:06:28.710
And we're going to cross that.

0:06:28.710,0:06:31.775
Now, we're going to cross[br]that with the partial of r,

0:06:31.775,0:06:34.900
the partial of r[br]in the v-direction.

0:06:34.900,0:06:37.280
Now, this right over here,[br]let's just conceptualize this.

0:06:37.280,0:06:39.920
And this goes back to[br]our original visions

0:06:39.920,0:06:42.330
of what a surface[br]integral was all about.

0:06:42.330,0:06:44.314
If we're on a surface--[br]and I'll draw surface.

0:06:44.314,0:06:45.480
Let me draw another surface.

0:06:45.480,0:06:47.970
I won't use the one that[br]I had already drawn on.

0:06:47.970,0:06:51.077
If we draw a surface, and for[br]a very small change in u--

0:06:51.077,0:06:52.910
and we're not going to[br]think about the rate.

0:06:52.910,0:06:56.220
We're just thinking about[br]kind of the change in r.

0:06:56.220,0:06:57.790
You're going in that direction.

0:06:57.790,0:07:01.220
So if that thing[br]looks like this,

0:07:01.220,0:07:05.310
this is actually kind of a[br]distance moved on the surface.

0:07:05.310,0:07:08.060
Because remember, this[br]isn't the derivative.

0:07:08.060,0:07:09.350
This is the differential.

0:07:09.350,0:07:11.840
So it's just a small[br]change along the surface,

0:07:11.840,0:07:13.510
that's that over there.

0:07:13.510,0:07:16.710
And that this is a small[br]change when you change v. So

0:07:16.710,0:07:19.050
it's also a change[br]along the surface.

0:07:19.050,0:07:21.310
When you take the cross[br]product of these two things,

0:07:21.310,0:07:23.430
you get a vector[br]that is orthogonal.

0:07:23.430,0:07:26.570
You get a vector that is[br]normal to the surface.

0:07:26.570,0:07:33.740
So it is normal to the[br]surface and its magnitude--

0:07:33.740,0:07:36.340
and we saw this when we first[br]learned about cross products.

0:07:36.340,0:07:40.080
Its magnitude is[br]equal to the area that

0:07:40.080,0:07:42.680
is defined by these two vectors.

0:07:42.680,0:07:49.302
So its magnitude[br]is equal to area.

0:07:49.302,0:07:50.760
So in a lot of[br]ways, you can really

0:07:50.760,0:07:52.420
think of it-- you[br]really could think

0:07:52.420,0:07:57.020
of it as a unit normal[br]vector times ds.

0:07:57.020,0:07:59.960
And so the way that we would,[br]I guess notationally do this,

0:07:59.960,0:08:02.470
is we can call this--[br]because this is kind of a ds,

0:08:02.470,0:08:04.045
but it's a vector[br]version of the ds.

0:08:04.045,0:08:06.760
Over here, this is just[br]an area right over here.

0:08:06.760,0:08:08.830
This is just a scalar value.

0:08:08.830,0:08:12.250
But now, we have a vector[br]that points normally

0:08:12.250,0:08:14.930
from the surface,[br]but its magnitude

0:08:14.930,0:08:17.300
is the same thing as[br]that ds that we were just

0:08:17.300,0:08:18.340
talking about.

0:08:18.340,0:08:22.070
So we can call this thing right[br]over here, we can call this ds.

0:08:22.070,0:08:25.140
And the key difference here[br]is this is a vector now.

0:08:25.140,0:08:28.240
So we'll call it ds with[br]a little vector over it

0:08:28.240,0:08:29.570
to know that this thing.

0:08:29.570,0:08:33.200
This isn't the scalar ds that[br]is just concerned with the area.

0:08:33.200,0:08:34.970
But when you view[br]things this way,

0:08:34.970,0:08:39.289
we just saw that this entire[br]thing simplifies to ds.

0:08:39.289,0:08:43.419
Then our whole surface[br]integral can be rewritten.

0:08:43.419,0:08:45.200
Instead of writing[br]it like this, we

0:08:45.200,0:08:50.480
can write it as the[br]integral or the surface

0:08:50.480,0:08:54.050
integral-- those integral[br]signs were too fancy.

0:08:54.050,0:08:57.794
The surface integral of F dot.

0:08:57.794,0:09:00.750
And instead of saying a[br]normal vector times the scalar

0:09:00.750,0:09:03.040
quantity, that little chunk[br]of area on the surface,

0:09:03.040,0:09:07.731
we can now just call that[br]the vector differential ds.

0:09:07.731,0:09:10.230
And I want to make it clear,[br]these are two different things.

0:09:10.230,0:09:12.280
This is a vector.

0:09:12.280,0:09:14.130
This is essentially[br]what we're calling it.

0:09:14.130,0:09:16.655
This right over here is a[br]scalar times a normal vector.

0:09:16.655,0:09:18.030
So these are three[br]different ways

0:09:18.030,0:09:19.800
of really representing[br]the same thing.

0:09:19.800,0:09:22.175
And in different contexts,[br]you will see different things,

0:09:22.175,0:09:25.110
depending on what the author of[br]whoever's trying to communicate

0:09:25.110,0:09:26.820
is trying to communicate.

0:09:26.820,0:09:29.990
This right over here is the one[br]that we'll use most frequently

0:09:29.990,0:09:34.192
as we actually try to calculate[br]these surface integrals.

0:09:34.192,0:09:34.692