1
00:00:00,000 --> 00:00:00,680


2
00:00:00,680 --> 00:00:03,240
So where we left off I was attempting to give you the

3
00:00:03,240 --> 00:00:05,419
intuition of divergence and then I ran out of time.

4
00:00:05,419 --> 00:00:09,330
But anyway, I had defined this fairly straightforward vector

5
00:00:09,330 --> 00:00:12,460
field that tells us the velocity of particles in a

6
00:00:12,460 --> 00:00:13,755
fluid at any given point.

7
00:00:13,755 --> 00:00:15,580
And let me clean it up a little bit.

8
00:00:15,580 --> 00:00:17,899
This one half I had all these scratch offs.

9
00:00:17,899 --> 00:00:23,099
The velocity, I'm just going to rewrite it, is equal to 1/2xi.

10
00:00:23,100 --> 00:00:25,750
So at any given point has no y component.

11
00:00:25,750 --> 00:00:28,719
So all of the velocity is only in the x direction -- there is

12
00:00:28,719 --> 00:00:31,339
no upwards movement in the xy plane.

13
00:00:31,339 --> 00:00:32,530
And I was drawing it out.

14
00:00:32,530 --> 00:00:35,359
I said OK, when x is equal to 1, the magnitude of the

15
00:00:35,359 --> 00:00:39,689
velocity is 1/2 maybe meters per second, if that's our unit.

16
00:00:39,689 --> 00:00:42,429
When x is equal to 2, the velocity to the right will

17
00:00:42,429 --> 00:00:45,250
be 1 meter per second, right -- 1/2 times 2.

18
00:00:45,250 --> 00:00:48,579
So the further we go to the right, or the more we go to the

19
00:00:48,579 --> 00:00:52,729
right, the faster the particles are moving to the right.

20
00:00:52,729 --> 00:00:54,229
So now let's try to get our handle on what

21
00:00:54,229 --> 00:00:55,129
divergence means.

22
00:00:55,130 --> 00:00:58,250
So first of all, let's take that the divergence

23
00:00:58,250 --> 00:01:01,200
of this function.

24
00:01:01,200 --> 00:01:09,549
So that divergence of v, of our velocity vector field -- you

25
00:01:09,549 --> 00:01:12,859
could also view that if you want to abuse some notation,

26
00:01:12,859 --> 00:01:18,340
is our del vector, dot v.

27
00:01:18,340 --> 00:01:19,939
But if we only have one dimension, so it's the

28
00:01:19,939 --> 00:01:26,390
partial derivative of the x magnitude with respect to x.

29
00:01:26,390 --> 00:01:27,400
So what's the partial derivative?

30
00:01:27,400 --> 00:01:30,469
So it's equal to the partial derivative with

31
00:01:30,469 --> 00:01:31,859
respect to x, of 1/2x.

32
00:01:31,859 --> 00:01:34,540


33
00:01:34,540 --> 00:01:36,460
So it's equal to -- well the derivative of this with respect

34
00:01:36,459 --> 00:01:38,140
to x is just equal to 1/2 .

35
00:01:38,140 --> 00:01:40,000
So that divergence of this vector field at

36
00:01:40,000 --> 00:01:42,299
any point is 1/2.

37
00:01:42,299 --> 00:01:44,599
Now what does that tell us?

38
00:01:44,599 --> 00:01:48,030
Well, if you just look at the definition, right, we

39
00:01:48,030 --> 00:01:51,460
essentially just took the -- how much does the magnitude

40
00:01:51,459 --> 00:01:53,689
of the field increase in the x direction?

41
00:01:53,689 --> 00:01:54,530
And we see it visually.

42
00:01:54,530 --> 00:01:57,549
As we go, increase in the x direction, the field gets

43
00:01:57,549 --> 00:01:58,509
stronger and stronger.

44
00:01:58,510 --> 00:02:01,130
Or since we know that this is the velocity of particles, as

45
00:02:01,129 --> 00:02:03,909
we go in the x direction, the particles go faster and

46
00:02:03,909 --> 00:02:05,229
faster to the right.

47
00:02:05,230 --> 00:02:07,689
Now what this tells us, what this positive divergence tells

48
00:02:07,689 --> 00:02:09,609
us is if we were to take -- let's just take an

49
00:02:09,610 --> 00:02:12,500
arbitrarily small circle.

50
00:02:12,500 --> 00:02:15,310
I think it'll start to make sense once I draw the circle.

51
00:02:15,310 --> 00:02:17,849
If I take an arbitrarily -- I'm going to draw it in a different

52
00:02:17,849 --> 00:02:20,620
color -- and this circle could be arbitrarily small, but I'm

53
00:02:20,620 --> 00:02:24,430
drawing it pretty large so it can include some of our

54
00:02:24,430 --> 00:02:26,409
vectors that I've drawn.

55
00:02:26,409 --> 00:02:27,030
What's happening?

56
00:02:27,030 --> 00:02:30,599


57
00:02:30,599 --> 00:02:33,639
On the right hand side, I have particles exiting really,

58
00:02:33,639 --> 00:02:34,639
really fast, right?

59
00:02:34,639 --> 00:02:41,659
And let's say in a given amount of time, let's say in one

60
00:02:41,659 --> 00:02:44,419
second, in one second out of the right side, since the

61
00:02:44,419 --> 00:02:46,449
particles are moving really fast, I'm going to have a

62
00:02:46,449 --> 00:02:49,629
bunch of particles leave the right hand side, right?

63
00:02:49,629 --> 00:02:51,819
And in the same amount of time, I will have some particles

64
00:02:51,819 --> 00:02:53,859
come in through the left hand side, but it's going to be a

65
00:02:53,860 --> 00:02:55,470
fewer number of particles.

66
00:02:55,469 --> 00:02:58,289
So the way you could think about it is in any given amount

67
00:02:58,289 --> 00:03:00,289
of time, what's happening?

68
00:03:00,289 --> 00:03:03,060
In this space, I have a few particles entering in through

69
00:03:03,060 --> 00:03:05,990
the left, and I have a much larger number of particle

70
00:03:05,990 --> 00:03:07,500
leaving through the right.

71
00:03:07,500 --> 00:03:10,110
So what's going to happen in this space?

72
00:03:10,110 --> 00:03:12,660
It's going to become less dense, right?

73
00:03:12,659 --> 00:03:14,849
Because in that space is going to be fewer particles after

74
00:03:14,849 --> 00:03:15,750
a certain amount of time.

75
00:03:15,750 --> 00:03:18,199
More are leaving than are coming in.

76
00:03:18,199 --> 00:03:22,699
So this positive divergence tells us that at that point, or

77
00:03:22,699 --> 00:03:24,479
really at any point in this vector field since the

78
00:03:24,479 --> 00:03:28,639
divergence is 1/2 everywhere, at any point in this vector

79
00:03:28,639 --> 00:03:31,439
field, the field is becoming less dense.

80
00:03:31,439 --> 00:03:34,099
Or you could say that more is flowing out of any

81
00:03:34,099 --> 00:03:36,629
point than flowing in.

82
00:03:36,629 --> 00:03:37,530
It makes sense, right?

83
00:03:37,530 --> 00:03:39,990
Because if as we move to the right, and it kind of gets

84
00:03:39,990 --> 00:03:43,950
funky if you go into the other quadrant, so we'll stick to the

85
00:03:43,949 --> 00:03:46,319
first quadrant while we're trying to get our intuition.

86
00:03:46,319 --> 00:03:49,139
But it makes sense, because as we move to the right

87
00:03:49,139 --> 00:03:52,239
our particles are getting faster and faster.

88
00:03:52,240 --> 00:03:56,629
And that kind of just falls out of the fact that our derivative

89
00:03:56,629 --> 00:03:58,389
with respect to x is positive.

90
00:03:58,389 --> 00:04:04,519
The slope of how much our x component is

91
00:04:04,520 --> 00:04:05,659
increasing is positive.

92
00:04:05,659 --> 00:04:09,189
So as we go to the right, our velocities are going getting

93
00:04:09,189 --> 00:04:11,270
faster and faster, which means if we were to draw a circle

94
00:04:11,270 --> 00:04:15,210
anywhere, we're always going to have more exiting the right

95
00:04:15,210 --> 00:04:17,420
than entering through the left.

96
00:04:17,420 --> 00:04:19,750
So we're going to be getting less dense at any given point.

97
00:04:19,750 --> 00:04:23,500
Or you could almost view it as any given point is almost a

98
00:04:23,500 --> 00:04:26,790
source of particles, or if you have a sphere, more particles

99
00:04:26,790 --> 00:04:29,000
are going to be coming out of the sphere through the right,

100
00:04:29,000 --> 00:04:31,069
than coming in through the sphere to the left.

101
00:04:31,069 --> 00:04:34,959
So you could view a positive divergence as you could kind of

102
00:04:34,959 --> 00:04:40,129
say well, the field is becoming less dense at that point, or

103
00:04:40,129 --> 00:04:45,269
the point is a source of the field, or it's a source of

104
00:04:45,269 --> 00:04:49,129
particles, depending on what model you want to use.

105
00:04:49,129 --> 00:04:51,956
Now, with that said, let's take the opposite situation.

106
00:04:51,956 --> 00:04:57,200


107
00:04:57,199 --> 00:05:00,649
Let's say that the vector field is equal to is

108
00:05:00,649 --> 00:05:06,169
minus 1/2x times i.

109
00:05:06,170 --> 00:05:10,120
And so the divergence -- I'll use this notation -- the

110
00:05:10,120 --> 00:05:13,189
divergence of our vector field is just a partial derivative

111
00:05:13,189 --> 00:05:16,569
with respect to x, which is just minus 1/2.

112
00:05:16,569 --> 00:05:21,242
If I were to graph it -- this is my y-axis,

113
00:05:21,242 --> 00:05:22,170
this is my x-axis.

114
00:05:22,170 --> 00:05:29,250


115
00:05:29,250 --> 00:05:32,620
So here at like, say, the point 1, my velocity is

116
00:05:32,620 --> 00:05:35,319
going to be the left 1/2.

117
00:05:35,319 --> 00:05:39,389
At the point 2, my velocity is going to be the left

118
00:05:39,389 --> 00:05:42,079
1 meter per second.

119
00:05:42,079 --> 00:05:47,959
At the velocity 3, it's going to be 3/2.

120
00:05:47,959 --> 00:05:49,930
You know it doesn't depend on y.

121
00:05:49,930 --> 00:05:50,660
It only depends on x.

122
00:05:50,660 --> 00:05:53,770


123
00:05:53,769 --> 00:05:56,829
So now let's draw a little circle and see

124
00:05:56,829 --> 00:05:58,050
what's happening.

125
00:05:58,050 --> 00:05:58,944
Let's draw it here.

126
00:05:58,944 --> 00:05:59,819
It could be anywhere.

127
00:05:59,819 --> 00:06:01,540
It's infinitely small, but we're just trying

128
00:06:01,540 --> 00:06:02,840
to get some intuition.

129
00:06:02,839 --> 00:06:05,609
So after a certain amount of time what's happening?

130
00:06:05,610 --> 00:06:06,850
Let's say after a second.

131
00:06:06,850 --> 00:06:10,520
Well, I'm having a few particles leave through the

132
00:06:10,519 --> 00:06:13,349
left hand side, right, but I have many more particles

133
00:06:13,350 --> 00:06:15,790
entering this little region that I've defined, this little

134
00:06:15,790 --> 00:06:18,129
circle, I'm having many more particles enter through the

135
00:06:18,129 --> 00:06:19,949
right in a given amount of time.

136
00:06:19,949 --> 00:06:24,689
So in any given amount of time, in my defined space, it's going

137
00:06:24,689 --> 00:06:25,639
to get denser and denser.

138
00:06:25,639 --> 00:06:27,180
There's going to be more and more particles in

139
00:06:27,180 --> 00:06:30,180
that space over time.

140
00:06:30,180 --> 00:06:33,689
So it's getting denser or you could almost view it as this

141
00:06:33,689 --> 00:06:35,125
space is sucking up particles.

142
00:06:35,125 --> 00:06:37,920


143
00:06:37,920 --> 00:06:40,580
In the previous example it was a source of particles -- more

144
00:06:40,579 --> 00:06:42,419
were coming out than going in.

145
00:06:42,420 --> 00:06:46,170
Now more going in through the right than coming out.

146
00:06:46,170 --> 00:06:48,569
And that's what a negative divergence.

147
00:06:48,569 --> 00:06:52,000
You could almost say -- let's think about the

148
00:06:52,000 --> 00:06:53,139
word, divergence.

149
00:06:53,139 --> 00:06:55,519
When it's positive, if I have a positive divergence, the

150
00:06:55,519 --> 00:06:58,879
particles or the field is diverting out of that point.

151
00:06:58,879 --> 00:07:01,240
If I have a negative divergence -- maybe

152
00:07:01,240 --> 00:07:02,139
let's define a new term.

153
00:07:02,139 --> 00:07:04,209
I've never actually heard it this way, but maybe a negative

154
00:07:04,209 --> 00:07:06,339
divergence we view as a convergence, right?

155
00:07:06,339 --> 00:07:08,500
Converge is the opposite of diverge.

156
00:07:08,500 --> 00:07:11,370
So here, even though some particles are leaving through

157
00:07:11,370 --> 00:07:14,480
the left, many more particles are coming through the

158
00:07:14,480 --> 00:07:17,340
right, so it's getting denser and denser.

159
00:07:17,339 --> 00:07:19,039
And that's this example here.

160
00:07:19,040 --> 00:07:21,470
And actually at every point in this field we have

161
00:07:21,470 --> 00:07:22,470
a negative divergence.

162
00:07:22,470 --> 00:07:25,660
So every point is getting denser and denser actually

163
00:07:25,660 --> 00:07:27,370
everywhere in this field.

164
00:07:27,370 --> 00:07:29,879
And then the classic example of a divergence, although I wanted

165
00:07:29,879 --> 00:07:32,750
to show you that what matters is the net that's coming

166
00:07:32,750 --> 00:07:34,060
in to a certain area.

167
00:07:34,060 --> 00:07:36,790
But the classic example of a divergence is a field that

168
00:07:36,790 --> 00:07:38,860
looks something like this.

169
00:07:38,860 --> 00:07:41,955
Where maybe that's the x -- that's the y, this is the x.

170
00:07:41,954 --> 00:07:45,279


171
00:07:45,279 --> 00:07:50,149
If you have a field that looks something like this, this is

172
00:07:50,149 --> 00:07:55,169
the classical example of a negative divergence, right?

173
00:07:55,170 --> 00:07:57,210
Where from every direction you have particles entering,

174
00:07:57,209 --> 00:07:58,019
nothing's leaving.

175
00:07:58,019 --> 00:08:01,979
So obviously, in any given amount of time, that point is

176
00:08:01,980 --> 00:08:04,140
getting more and more dense.

177
00:08:04,139 --> 00:08:12,120
And the classic example of a positive divergence is a point

178
00:08:12,120 --> 00:08:16,970
where from every direction things are leaving it.

179
00:08:16,970 --> 00:08:20,160
So clearly this area is going to become less dense.

180
00:08:20,160 --> 00:08:23,660
If we're talking about velocity of particles, after any moment

181
00:08:23,660 --> 00:08:26,689
in time, more particles are leaving than coming in because

182
00:08:26,689 --> 00:08:29,500
no particles are coming in.

183
00:08:29,500 --> 00:08:36,580
Now what does it mean if we have a 0 divergence?

184
00:08:36,580 --> 00:08:43,500
So let's try to create a vector field that has a 0 divergence.

185
00:08:43,500 --> 00:08:45,210
And we'll just stay at a one-dimension just

186
00:08:45,210 --> 00:08:45,889
for the intuition.

187
00:08:45,889 --> 00:08:47,269
So that means that the partial derivative with

188
00:08:47,269 --> 00:08:48,960
respect to x is 0.

189
00:08:48,960 --> 00:08:51,879
So let's say my vector field is 5i.

190
00:08:51,879 --> 00:08:54,350
So the magnitude is always 5 in the i direction.

191
00:08:54,350 --> 00:08:55,420
So let me draw that.

192
00:08:55,419 --> 00:09:02,490


193
00:09:02,490 --> 00:09:04,659
Vector field is always 5.

194
00:09:04,659 --> 00:09:07,169


195
00:09:07,169 --> 00:09:08,750
Another way to think of it if you have a constant

196
00:09:08,750 --> 00:09:10,580
vector field.

197
00:09:10,580 --> 00:09:13,230
So the magnitude of the vectors, no matter what

198
00:09:13,230 --> 00:09:17,139
my value of x, is always going to be the same.

199
00:09:17,139 --> 00:09:18,330
It's always going to be 5.

200
00:09:18,330 --> 00:09:30,509


201
00:09:30,509 --> 00:09:36,590
So if I were to draw a region, what's happening here?

202
00:09:36,590 --> 00:09:39,149
Are more particles entering than leaving or

203
00:09:39,149 --> 00:09:40,009
leaving than entering?

204
00:09:40,009 --> 00:09:40,269
No.

205
00:09:40,269 --> 00:09:43,220
For any amount that's coming in, an equal amount are coming

206
00:09:43,220 --> 00:09:45,820
out in a certain amount of time, if we use velocity

207
00:09:45,820 --> 00:09:47,010
as our example.

208
00:09:47,009 --> 00:09:49,960
So when you have a divergence of 0, that means that that part

209
00:09:49,960 --> 00:09:54,950
of the field is not becoming any more or less dense.

210
00:09:54,950 --> 00:09:56,930
And you could have done it -- let me show you another.

211
00:09:56,929 --> 00:10:03,579
If my function was, let's say it equals 2i plus 2j.

212
00:10:03,580 --> 00:10:04,910
It's still a constant, right?

213
00:10:04,909 --> 00:10:08,370
So this velocity field or vector field will look

214
00:10:08,370 --> 00:10:10,269
something like this.

215
00:10:10,269 --> 00:10:14,019
All the points would be, the vectors would

216
00:10:14,019 --> 00:10:15,049
have a slope of 1.

217
00:10:15,049 --> 00:10:17,769
But I just wanted you to see something in two dimensions.

218
00:10:17,769 --> 00:10:21,679
I'll do a fancier example in the next video.

219
00:10:21,679 --> 00:10:25,269
But even here, if I were to draw some region, the same

220
00:10:25,269 --> 00:10:27,649
amount is entering as exiting.

221
00:10:27,649 --> 00:10:29,689
So it's not getting any denser at any point.

222
00:10:29,690 --> 00:10:33,510
And that makes sense because the divergence of this vector

223
00:10:33,509 --> 00:10:36,980
field -- well, both of them actually, the divergence

224
00:10:36,980 --> 00:10:37,820
of that vector field.

225
00:10:37,820 --> 00:10:39,685
The partial derivative of 2 with respect to

226
00:10:39,684 --> 00:10:41,319
x, well that's 0.

227
00:10:41,320 --> 00:10:44,120
Plus the partial derivative of 2 with respect to y.

228
00:10:44,120 --> 00:10:46,500
Well, that's also 0.

229
00:10:46,500 --> 00:10:48,179
Anyway, I've run out of time again.

230
00:10:48,179 --> 00:10:51,169
I will see you in the next video.

231
00:10:51,169 --> 00:10:51,399