1
00:00:00,000 --> 00:00:00,870


2
00:00:00,870 --> 00:00:03,490
Now let's do a slightly fancier
example, then we'll try to

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00:00:03,490 --> 00:00:04,610
analyze the vector field.

4
00:00:04,610 --> 00:00:07,780
And hopefully this will
make everything a little

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00:00:07,780 --> 00:00:08,580
bit more tangible.

6
00:00:08,580 --> 00:00:12,710
So let's say that the velocity
of the fluid, or the particles

7
00:00:12,710 --> 00:00:18,150
in the fluid, at any given
point in the x-y plane, let's

8
00:00:18,150 --> 00:00:28,490
say in the x-direction, it is x
squared, x squared minus 3x

9
00:00:28,490 --> 00:00:38,170
plus 2 in the x-direction, plus
y squared minus 3y plus

10
00:00:38,170 --> 00:00:39,770
2 in the y-direction.

11
00:00:39,770 --> 00:00:44,250
Make it simple so that we only
have one thing to factor.

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00:00:44,250 --> 00:00:45,910
So let's just do
the math first.

13
00:00:45,910 --> 00:00:51,730
Let's figure out the divergence
of our vector field, the

14
00:00:51,730 --> 00:00:54,050
divergence of our field.

15
00:00:54,050 --> 00:00:56,280
And going to show you a graph
of this field soon, so we'll

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00:00:56,280 --> 00:00:58,270
get an intuition of what it
actually looks like, instead of

17
00:00:58,270 --> 00:01:01,870
my not-so-accurate drawings.

18
00:01:01,870 --> 00:01:02,880
So what's the divergence?

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00:01:02,880 --> 00:01:05,200
We take the partial derivative
of the x-component

20
00:01:05,200 --> 00:01:06,940
with respect to x.

21
00:01:06,940 --> 00:01:09,430
So that's just, there's only an
x-variable here, so we don't

22
00:01:09,430 --> 00:01:11,645
really have to worry about
keeping y or z constant.

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00:01:11,645 --> 00:01:13,650
It's really just a derivative
of this expression

24
00:01:13,650 --> 00:01:15,150
with respect to x.

25
00:01:15,150 --> 00:01:19,820
So it's 2x minus 3.

26
00:01:19,820 --> 00:01:22,520
And then we add that to the
partial derivative of the

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00:01:22,520 --> 00:01:25,290
y-component, or the y-function,
with respect to y.

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00:01:25,290 --> 00:01:27,800
There's only y's in the
y-component, so we just

29
00:01:27,800 --> 00:01:29,000
take the derivative
with respect to y.

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00:01:29,000 --> 00:01:34,910
So it's plus 2y minus 3.

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00:01:34,910 --> 00:01:39,300
Or we could just say that the
divergence of v, at any point

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00:01:39,300 --> 00:01:48,120
xy, so this is a function of x
and y, is 2x plus 2y minus 3.

33
00:01:48,120 --> 00:01:52,750
Now before I show you the
graph, let's analyze this

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00:01:52,750 --> 00:01:54,010
function a little bit.

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00:01:54,010 --> 00:01:56,270
First of all, let's just look
at the original vector field,

36
00:01:56,270 --> 00:02:01,750
and think about when does that
vector field have some

37
00:02:01,750 --> 00:02:03,400
interesting points?

38
00:02:03,400 --> 00:02:06,120
Well, I think some interesting
points are when either the x-

39
00:02:06,120 --> 00:02:08,020
or the y-components
are equal to 0.

40
00:02:08,020 --> 00:02:10,590
So when is the
x-component equal to 0?

41
00:02:10,590 --> 00:02:13,480
Well, if we factor the
x-component, that's the

42
00:02:13,480 --> 00:02:17,420
same thing as, we could
rewrite our vector field.

43
00:02:17,420 --> 00:02:23,010
If we just factored that,
that's x minus 1 times x minus

44
00:02:23,010 --> 00:02:29,910
2i plus, and it's the same
polynomial, just with y, for

45
00:02:29,910 --> 00:02:38,500
the y-component, so y minus
1 times y minus 2 times j.

46
00:02:38,500 --> 00:02:43,130
So the x-component is 0 when x
is equal to 1, these are just

47
00:02:43,130 --> 00:02:45,060
the roots of this polynomial,
when x is equal to

48
00:02:45,060 --> 00:02:47,050
1 or 2, right?

49
00:02:47,050 --> 00:02:51,980
And the y-component is 0
when y is equal to 1 or 2.

50
00:02:51,980 --> 00:02:54,790
And they're both equal to 0
if we have any combination

51
00:02:54,790 --> 00:02:55,420
of these points.

52
00:02:55,420 --> 00:03:02,240
So the points where they're
both equal to 0 are 1, 1, x's

53
00:03:02,240 --> 00:03:04,690
is 1, y is 2, right, because
then both components

54
00:03:04,690 --> 00:03:10,340
are 0, 2, 1, or 2, 2.

55
00:03:10,340 --> 00:03:15,090
So these are the points where
the magnitude of the velocity

56
00:03:15,090 --> 00:03:17,490
of our fluid, or the particles
of the fluid, are 0.

57
00:03:17,490 --> 00:03:20,200
And we'll see that on
our graph in a second.

58
00:03:20,200 --> 00:03:21,670
And let me ask
another question.

59
00:03:21,670 --> 00:03:26,960
At what points are, well, let's
first decide, what point is

60
00:03:26,960 --> 00:03:29,290
the divergence equal to 0?

61
00:03:29,290 --> 00:03:31,500
Let's say, at what point is
a divergence equal to 0.

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00:03:31,500 --> 00:03:34,460
Let me click clear
up some space.

63
00:03:34,460 --> 00:03:35,570
I think I can delete this.

64
00:03:35,570 --> 00:03:40,140
We got our points, we figured
out what coordinates are the

65
00:03:40,140 --> 00:03:43,072
magnitude of the
vector field 0.

66
00:03:43,072 --> 00:03:47,790
So let's try to figure out,
when is the divergence

67
00:03:47,790 --> 00:03:49,870
equal to 0?

68
00:03:49,870 --> 00:03:50,940
So this is divergence.

69
00:03:50,940 --> 00:03:57,610
So if we set that equal
to 0, 2x plus 2y, 2x

70
00:03:57,610 --> 00:03:59,250
plus 2y, oh, sorry.

71
00:03:59,250 --> 00:04:01,330
You know what, this is 2x
minus 3 plus 2y minus 3.

72
00:04:01,330 --> 00:04:03,640
So this is minus 6, right?

73
00:04:03,640 --> 00:04:06,480
Minus 3 minus 3,
that's minus 6.

74
00:04:06,480 --> 00:04:09,880
That's my major flaw,
adding and subtracting.

75
00:04:09,880 --> 00:04:11,560
Anyway, so the divergence.

76
00:04:11,560 --> 00:04:14,710
2x plus 2y, minus 6.

77
00:04:14,710 --> 00:04:16,860
And we want to know,
when does that equal 0?

78
00:04:16,860 --> 00:04:19,950
So let's set it equal to 0.

79
00:04:19,950 --> 00:04:22,090
And we can simplify
this a little bit.

80
00:04:22,090 --> 00:04:25,760
We can divide both sides of the
equation by 2, and you get x

81
00:04:25,760 --> 00:04:28,940
plus y minus 3 is equal to 0.

82
00:04:28,940 --> 00:04:32,640
You get x plus y is equal to 3.

83
00:04:32,640 --> 00:04:35,190
We could be finished there, or
we could just put it in our

84
00:04:35,190 --> 00:04:39,690
traditional mx plus b form,
that's the way I find it

85
00:04:39,690 --> 00:04:41,330
easier to visualize a line.

86
00:04:41,330 --> 00:04:45,930
We could say y is
equal to 3 minus x.

87
00:04:45,930 --> 00:04:50,960
So along this line, the
divergence of the vector fields

88
00:04:50,960 --> 00:04:55,830
b is equal to 0, along the line
y is equal to 3 minus x.

89
00:04:55,830 --> 00:04:58,990
And if we're above that line,
the divergence is going to be

90
00:04:58,990 --> 00:05:02,210
positive, right, because if you
just made this a greater than

91
00:05:02,210 --> 00:05:03,520
sign, that would carry over.

92
00:05:03,520 --> 00:05:05,890
You'd have y is
greater than 3-x.

93
00:05:05,890 --> 00:05:13,640
So y greater than 3-x, the
divergence is positive.

94
00:05:13,640 --> 00:05:19,420
And y is less than 3 minus x,
the divergence is negative.

95
00:05:19,420 --> 00:05:21,880
And you could just make this a
less than sign then solve, and

96
00:05:21,880 --> 00:05:24,310
you'll get y is less than 3-x,
you want to know when the

97
00:05:24,310 --> 00:05:25,640
divergence is negative.

98
00:05:25,640 --> 00:05:28,510
So I think we've done all of
the analyzing we can do.

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00:05:28,510 --> 00:05:31,020
So let's take a look at the
graph and see if it if it's

100
00:05:31,020 --> 00:05:33,760
consistent with our intuition
of what a divergence is,

101
00:05:33,760 --> 00:05:34,755
and the numbers we found.

102
00:05:34,755 --> 00:05:37,480


103
00:05:37,480 --> 00:05:39,280
I hope you can see this.

104
00:05:39,280 --> 00:05:41,450
So this is the vector field.

105
00:05:41,450 --> 00:05:45,630
I don't have space to show it,
but I think you remember, this

106
00:05:45,630 --> 00:05:47,620
is, you know, x squared
minus 3x plus 2.

107
00:05:47,620 --> 00:05:50,370
This is the definition
of our vector field.

108
00:05:50,370 --> 00:05:51,590
Have it graphed here.

109
00:05:51,590 --> 00:05:55,040
And we figured out, we just
figured out when the

110
00:05:55,040 --> 00:05:57,130
x-components and the
y-components are equal to 0,

111
00:05:57,130 --> 00:05:58,980
and then we said, when are
both of them equal to 0?

112
00:05:58,980 --> 00:06:01,370
And we said, oh, well
at the point 1, 1.

113
00:06:01,370 --> 00:06:04,290
Well, this is the point 1, 1,
and we that the magnitudes of

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00:06:04,290 --> 00:06:06,210
the vectors are 0
at that point.

115
00:06:06,210 --> 00:06:09,600
And actually, I could
zoom in a little bit.

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00:06:09,600 --> 00:06:11,750
Right around here, they're all
pointing inwards,, but they get

117
00:06:11,750 --> 00:06:16,130
smaller and smaller as you
approach the point 1, 1, right?

118
00:06:16,130 --> 00:06:19,690
We also said at the point
1, 2. x is 1, y is 2.

119
00:06:19,690 --> 00:06:22,290
And here, too, we see that
the magnitude of the vectors

120
00:06:22,290 --> 00:06:24,680
get very, very, very small.

121
00:06:24,680 --> 00:06:26,480
We could zoom in again.

122
00:06:26,480 --> 00:06:29,020
And we see, the magnitude
gets very small.

123
00:06:29,020 --> 00:06:31,460
The other point, 2, 1, once
again, we see the magnitude

124
00:06:31,460 --> 00:06:33,060
get small, and then 2, 2.

125
00:06:33,060 --> 00:06:35,590
So that's consistent with what
we found out, that the vector

126
00:06:35,590 --> 00:06:37,980
field gets very small
at this point.

127
00:06:37,980 --> 00:06:39,900
And the other interesting
thing we said, OK, when does

128
00:06:39,900 --> 00:06:41,500
the divergence equal 0?

129
00:06:41,500 --> 00:06:46,160
Well, the divergence equaled
0 along the line y is

130
00:06:46,160 --> 00:06:48,800
equal to 3 minus x.

131
00:06:48,800 --> 00:06:52,250
So the line y is equal to 3
minus x starts, the y-intercept

132
00:06:52,250 --> 00:06:55,640
is going to be 3, and it's
going to come down like this.

133
00:06:55,640 --> 00:06:56,020
Right?

134
00:06:56,020 --> 00:06:58,730
So anything along, at any
point along that line,

135
00:06:58,730 --> 00:07:00,270
the divergence is 0.

136
00:07:00,270 --> 00:07:03,310
And if we actually look at
the graph, it makes sense.

137
00:07:03,310 --> 00:07:06,570
Because I can't draw on this
graph, but if we drew a circle

138
00:07:06,570 --> 00:07:11,450
right there, let's assume that
that's on that, the line y is

139
00:07:11,450 --> 00:07:14,920
equal to 3 minus x, we would
see that in a given amount of

140
00:07:14,920 --> 00:07:18,000
time, just as many particles
are entering through the top

141
00:07:18,000 --> 00:07:20,990
right as leaving through
the bottom left, right?

142
00:07:20,990 --> 00:07:23,670
A lot of entering through the
top right and a lot of leaving

143
00:07:23,670 --> 00:07:24,340
through the bottom left.

144
00:07:24,340 --> 00:07:26,740
And the vectors, though,
are about the same.

145
00:07:26,740 --> 00:07:32,030
And if we go to the bottom, if
we go here on the line, it

146
00:07:32,030 --> 00:07:35,860
looks like maybe there are less
entering, but there's

147
00:07:35,860 --> 00:07:36,970
also less leaving.

148
00:07:36,970 --> 00:07:39,520
I know it's hard to see, but
anywhere along the line,

149
00:07:39,520 --> 00:07:41,450
you see just as much
entering as leaving.

150
00:07:41,450 --> 00:07:45,240
And that's why the
divergence is 0.

151
00:07:45,240 --> 00:07:47,650
Now, let's look at some
of the other points.

152
00:07:47,650 --> 00:07:50,630
Up here, we figure the
diverge is positive.

153
00:07:50,630 --> 00:07:51,740
And does that make
sense?

154
00:07:51,740 --> 00:07:53,310
Let's pick an arbitrary point.

155
00:07:53,310 --> 00:07:57,150
If we were to draw a circle
around here, we see the vectors

156
00:07:57,150 --> 00:08:00,080
on the left-hand side of that
circle that you can't see,

157
00:08:00,080 --> 00:08:02,050
because I can't draw
on this graph.

158
00:08:02,050 --> 00:08:04,790
But actually, let's
just say the square.

159
00:08:04,790 --> 00:08:07,690
Let's say the square
is my region, right?

160
00:08:07,690 --> 00:08:09,210
This one right here.

161
00:08:09,210 --> 00:08:12,320
If that square is my region,
we see the vectors on the

162
00:08:12,320 --> 00:08:15,980
left-hand side are larger, than
factors leaving are larger than

163
00:08:15,980 --> 00:08:18,830
the vectors entering it, right?

164
00:08:18,830 --> 00:08:22,540
So if, in a given amount of
time, more is leaving than

165
00:08:22,540 --> 00:08:27,180
entering, then I'm becoming
less dense, or you could say

166
00:08:27,180 --> 00:08:28,860
that the particles
are diverging.

167
00:08:28,860 --> 00:08:32,070
And that makes sense, because
I have a positive divergence.

168
00:08:32,070 --> 00:08:35,210
And if we go here, where the
divergence is negative, let's

169
00:08:35,210 --> 00:08:36,880
pick an arbitrary spot.

170
00:08:36,880 --> 00:08:39,600
Let's say this
square right here.

171
00:08:39,600 --> 00:08:43,710
We see that the vectors
entering it, the magnitude of

172
00:08:43,710 --> 00:08:47,470
the vectors entering it, are
larger than the magnitude of

173
00:08:47,470 --> 00:08:48,800
the vectors exiting it.

174
00:08:48,800 --> 00:08:51,390
So in any given amount of time,
more is coming in than leaving.

175
00:08:51,390 --> 00:08:53,850
So it's getting denser,
or it's converging.

176
00:08:53,850 --> 00:08:56,600
So negative divergence, you can
view it as getting denser,

177
00:08:56,600 --> 00:08:58,610
or it's actually converging.

178
00:08:58,610 --> 00:09:00,770
Actually, something
interesting is happening

179
00:09:00,770 --> 00:09:04,240
at these two points.

180
00:09:04,240 --> 00:09:09,220
So we said that at the point
2, 1, we see that in the

181
00:09:09,220 --> 00:09:13,580
y-direction it's actually
converging, right?

182
00:09:13,580 --> 00:09:18,030
Above y is equal to 1, the
arrows are pointing down,

183
00:09:18,030 --> 00:09:20,630
and below it the arrows
are pointing up, right?

184
00:09:20,630 --> 00:09:23,310
So in the y-direction, we're
actually converging, or we

185
00:09:23,310 --> 00:09:24,490
have a negative divergence.

186
00:09:24,490 --> 00:09:26,990
Things are entering
any given spot.

187
00:09:26,990 --> 00:09:30,690
But in the x-direction, things
are getting pushed out, right?

188
00:09:30,690 --> 00:09:33,890
So the reason why the
divergence is 0 here, you might

189
00:09:33,890 --> 00:09:38,020
have particles entering above
and below a certain space, but

190
00:09:38,020 --> 00:09:41,090
you have just as many particles
exiting to the left

191
00:09:41,090 --> 00:09:41,650
and the right.

192
00:09:41,650 --> 00:09:44,160
So it's kind of like particles
are getting deflected out.

193
00:09:44,160 --> 00:09:48,710
So on a net basis, between both
dimensions, you have no

194
00:09:48,710 --> 00:09:52,030
increase or decrease in density
along that line y is

195
00:09:52,030 --> 00:09:53,680
equal to 3 minus x.

196
00:09:53,680 --> 00:09:56,180
And before I run out of time, I
just want to give you that core

197
00:09:56,180 --> 00:10:00,880
intuition again, of why the
divergence is positive, and why

198
00:10:00,880 --> 00:10:04,930
that means that things are
flowing out, when the rate

199
00:10:04,930 --> 00:10:06,300
of change is positive.

200
00:10:06,300 --> 00:10:07,590
So we said the
divergence is positive.

201
00:10:07,590 --> 00:10:09,420
Let's say at the spot, right?

202
00:10:09,420 --> 00:10:12,350
So it makes sense, if our
partial derivatives are

203
00:10:12,350 --> 00:10:15,710
positive, that means that the
magnitude of our vector is

204
00:10:15,710 --> 00:10:18,690
getting larger and larger for
larger values of our

205
00:10:18,690 --> 00:10:20,510
x's and y's, right?

206
00:10:20,510 --> 00:10:22,760
So if the magnitude of our
vectors are getting larger and

207
00:10:22,760 --> 00:10:26,120
larger for larger values of x
and y, the vectors on the

208
00:10:26,120 --> 00:10:28,310
right are going to have
a larger magnitude than

209
00:10:28,310 --> 00:10:29,290
the factors on the left.

210
00:10:29,290 --> 00:10:31,190
They're increasing
in magnitude.

211
00:10:31,190 --> 00:10:34,390
And so if I were to draw a
boundary, more is going to

212
00:10:34,390 --> 00:10:36,290
be coming out of the right
than entering to the left.

213
00:10:36,290 --> 00:10:39,220
And so you have a positive
diverge, or you're

214
00:10:39,220 --> 00:10:40,520
getting less dense.

215
00:10:40,520 --> 00:10:43,840
Anyway, I hope I haven't
confused you too much,

216
00:10:43,840 --> 00:10:45,690
but I have run out
of time once again.

217
00:10:45,690 --> 00:10:48,080
I'll see you in the next video.