1
00:00:07,030 --> 00:00:07,600
Hi.

2
00:00:07,600 --> 00:00:09,110
Welcome back to recitation.

3
00:00:09,110 --> 00:00:11,630
In lecture you've been
learning about vector calculus,

4
00:00:11,630 --> 00:00:15,570
Stokes' theorem, all sorts of
cool stuff like that, curl.

5
00:00:15,570 --> 00:00:17,920
So I have a nice
problem here for you.

6
00:00:17,920 --> 00:00:21,980
So let F be the
following vector field.

7
00:00:21,980 --> 00:00:25,000
So it's the vector
field whose direction

8
00:00:25,000 --> 00:00:27,620
is x i hat plus y
j hat plus z k hat,

9
00:00:27,620 --> 00:00:30,030
but in addition I want to
multiply it by rho to the n.

10
00:00:30,030 --> 00:00:32,760
So this is your usual rho
from spherical coordinates.

11
00:00:32,760 --> 00:00:34,350
This is the square
root of x squared

12
00:00:34,350 --> 00:00:35,610
plus y squared plus z squared.

13
00:00:35,610 --> 00:00:37,130
And n can be any number.

14
00:00:37,130 --> 00:00:39,610
So it might be positive,
it might be negative,

15
00:00:39,610 --> 00:00:41,940
it might be 0, doesn't
have to be an integer.

16
00:00:41,940 --> 00:00:45,040
Some number, though,
it's some constant.

17
00:00:45,040 --> 00:00:49,320
So what I'd like you to do is
to show that for this field,

18
00:00:49,320 --> 00:00:51,600
regardless of what the
value of n happens to be,

19
00:00:51,600 --> 00:00:54,220
that it is a gradient field.

20
00:00:54,220 --> 00:00:57,370
So it is the gradient
of some function.

21
00:00:57,370 --> 00:01:00,122
So why don't you pause the
video, have a go at that,

22
00:01:00,122 --> 00:01:01,830
come back and we can
work on it together.

23
00:01:10,750 --> 00:01:14,420
So recall that for something to
be a gradient field, what that

24
00:01:14,420 --> 00:01:16,130
means-- well, first
of all, that means

25
00:01:16,130 --> 00:01:18,700
that there is some function
that has it as the gradient,

26
00:01:18,700 --> 00:01:21,010
and we know a lot of other
characterizations of it.

27
00:01:21,010 --> 00:01:24,610
And one of them that we
know involves the curl.

28
00:01:24,610 --> 00:01:26,520
So let's talk about that one.

29
00:01:26,520 --> 00:01:29,245
So in order to look at
the curl of this field,

30
00:01:29,245 --> 00:01:31,120
I'm going to want to
take partial derivatives

31
00:01:31,120 --> 00:01:32,300
of its components.

32
00:01:32,300 --> 00:01:34,910
And so I'm going to want to
take partial derivatives of rho.

33
00:01:34,910 --> 00:01:43,430
So let's just remember or
recall that partial rho

34
00:01:43,430 --> 00:01:47,230
partial x equals x over rho.

35
00:01:47,230 --> 00:01:50,760
And similarly, partial rho
partial y is y over rho.

36
00:01:50,760 --> 00:01:53,650
And partial z partial--
sorry, partial rho partial z

37
00:01:53,650 --> 00:01:55,720
is z over rho.

38
00:01:55,720 --> 00:01:58,440
So you can just check that using
the fact that you know what rho

39
00:01:58,440 --> 00:02:00,050
is.

40
00:02:00,050 --> 00:02:00,720
Et cetera.

41
00:02:00,720 --> 00:02:02,020
I'm going to write et
cetera because it's

42
00:02:02,020 --> 00:02:03,220
the same for the other two.

43
00:02:08,460 --> 00:02:13,540
So let's call-- for
shorthand-- let's call F,

44
00:02:13,540 --> 00:02:20,380
M*i plus N*j plus P times k.

45
00:02:20,380 --> 00:02:22,720
So M, N, and P are
its components.

46
00:02:22,720 --> 00:02:31,270
If F is this, then
we know that curl F

47
00:02:31,270 --> 00:02:33,580
is equal to-- what is it?

48
00:02:33,580 --> 00:02:38,200
So it's P, the y-th
partial derivative of P,

49
00:02:38,200 --> 00:02:43,900
minus the z-th partial
derivative of N,

50
00:02:43,900 --> 00:02:51,250
i hat, plus-- OK, so it's the
z-th partial derivative of M

51
00:02:51,250 --> 00:02:58,360
minus the x-th partial
derivative of P, j hat, plus--

52
00:02:58,360 --> 00:03:00,380
it's going to be,
what's the last one?

53
00:03:00,380 --> 00:03:06,200
The x partial derivative of N,
minus the y partial derivative

54
00:03:06,200 --> 00:03:09,480
of M, k hat.

55
00:03:09,480 --> 00:03:13,080
So if this is our
formula for F, then this

56
00:03:13,080 --> 00:03:17,870
is our formula for
the curl of F. And OK,

57
00:03:17,870 --> 00:03:21,050
so now we just have to compute
these various different partial

58
00:03:21,050 --> 00:03:23,720
derivatives in order to
see what the curl is.

59
00:03:23,720 --> 00:03:27,630
And then hopefully that'll tell
us something about this field.

60
00:03:27,630 --> 00:03:28,200
So let's see.

61
00:03:28,200 --> 00:03:31,040
So P_y.

62
00:03:31,040 --> 00:03:38,540
In our case, so M is equal to
rho to the little n times x.

63
00:03:38,540 --> 00:03:43,000
Big N is equal to rho
to the little n times y.

64
00:03:43,000 --> 00:03:49,140
And big P is equal to rho
to the little n times z.

65
00:03:49,140 --> 00:03:53,030
So let's look at
our components here.

66
00:03:53,030 --> 00:04:02,590
So P sub y-- well, z is a
constant with respect to y.

67
00:04:02,590 --> 00:04:05,340
So this is just rho to the n.

68
00:04:05,340 --> 00:04:08,990
Well, z times the y-th
partial of rho to the n.

69
00:04:08,990 --> 00:04:10,750
So that's by the
chain rule, so we

70
00:04:10,750 --> 00:04:23,110
got n rho to the n minus 1
times y over rho times z.

71
00:04:23,110 --> 00:04:34,450
So we can rewrite this as
n*y*z rho to the n minus 2.

72
00:04:34,450 --> 00:04:42,350
And similarly-- so
that was P sub y,

73
00:04:42,350 --> 00:04:45,730
so let's look at N sub z.

74
00:04:45,730 --> 00:04:51,000
So N is rho to the n times y.

75
00:04:51,000 --> 00:04:54,780
So you take the z-th partial
derivative, so y is a constant.

76
00:04:54,780 --> 00:04:57,460
So we need to look at the
z-th partial derivative of P

77
00:04:57,460 --> 00:04:57,960
to the n.

78
00:04:57,960 --> 00:05:02,720
So this is, again,
it's n-- sorry,

79
00:05:02,720 --> 00:05:04,130
I think I said P to the n.

80
00:05:04,130 --> 00:05:07,360
But of course this isn't
a P, this is a rho.

81
00:05:07,360 --> 00:05:12,600
So it's n rho to the n minus 1
times partial rho partial z--

82
00:05:12,600 --> 00:05:16,610
so that's z over rho-- times y.

83
00:05:16,610 --> 00:05:25,490
And again, this is equal to
n*y*z rho to the n minus 2.

84
00:05:25,490 --> 00:05:30,780
OK, so P sub y,
the y partial of P,

85
00:05:30,780 --> 00:05:34,160
is n*y*z rho to the n minus 2.

86
00:05:34,160 --> 00:05:39,930
And the z partial of n is
n*y*z rho to the n minus 2.

87
00:05:39,930 --> 00:05:41,422
And they're the same.

88
00:05:41,422 --> 00:05:42,380
So what does that mean?

89
00:05:42,380 --> 00:05:45,890
So that means the first
component of curl of F is 0.

90
00:05:48,980 --> 00:05:52,670
So if you do a little
bit more arithmetic

91
00:05:52,670 --> 00:05:55,310
of exactly the same
sort-- you have two more

92
00:05:55,310 --> 00:05:58,510
components to check--
what you're going to find

93
00:05:58,510 --> 00:06:00,550
is that the other
ones are 0 also.

94
00:06:00,550 --> 00:06:03,030
I'm not going to do out all
those partial derivatives

95
00:06:03,030 --> 00:06:06,340
for you, but I trust that you
can compute the similar looking

96
00:06:06,340 --> 00:06:10,610
partial derivatives that appear
in these other two components--

97
00:06:10,610 --> 00:06:17,340
this j should have had a hat--
the other partial derivatives

98
00:06:17,340 --> 00:06:20,610
that appear in these components
and show that they're all also

99
00:06:20,610 --> 00:06:21,260
equal to 0.

100
00:06:21,260 --> 00:06:29,360
So in our case, curl of F is
just equal to the zero vector.

101
00:06:29,360 --> 00:06:30,670
OK, great.

102
00:06:30,670 --> 00:06:31,840
So what does that mean?

103
00:06:31,840 --> 00:06:40,000
Well, we want to show that
something is a gradient field.

104
00:06:40,000 --> 00:06:42,580
So we know that we can do that.

105
00:06:42,580 --> 00:06:47,185
We know that that happens
precisely when its curl is 0,

106
00:06:47,185 --> 00:06:50,790
if it's defined on a
simply connected domain.

107
00:06:50,790 --> 00:06:52,280
On a simply connected region.

108
00:06:52,280 --> 00:06:55,090
So we know that a function
is a gradient field

109
00:06:55,090 --> 00:06:57,570
if it's defined on a
simply connected domain

110
00:06:57,570 --> 00:06:59,320
and has curl 0.

111
00:06:59,320 --> 00:07:00,720
Well, let's see.

112
00:07:00,720 --> 00:07:02,700
So where is this thing defined?

113
00:07:02,700 --> 00:07:04,130
Where is F defined?

114
00:07:04,130 --> 00:07:11,260
Well, if n is bigger than
or equal to 0-- well,

115
00:07:11,260 --> 00:07:13,205
I guess I want strictly
bigger than 0 just

116
00:07:13,205 --> 00:07:14,400
to be on the safe side.

117
00:07:14,400 --> 00:07:17,260
If n is positive, this
is defined everywhere

118
00:07:17,260 --> 00:07:18,300
and we're great.

119
00:07:18,300 --> 00:07:22,230
If n is negative, then
we have a problem at 0.

120
00:07:22,230 --> 00:07:25,420
Because then we have
division by rho.

121
00:07:25,420 --> 00:07:27,560
So we don't want to divide by 0.

122
00:07:27,560 --> 00:07:31,560
So if n is negative, there's
a problem at the origin.

123
00:07:31,560 --> 00:07:35,250
So in the worst case,
F is defined everywhere

124
00:07:35,250 --> 00:07:37,100
except the origin.

125
00:07:37,100 --> 00:07:39,320
But that's simply connected.

126
00:07:39,320 --> 00:07:40,340
Because we're in space.

127
00:07:40,340 --> 00:07:42,715
If we were in the plane, this
would be a different story.

128
00:07:42,715 --> 00:07:44,519
But in space, just
removing a point

129
00:07:44,519 --> 00:07:46,060
doesn't destroy
simply connectedness.

130
00:07:48,750 --> 00:07:51,030
So this field F is
defined everywhere,

131
00:07:51,030 --> 00:07:54,090
except the origin
perhaps, so we're

132
00:07:54,090 --> 00:07:55,700
in a simply connected region.

133
00:07:55,700 --> 00:07:58,210
So when you have curl F in
a simply connected region,

134
00:07:58,210 --> 00:08:00,520
your field is, in
fact, a gradient field.

135
00:08:00,520 --> 00:08:02,105
Now I leave it as
an exercise to you

136
00:08:02,105 --> 00:08:04,290
to come up with the
actual function,

137
00:08:04,290 --> 00:08:08,651
or one of the actual functions,
of which this is a gradient.

138
00:08:08,651 --> 00:08:10,400
But we've just shown
that because its curl

139
00:08:10,400 --> 00:08:13,320
is 0, and because it's defined
in a simply connected region

140
00:08:13,320 --> 00:08:16,750
of space, that this field
really is the gradient field

141
00:08:16,750 --> 00:08:17,760
of some function.

142
00:08:17,760 --> 00:08:19,401
So I'll stop there.