Remark 176
In physics
B
"
U
f
(
x
)
·
$
(
x
)
d
H
N
#
1
(
x
)
represents the
outward ±ux
of a vector Feld
f
across the boundary of a region
U
.
If
E
)
R
N
and
f
:
E
*
R
N
is di
"
erentiable, then
f
is called a
divergencefree
Feld
or
solenoidal Feld
if
div
f
=0
.
Thus for a smooth solenoidal ²eld, the outward ±ux across the boundary of a
regular set
U
is zero.
104
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View Full DocumentCorollary 177
The theorem continues to hold if
U
)
R
N
is open, bounded, and
its boundary consists of two sets
E
1
and
E
2
, where
E
1
is a closed set contained
in the Fnite union of compact surfaces of class
C
1
and dimension less than
N
'
1
, while for every
x
0
!
E
2
there exist a ball
B
(
x
0
,r
)
and a function
g
!
C
1
(
B
(
x
0
))
such that with
,
g
(
x
)
&
=
0
for all
x
!
B
(
x
0
)
+
#
U
,such
that
B
(
x
0
)
+
U
=
{
x
!
B
(
x
0
):
g
(
x
)
<
0
}
,
B
(
x
0
)
\
U
=
{
x
!
B
(
x
0
g
(
x
)
>
0
}
,
B
(
x
0
)
+
#
U
=
{
x
!
B
(
x
0
g
(
x
)=0
}
.
Note that the radius of the ball and the function
g
depend on
x
0
.
Example 178
Let’s calculate the outward ±ux of the function
f
(
x,y,z
):=(0
,yz,x
)
across the boundary of the region
U
:=
$
(
)
!
R
3
:
x
2
+
y
2
<z
2
,x
2
+
y
2
+
z
2
<
2
y, z >
0
%
.
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 Spring '14
 Topology, Vector Calculus, Topological space, Closed set

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