The University of Sydney
School of Mathematics and Statistics
Solutions to Practice Session 6
MATH2061: Vector Calculus
Summer School 2014
1.
Evaluate
ii
S
F
·
n
dS
, where
F
= 2
x
i
+3
xy
j
−
yz
2
k
and
S
is the surface bounded
by the planes
x
= 0
, x
+
y
= 2
, y
= 0
, z
= 0 and
z
= 2
.
(
n
is the unit outward
normal to the surface
S.
)
Solution:
Let
V
be the solid enclosed by
S
.
Then
V
is described by:
0
≤
x
≤
2
0
≤
y
≤
2
−
x
0
≤
z
≤
2
z
y
x
2
2
2
Now, div
F
=
∇·
F
= 2 + 3
x
−
2
yz
, and so by the divergence theorem,
S
F
·
n
dS
=
iii
V
F
dV
=
V
(2 + 3
x
−
2
yz
)
dV
=
i
2
0
i
2
0
i
2

x
0
(2 + 3
x
−
2
yz
)
dy dx dz
=
i
2
0
i
2
0
((2 + 3
x
)(2
−
x
)
−
(2
−
x
)
2
z
)
dx dz
=
i
2
0
b
4
x
+ 2
x
2
−
x
3
+
z
(2
−
x
)
3
3
B
2
0
dz
=
i
2
0
p
8
−
8
z
3
P
dz
=
32
3
.
2.
Find the ±ux of
F
=
xz
2
i
+(
x
2
y
−
z
3
)
j
+(2
xy
+
y
2
z
)
k
outwards across the entire
surface of the hemispherical region bounded by
z
= (
a
2
−
x
2
−
y
2
)
1
/
2
and
z
= 0
.
Solution:
Let
S
be the entire surface of the given
region, and let
V
be the solid enclosed by
S
. Then
y
x
z
a
S
F
·
n
dS
=
V
F
dV
=
V
(
z
2
+
x
2
+
y
2
)
dV .
Copyright c
c
2014 The University of Sydney
1
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View Full DocumentUsing the spherical coordinates
x
=
r
cos
θ
sin
ϕ , y
=
r
sin
θ
sin
ϕ , z
=
r
cos
ϕ ,
V
is described by:
0
≤
r
≤
a,
0
≤
θ
≤
2
π,
0
≤
ϕ
≤
π/
2
.
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 One '14
 Statistics, Vector Calculus, Stokes' theorem, dr dθ dφ

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