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Page 4
4
Question 3:
Part (i)
Consider the vector field
F
(
x, y
)=(
ye
xy
+2
x
)
i
+(
xe
xy
-
2
y
)
j
.
a) Show that
F
is conservative.
Solution.
F
=
φ
, where
φ
(
x, y
)=
e
xy
+
x
2
-
y
2
, so
F
is conservative.
b) Using the fundamental theorem of line integration or otherwise, find the line
integral
Z
γ
F
·
d
r
,
where
γ
:
r
(
t
)=
t
i
+
t
2
j
,
0
t
2
.
Solution.
By the fundamental theorem of line integration, the integral is given by
φ
(
r
(2))
-
φ
(
r
(0)) =
φ
((2
,
4))
-
φ
((0
,
0)) =
e
8
-
13
.
Part (ii)
Let
F
(
x, y, z
)=2
x
i
+
xz
j
+ (1 +
z
2
)
k
.
a) Find div
F
.
Solution.
div
F
=
∂P
∂x
+
∂Q
∂y
+
∂R
∂z
=2+0+2
z
=2+2
z.
b) Using Gauss’ Divergence Theorem or otherwise, calculate the total flux of
F
out of the solid bounded by the paraboloid
z
=9
-
x
2
-
y
2
and the
xy
-plane.
Solution.
By Gauss’ divergence theorem, the flux is given by the triple integral of
the divergence over the solid.
Using cylindrical coordinates (as in Question 2) we can express this integral as
Z
2
π
0
Z
3
0
Z
9
-
r
2
0
(2+2
z
)
r dz dr dθ
=2
π
Z
3
0
r
(2(9
-
r
2
)+(9
-
r
2
)
2
)
dr
=
π
Z
9
0
2
u
+
u
2
du
= 324
π
where we have used the substitution
u
=9
-
r
2
.


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