exam-2011 solutions vector.pdf-Solutions...
exam-2011_solutions_vector.pdf-Solutions to Math 2069 Exam 2011,
Showing 4 out of 4
exam-2011 solutions vector.pdf-Solu...
exam-2011_solutions_vector.pdf-Solutions to Math 2069 Exam 2011,
##### 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
.

Browse thousands of Study Materials & Solutions from your Favorite Schools
Pepperdine University
Pepperdine_University
School:
Mathematics_2A
Course:
Great resource for chem class. Had all the past labs and assignments
Leland P.
Santa Clara University
Introducing Study Plan
Using AI Tools to Help you understand and remember your course concepts better and faster than any other resource.
Find the best videos to learn every concept in that course from Youtube and Tiktok without searching.
Save All Relavent Videos & Materials and access anytime and anywhere
Prepare Smart and Guarantee better grades