Calculus of Several Variables (MATS3121)
Mathematics and Computer Science - MCS
Semester: Resit
Level: 300
Year: 2016
THE UNIVERSITY OF BAMENDA
FACULTY OF SCIENCE
DEPARTMENT OF MATHEMATICS AND COMPUTER SCIENCE
RESIT FOR FIRST SEMESTER EXAMINATION 2016MATS3121: CALCULUS OF SEVERAL VARIABLES
ANSWER ALL QUESTIONS TIME ALLOWED: 3 H
EXAMINER: Snr MENGUE MENGUE
(Ph.D)
Exercice:1 (1 Opts)
1) What is the difference between a metric and a norm. 1 pt
2) Define open subset. When do we say that two norms are equivalent, give an example.1,5pts
3) Let x
do you think||x||=
, 1 p is a norm? justify 1,5pts
4) Set F (x, y) =
, for (x, y)(0,0)
find
ln(
1pt
5) Prove that lim(
x
,
y
)_
+
(
0
,
0
)
x
7
+
;
2
= 0 and find lim(
zy
)_,(
0
,
0
)(x
2
+ y
2
)1n (x
2
+ y
2
) 1pt
6) Let F(x,y) =
!
"
be a vector field and C
1
, C
2
be the parametric curves defined as: C
1
:r(t) =
(5t - 5)i + (5t - 3)j, 0
t C
2
:r(t) = (4 - t
2
)i + tj, - 3
t 2
a) Sketch the curves C
1
and C2, verify that they have the same initial and terminal point. 1pt
b) Show that
#
$%&
'
#
$%&
'
1 pt
7) Evaluate the surface integral
(
)
*
dS where S is the part of the cylinder x
2
+ y
2
=
9 that lies between the plane z = O and z = 2. 2pts
Exercice: 2 (1Opts)
a) Let V(x, y, z)= (2xy
2
+ z) e
1
+ 2y
2
e
2
+ xe
3
be a vector field define on
3
, check if V is
conservative and find if it exist the potential. 2,5pts
b) For which value(s) of r is F = (4x
2
+ rxy)i + (3y
2
+ 4x
2
)j is a gradient field? Find the
potential 1,5pts
c) State Green's and Stoke's Theorems 1,5pts
d) State the Chain rule in terms of Jacobian matrices 1,5pts
e) Evaluate
+
%
'
; where C is the circle x = cost, y = sint and 0 t 27,. 1pt
f) Find k where
# #
-
.
/
dydx =
.
(-
– 1)
g) Compute
# # #
01%)%%
.
/
(
# #
%)%%
2
3
/
1pt
Bonus: Find the directional derivative of
G
(
x,y,z
) = arcsin (
.
4
45
) at the point
P
(1,2,0)
in the direction of the point Q(-1,3,1).
1,5pts
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