Multivariable Calculus (FSCT3205)
Mechanical Engineering - MEN
Semester: Second Semester
Level: 300
Year: 2013
1
UNIVERSITY OF BAMENDA
H.T.T.T.C BAMBILI
END OF SECOND SEMESTER EXAMINATION
COURSE CODE & NUMBER:
MAT211
COURSE TITLE:
Numerical Functions with Several Variables
CREDIT VALUE: Three
Credits
DEPARTMENT:
Fundamental Science
COURSE INSTRUCTOR:
Tanyu Vivian
MONTH:
June
YEAR:
2013
DATE :02/07/2013 TIME ALLOWED : 2. hours
INSTRUCTION: Answer all the questions. All necessary work must be shown and must be neatly
and orderly presented.
1. i)a) Define the limit of a function f
b) Use your definition above to show that the function f(x,y) =
does have a limit at the
origin
ii)The transformation to polar coordinate, x = r cos , y = r sin implies that r
2
= x
2
+ y
2
and
tan =
. Use this information to show that
= ,
=
=
,
=
2.a) -Evaluate the following integrals
i)
∫ ∫
(xy — x
2
)dA, where R is the region bounded by y = x and y = 3x — x
2
ii)
∫ ∫
(6x+ 2y
2
)dA, where R is the region bounded by the parabola x = y
2
and x + y = 2
b) Show that the differential y
2
z
2
dx + 2xyz
2
dy + 2xy
2
zdz is an exact differential in the interior of
a rectangular parallelepiped and find its potential.
3. a) i) State Green’s theorem
ii) Evaluate ∮ (x — y)
3
dx + (y
3
+ x
3
)dy where c is the positively oriented boundary of the quarter
disk R, 0 ≤x
2
+ y
2
≤ a
2
, x ≥0, y ≥ 0, a > 0
b) Evaluate I = ∫
ydx + zdy + xdz where c is the parametric curve x = t , y = t
2
,z = t
3
, 0 ≤t ≤ 1
GOOD LUCK
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