1. a) (5 marks) Show that
f ( x , y ) =
not continuous at ( 0 , 0 ) .
b) (7 marks) Let r
= x i + y j + z k , r
w = r 3 .
Find the value of the
constant k , if any, for which
2 w =
k r .
2. a) (6 marks) Suppose
that the position of a bug moving in space at any time t
0 is given by the path
c ( t ) =
( 2 t 3 , t
2 2 , 7 t 2 ) . Suppose also that the
temperature at any point ( x , y , z ) is given
by the function
( x , y , z
) = x z y + z
What is the rate of change
of temperature that the bug experiences at the point ( 1 , 2 , 3 ) ?
b) (7 marks) Suppose that
the equation z 3 = 2 + x 2 3
y + z is used to implicitly defines z as a
function of ( x ,
y ) near ( 1 , 1 , 1 ) . Compute
( x , y ) = ( 1 , 1 ) .
all the critical points of the function f ( x , y
) = x 3 4 x y + 2 y 2
+ y + 5 and use
test to classify each of them as a local maximum, a local minimum or a
4. (10 marks) Determine the
absolute minimum and absolute maximum values for f ( x ,
y ) = x 2 x + 3 y 2
on the elliptic region
x 2 + 2 y 2 ≤ 4 . Find the coordinates
of all the points on the given region where the
function reaches its
absolute maximum or its absolute minimum.
5. Evaluate each of the
a) (7 marks) .
b) (8 marks)
(15 marks) Let F
( x , y , z ) = y
i 2 x z j + x y k
and let S be the part of the parabolic cylinder
lies inside the elliptic cylinder
2 + 4 y
2 = 4 .
Verify Stokes theorem for the vector field
and the surface S .
(15 marks) Let W be the solid region consisting of all
points ( x , y , z ) for which x
2 + y 2 ≤
1 and 0 ≤ z ≤ y . Let
W denote the closed surface that bounds the region W ,
oriented by the
outward pointing normal vector. Suppose that g
= div F , where F denotes any C
1 vector field such
( x , y , z ) ≤ 6 for every point ( x ,
y , z ) in the region W . What is the minimum
value of the
for which ≤
(10 marks) Let
( x , y ) = x
2 be the
density at any point ( x , y )
of the region D
Suppose that the boundary of the region D is
a simple closed curve C , positively oriented and such
for any two positive integer numbers m ≤ 2 and n ≤ 2 .
Find the coordinates
the centroid of the region D .