Department of Mathematics and Philosophy of Engineering
MHZ4530- Engineering Mathematics II
Assignment No.01
Due date: 04/07/2018
Academic Year: 2017/18
1. Find & classify the stationary points of the function
, ,
2. Let
a) Find ∇
,
=
+
− 36
.
=
at the point
2, −1,2 .
b) Find the directional derivatives in the direction
3. a) The rate of flow of fluid through a cylindrical tube
=2 − −2 .
with radius and height is given
by
=
where
8
is the viscosity of the fluid and
is the difference in pressure at the two ends
of the tube. Suppose the length the tube remains constant. While the radius increases at
the rate of 0.1 and the pressure decrease at the rate 0.2. Find the rate of change of
with respect to time.
4. Find the value of
"
! +
!
in the clockwise sense along the rectangle
±3$, ±3$
Page 1 of 2
5. Let % , & =
cos 2& − cos & + 2
a) Is % , & harmonic function?
= * , & + % , & , whose imaginary part is
b) Determine the analytic function
% ,& =
cos 2& − cos & + 2 .
c) Express
6. a) Evaluate ∮
4
b) Evaluate
= + ,- .
in terms of ,where
9
:
/0
! , where 5 is the circle | − 2| = 2.
123 12
,
7 − −
8!
i)
Along the straight line from
ii)
Along the real axis from
imaginary axis from
iii)
= 0 to
= 0 to
= 4 to
Along the imaginary axis from
to real axis from
= 4 to
=4+4 .
= 4 and then along a line parallel to
=4+4 .
= 0 to
=4+4 .
Page 2 of 2
= 4 and then along a line parallel