The University of the West Indies
St. Augustine
Examinations of July 2011
Code and Name of Course: MATH2230 Engineering Mathematics II
Paper:
Date and Time:
Duration: 3 hours
INSTRUCTIONS TO CANDIDATES: This paper has 7 pages and 7 questions
ATTEMPT ANY FIVE (5) QUESTIONS
Only non-programmable calculators are allowed
A table of Laplace transforms is provided
c
The
University of the West Indies
Course Code MATH2230
2010/2011
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University
Examiner and/or the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First and Second Examiners. Completed forms should be handed
Page 2
(1) (a) Let C be the parametric curve
r(t) = ti + tj + (1 − t)k
0 ≤ t ≤ 1.
(i) Sketch the curve C.
(ii) Determine the arc length of C.
[2]
[3]
(b) Let F = P i + Qj + Rk be a vector field where the component functions P, Q and
R satisfy the conditions of Clairaut’s theorem. Prove that
div(curl F) = 0.
[5]
(c) Determine the equation of the plane that is tangent to the surface 3xy + z 2 = 4
at the point (1, 1, 1).
[5]
(2) (a)
(i) Show that
∂
∂x
1
(x2
+
y2
+
1
z 2 ) /2
!
=
(x2
−x
+ y 2 + z 2 ) /2
3
[2]
(ii) The gravitational force exerted on a mass m at the point (x, y, z) by a mass
M at the origin is, by Newton’s law of gravitation,
GM m
F=−
(xi + yj + zk) .
3
(x2 + y 2 + z 2 ) /2
where G is the universal gravitational constant. Use part (i) above and
similar formulae to show that
GM m
f=
1
(x2 + y 2 + z 2 ) /2
is a potential function for F.
[3]
c
The
University of the West Indies
Course Code MATH2230
2010/2011
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University
Examiner and/or the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First and Second Examiners. Completed forms should be handed
Question 2 is continued on the next
page
2
4y 2 dA
3x
0
1
0
1
3r cos
4r 2 si
3r 2 cos
4r 3 s
R
2
Page 3
[r 3 cos
0
[7 cos
0
sin2
1
2
1
15
2
cos 2
1
15
2
7 sin
]
r 4 sin 2
co
15
si
4
Find thez volume
of the solid bounded
(b) By evaluating a double integral, find the volume enclosed by the paraboloid
=
1 x 2 y 2.
1 − x2 − y 2 and the plane z = 0.
If we put
0 in the equation of the p
means that the plane intersects the paraboloid
under the paraboloid and abov e the circular dis
and 1(a)]. In polar coordinates D is given by 0
1 x 2 y 2 1 r 2, the volume is
x2
1
V
y 2 dA
D
2
[5]
(c) Use Green’s theorem to evaluate the line integral
Z
y 3 i − x3 j · dr
0
1
d
0
r
r 3 dr
2
If we had used rectangular coordinates instead
obtained
C
V
1
where C is the circle x2 + y 2 = 4 with positive orientation.
x2
1
y 2 dA
1
D
[5]
which is not easy to evaluate because it involve
x2 1
x2 dx
1
(3) (a) The surface S
x2
What we have done so far can be extended
shown in Figure 7. It’s similar to the type IIrectan
In fact, by combining Formula 2 in this section
lowing formula.
If f is continuous on a polar region of th
D
f x, y dA
then
can be described parametrically as
D
r(θ, z) = cos(θ)i + sin(θ)j + zk
where the region
, h
h2
h1
f rc
(θ, z) ∈ R
R = {(θ, z)|0 ≤ θ ≤ 2π, 0 ≤ z ≤ 1 + cos(θ)}.
Determine the surface area of S by evaluating the surface integral
c
The
University of the West Indies
r,
Course Code MATH2230
ZZ
1dS. [8]
S
2010/2011
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University
Examiner and/or the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First and Second Examiners. Completed forms should be handed
Question 3 is continued on the next
page
Page 4
(b) Use the Divergence theorem to evaluate the surface integral
ZZ
(xyi + yzj + zxj) · n dS
S
where S is the surface of the solid that is bounded by the cylinder x2 + y 2 = 1
and the planes z = 0 and z = 1. Assume that S has an outward orientation. [7]
(4) (a) Use the Laplace transform to solve the differential equation
x00 + 4x = cos(2t) − H(t − 2π) cos(2(t − 2π))
subject to the initial conditions x(0) = x0 (0) = 0.
Hint: L−1
1
s
=
t sin(kt)
(s2 + k 2 )2
2k
[6]
(b) Consider the following system of differential equations
x01 = x2
x02 = −x1
(i) Express the given system in matrix form
0
x1
x
=A 1
0
x2
x2
where A is a 2 × 2 matrix.
[1]
(ii) Show that the Laplace transform of the given system can be written as
x1 (0)
X1 (s)
=
(sI − A)
X2 (s)
x2 (0)
where I is the identity 2×2 matrix, L[xi ] = Xi (s) (i = 1, 2) and x1 (0), x2 (0)
[4]
are the initial values of the functions x1 (t) and x2 (t).
(iii) Use a formula for the inverse of a 2 × 2 matrix to show that
s
1 X1 (s)
x1 (0)
2 +1
2 +1
s
s
= −1
s
X2 (s)
x2 (0)
s2 +1
s2 +1
c
The
University of the West Indies
Course Code MATH2230
[2]
2010/2011
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University
Examiner and/or the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First and Second Examiners. Completed forms should be handed
Question 4 is continued on the next
page
Page 5
(iv) Hence obtain the solution
x1 (t) = x1 (0) cos t + x2 (0) sin t
x2 (t) = −x1 (0) sin t + x2 (0) cos t
of the given system.
(5) Let the function f (x) be defined as
(
−x −2 ≤ x < 0
f (x) =
x 0≤x<2
[2]
f (x + 4) = f (x).
(a) Sketch the function f (x) for at least two periods.
[2]
(b) Is the function f (x) even or odd?
[1]
(c) Show that
∞
8 X
1
(2k − 1)πx
1− 2
cos
π
(2k − 1)2
2
k=1
is the Fourier series of f (x).
[8]
(d) For which values of x does the Fourier series in (c) converge to f (x)?
(e) Use parts (c) and (d) to show that
∞
π2 X
1
=
8
(2k − 1)2
[1]
k=1
[3]
(6) Solve the heat equation
∂ 2u
∂u
= 100 2
∂t
∂x
subject to the boundary conditions
u(0, t) = 0
u(1, t) = 0
t≥0
and initial condition
u(x, 0) = sin(2πx) − sin(5πx)
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The
University of the West Indies
0 ≤ x ≤ 1.
Course Code MATH2230
[15]
2010/2011
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University
Examiner and/or the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First and Second Examiners. Completed forms should be handed
Page 6
(7) A thin rectangular plate {(x, y)|0 ≤ x ≤ a, 0 ≤ y ≤ b} of length a and height b
has each of its four sides at temperatures that are given by the following boundary
conditions
u(0, y) = 0
0≤x≤a
u(a, y) = 0
.
where
0≤y≤b
u(x, 0) = 0
u(x, b) = f (x)
By solving Laplace’s equation
∂ 2u
∂ 2u
+
=0
∂x2 ∂y 2
in the rectangular domain
{(x, y)|0 ≤ x ≤ a, 0 ≤ y ≤ b},
show that the steady-state temperature u(x, y) of this rectangular plate is given by
∞
nπy nπx X
u(x, y) =
cn sinh
sin
a
a
n=1
where
cn =
2
a sinh nπb/a
Z
a
f (x) sin
0
nπx a
dx
[15]
END OF PAPER
c
The
University of the West Indies
Course Code MATH2230
2010/2011
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University
Examiner and/or the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First and Second Examiners. Completed forms should be handed
Page 7
y(t)
L[y(t)]
n!
tn
sn+1
(n = 0, 1, 2, . . .)
Γ(x + 1)
sx+1
tx
eat
eat y(t)
cos(kt)
sin(kt)
cosh(kt)
sinh(kt)
H(t − t0 )
(x > −1)
1
s−a
Y (s − a)
s
2
s + k2
k
2
s + k2
s
2
s − k2
k
2
s − k2
e−st0
s
H(t − t0 )y(t − t0 )
e−st0 Y (s)
δ(t − t0 )
e−st0
y 0 (t)
sY (s) − y(0)
y 00 (t)
s2 Y (s) − sy(0) − y 0 (0)
Z
t
0
f (t − β)g(β) dβ
F (s)G(s)
Table of Laplace transforms
c
The
University of the West Indies
Course Code MATH2230
2010/2011
DO NOT WRITE OR TYPE ON THE BACK OF THIS SHEET: USE ONE SIDE ONLY
INSTRUCTIONS: Each page must be signed by the Examiners and where applicable, the University
Examiner and/or the External Examiner. Where the examination does not require a University Examiner, the form must be signed by the First and Second Examiners. Completed forms should be handed