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MATH3023 SECOND SEMESTER EXAMINATION 2020 1
Department of Mathematics and Statistics
MATH3023 ADVANCED MATHEMATICS APPLICATIONS
FINAL EXAMINATION br>SEMESTER 2, 2020
Examination duration: 2 hours.
This examination constitutes 55 % of the total assessment for MATH3023.
There are 7 questions in this examination, worth a total of 65 marks.
Marks for each part of each question are shown in square brackets.
This is an open-book examination.
Your answers to this examination will require written solutions supported by
adequate working, and marks will be awarded for clarity and correctness of
method, not just for correct answers.
You must upload a scan of your working to Blackboard.
You will have an additional 30 minutes to scan and upload your working.
To ensure your working is clear and legible, you must use a black or blue pen
and light colored (preferably white) paper.
Your full name, your student ID number and the page number must be written
on the top of every page of your scanned working.
Your scanned working must be in PDF format as a single PDF file.
It is your responsibility to ensure that your scanned working is legible.
It is strongly recommended that you use the Microsoft Office Lens app to scan
your working.
The file size of your scanned working should not exceed 20 MB to ensure
successful upload within the 30 minute time frame.
MATH3023 SECOND SEMESTER EXAMINATION 2020 2
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MATH3023 SECOND SEMESTER EXAMINATION 2020 3
1. Consider the vector field
F(x, y) = (y − sinx, cosx),
on the triangular region R bounded by the straight lines y = 0, x =
pi
2
and y =
2
pi
x.
a) Sketch the triangular region R, and find suitable parameterisations for each of the
straight lines y = 0, x =
pi
2
and y =
2
pi
x such that the boundary of R is traversed
anti-clockwise. [3 marks]
b) Calculate the circulation of the vector field F along each of the straight lines y = 0,
x =
pi
2
and y =
2
pi
x using the parameterisations you found in Part a, and hence find
the circulation around the entire boundary of R. [6 marks]
c) Calculate the circulation of the vector field F around the boundary of R using Green’s
theorem, and confirm you answer is the same as Part b.
HINT: You may find it easier to integrate first with respect to x. [4 marks]
2. Consider the vector field
F(x, y, z) =
(
2x cos y + z sin y, xz cos y − x2 sin y, x sin y) .
a) Show that F is a conservative vector field. [3 marks]
b) Find a potential function for the conservative vector field F. [3 marks]
c) Calculate the circulation of the conservative vector field F along any curve C between
the points (2,−pi, 1) and
(
4,
pi
2
, 3
)
. [1 mark]
3. Evaluate the flux of the vector field
F(x, y, z) = (18z,−12, 3y),
through the triangle formed by the plane 2x+ 3y + 6z = 12 in the first octant. [5 marks]
MATH3023 SECOND SEMESTER EXAMINATION 2020 4
4. Use Gauss’ theorem to evaluate the total flux of the vector field
F(x, y, z) =
(
2xy, yz2, xz
)
,
through all the surfaces of the parallelepiped bounded by the planes x = 0, y = 0, z = 0,
x = 2, y = 1 and z = 3. [4 marks]
5. Consider the function
u(x, y) = e−x (x sin y − y cos y) .
a) Show that u is a harmonic function. [3 marks]
b) Find a function v(x, y) such that the complex function f(z) = u(x, y) + iv(x, y) is an
analytic function of the complex variable z = x+ iy.
HINT: You may find it easier to integrate first with respect to x, and the following
integration by parts formula may be useful:∫
xe−x dx = −xe−x − e−x + C. [4 marks]
c) Show that the complex function f(z) = u(x, y) + iv(x, y) is given by
f(z) = ize−z,
in terms of the complex variable z = x+ iy. [2 marks]
6. a) Use Cauchy’s integral formula to evaluate the contour integral∮
C
zez
(z + 1)3
dz,
where C is any closed contour that contains the point z = −1 in its interior.
[3 marks]
b) Prove that ∮
C
zeaz
(z + 1)3
dz = pia(2 − a)e−a · i,
where a is any real constant and C is any closed contour that contains the point z = −1
in its interior. [3 marks]
MATH3023 SECOND SEMESTER EXAMINATION 2020 5
7. Consider the following boundary value problem describing the heat flow in a bar of unit
length with thermal diffusivity equal to 1 and an initial temperature distribution equal to
2T0 where T0 is a constant, and with one end held at a constant temperature T0 and the
other end insulated:
∂u
∂t
=
∂2u
∂x2
, 0 < x < 1 , t > 0,
u(0, t) = T0 ,
∂u
∂x
∣∣∣∣
x=1
= 0 , t > 0,
u(x, 0) = 2T0 , 0 < x < 1.
a) Show that the equilibrium solution uE(x) that satisfies the ordinary differential equation
and boundary conditions
d2uE
dx2
= 0 , uE(0) = T0 ,
duE
dx
∣∣∣∣
x=1
= 0,
is given by
uE(x) = T0. [2 marks]
b) By substituting
v(x, t) = u(x, t) − uE(x) = u(x, t) − T0,
into the original boundary value problem, show that the function v(x, t) satisfies
∂v
∂t
=
∂2v
∂x2
, 0 < x < 1 , t > 0,
v(0, t) = 0 ,
∂v
∂x
∣∣∣∣
x=1
= 0 , t > 0,
v(x, 0) = T0 , 0 < x < 1. [2 marks]
c) Assuming a solution of the form v(x, t) = X(x)T (t), show that the partial differential
equation
∂v
∂t
=
∂2v
∂x2
,
results in the functions X(x) and T (t) satisfying the ordinary differential equations
X ′′(x) + λX(x) = 0 and T ′(t) + λT (t) = 0,
where −λ is the separation constant. [2 marks]
d) Show that assuming a solution of the form v(x, t) = X(x)T (t) implies that the boundary
conditions for the ordinary differential equation for the function X(x) are
X(0) = 0 and X ′(1) = 0. [1 mark]
MATH3023 SECOND SEMESTER EXAMINATION 2020 6
e) Setting λ = −β2, show that the solution of
X ′′(x) − β2X(x) = 0,
subject to
X(0) = 0 and X ′(1) = 0,
is the trivial solution X(x) = 0. [2 marks]
f) Setting λ = 0, show that the solution of
X ′′(x) = 0,
subject to
X(0) = 0 and X ′(1) = 0,
is the trivial solution X(x) = 0. [2 marks]
g) Setting λ = β2, show that the solution of
X ′′(x) + β2X(x) = 0,
subject to
X(0) = 0 and X ′(1) = 0,
requires β =
(
n− 1
2
)
pi for positive integers n, and
X(x) = Bn sin
[(
n− 1
2
)
pix
]
,
where Bn are the arbitrary integration constants for each positive integer n.
[3 marks]
h) Show that the solution of the first-order ordinary differential equation
T ′(t) +
(
n− 1
2
)2
pi2T (t) = 0,
is given by
T (t) = Cne
−(n− 12)
2
pi2t,
where Cn is an arbitrary integration constant for each positive integer n. [3 marks]
i) By applying the initial condition v(x, 0) = T0 to the general solution
v(x, t) = X(x)T (t) =
∞∑
n=1
Ane
−(n− 12)
2
pi2t sin
[(
n− 1
2
)
pix
]
,
where An = BnCn, show that we must have
An =
2T0(
n− 1
2
)
pi
.
Hence write down the solution u(x, t) of the original boundary value problem.
[4 marks]
[END OF EXAMINATION]

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