JUNE 2017 MATH2069 Page 2
Use a separate book clearly labelled Question 1
1. [20 marks]
a) [5 marks]
A particle moves along a curve C in R3 with its position at time t seconds
(t ≥ 0) given by
r(t) = (t2 − 6)i+ (t2 − 12)j+ (5t)k.
i) Calculate the acceleration vector a(t).
ii) Find the point Q on the curve where the acceleration vector a(t) is
perpendicular to the position vector r(t).
iii) Calculate the speed of the particle at the point Q from part ii).
b) [5 marks]
i) Sketch the surface S1 in R
3 with equation −x2 + y2 − z2 = 4.
ii) Sketch the surface S2 in R
3 with equation 9x2 + 9y2 + z2 = 36.
iii) Hence or otherwise find the set of all points in R3 which lie on both
S1 and S2.
c) [5 marks]
Consider the double integral
I =
ˆ
2
0
ˆ
4
x2
ey√
y
dy dx.
i) Sketch the region of integration.
ii) Evaluate I by first reversing the order of integration.
d) [5 marks]
By setting up and evaluating an appropriate triple integral in spherical
coordinates prove that the volume V of a sphere of radius R is given by
V =
4
3
piR3.
Please see over . . .
JUNE 2017 MATH2069 Page 3
Use a separate book clearly labelled Question 2
2. [20 marks]
a) [6 marks] Given that
u(x, y) = sinh x cos y − 4x3y + 4xy3
is harmonic (you do not need to prove this), find a harmonic conjugate
v(x, y) of u and write f = u+ iv as a function of z.
b) [6 marks] Let
f(z) = z1/2
and Γ = {eiθ : pi
2
≤ θ ≤ 3
2
pi}.
i) Calculating powers by taking arguments in the range from 0 to 2pi,
or otherwise, evaluate
´
Γ
f(z) dz.
ii) Why would the Principal Value not be a suitable branch to use?
c) [8 marks] Let
g(z) =
sin(z)
z3 − pi2
16
z
i) Find and classify the singular points as either essential, removable or
a pole and give the order of each pole.
ii) Determine the residue at each of these singular points.
iii) Hence, or otherwise, find the value of
˛
|z|=pi
g(z) dz.
Please see over . . .
JUNE 2017 MATH2069 Page 4
Use a separate book clearly labelled Question 3
3. [20 marks]
a) [4 marks]
Let φ be the scalar field given by φ(x, y, z) = xy2z3.
i) Calculate gradφ.
ii) Hence find div(gradφ).
iii) Is div(gradφ) = grad(divφ)? Explain your answer.
b) [4 marks]
The solid S in the first octant is bounded by the coordinate planes and
the plane 4x+ 2y + z = 12.
i) Sketch the solid S in R3.
ii) Set up a triple integral in Cartesian coordinates which could be used
to determine the volume of S.(You do not need to evaluate the triple
integral).
c) [12 marks]
In R3 let F be the vector field F(x, y, z) = (2x − y) i + (5) j + 3z k and
let the surface S be the portion of the paraboloid z = 4 − x2 − y2 for
which z ≥ 0. Denote the boundary curve of S by C and assume that C
is traversed in an anticlockwise direction when viewed from the positive
z axis.
i) Sketch the surface S and the curve C in R3, indicating the specified
orientation around C and the corresponding direction of the unit
normal nˆ on S, forced by the right-hand rule.
ii) Write down a parametric description of C and hence evaluate the line
integral ˛
C
F · dr.
[ You are given sin(2θ) = 2 sin(θ) cos(θ) and sin2(θ) =
1
2
(1−cos(2θ))].
iii) Evaluate ∇×F, the curl of F.
iv) Stokes’ theorem states that
˛
C
F · dr =
¨
S
(∇× F) · nˆ dS
By evaluating an appropriate surface integral , verify Stokes’ theorem
for this example.
Please see over . . .
JUNE 2017 MATH2069 Page 5
Use a separate book clearly labelled Question 4
4. [20 marks]
a) [7 marks] Suppose that
f(z) =
5
(z + 2)(z − 3) .
i) Write down the three (maximal) regions with centre 2 on which f(z)
has a convergent Laurent (or Taylor) expansion in powers of z − 2.
ii) Find the Laurent series expansion in powers of z − 2 which is con-
vergent at the point z = i.
b) [7 marks] Use complex analysis methods to find
I1 =
ˆ ∞
−∞
cos 2x
x2 + 4x+ 5
dx
and
I2 =
ˆ ∞
−∞
sin 2x
x2 + 4x+ 5
dx.
c) [6 marks] Suppose f is an entire function and the contour C is the circle
of radius r and centre a.
i) Write down the Cauchy Integral Formula for f(a) in terms of f(z).
ii) By parametrizing the contour C, show that
f(a) =
1
2pi
ˆ
2pi
0
f(a+ reiθ) dθ.
iii) Hence, or otherwise, evaluate
ˆ
2pi
0
cos(reiθ) dθ
for any real number r > 0.































































































































































学霸联盟