Student
Number
Semester 1 Assessment, 2019
School of Mathematics and Statistics
MAST20009 Vector Calculus
Writing time: 3 hours
Reading time: 15 minutes
This is NOT an open book exam
This paper consists of 5 pages (including this page)
Authorised Materials
• Mobile phones, smart watches and internet or communication devices are forbidden.
• No written or printed materials may be brought into the examination.
• No calculators of any kind may be brought into the examination.
Instructions to Students
• You must NOT remove this question paper at the conclusion of the examination.
• There are 11 questions on this exam paper.
• All questions may be attempted.
• Marks for each question are indicated on the exam paper.
• Start each question on a new page.
• Clearly label each page with the number of the question that you are attempting.
• There is a separate 3 page formula sheet accompanying the examination paper, which you
may use in this examination.
• The total number of marks available is 120.
Instructions to Invigilators
• Students must NOT remove this question paper at the conclusion of the examination.
• Initially students are to receive the exam paper, the 3 page formula sheet, and two 14
page script books.
This paper may be held in the Baillieu Library
Blank page (ignored in page numbering)
MAST20009 Semester 1, 2019
Question 1 (9 marks)
Consider the function
f(x, y) =

x2 − 3y3
5x2 + 2y2
, (x, y) 6= (0, 0)
0, (x, y) = (0, 0).
(a) Calculate
∂f
∂y
if (x, y) 6= (0, 0).
(b) Calculate
∂f
∂y
if (x, y) = (0, 0).
(c) Evaluate
lim
(x,y)→(0,0)
∂f
∂y
if it exists. If the limit does not exist, explain why it does not exist.
(d) Is
∂f
∂y
continuous at (0, 0)? Explain briefly.
(e) Is f of order C1 at (0, 0)? Explain briefly.
Question 2 (8 marks)
Consider the function
g(x, y) = log (4− 2x+ y) .
(a) Determine the first order Taylor polynomial for g about the point (1,−1).
(b) Using part (a), approximate g(1.1,−0.9).
(c) Determine an upper bound for the error in your approximation in part (b).
Question 3 (12 marks)
(a) Consider the vector field
F(x, y) = x3i− y3j.
(i) Sketch the vector field at the points (1, 1), (−1, 2) and (0,−1).
(ii) Determine the equation for the flow line of F passing through the point (2, 1) in
terms of x and y.
(b) Let T be the unit tangent vector, N be the unit principal normal vector and B be the
unit binormal vector to a C3 path. Prove the Frenet-Serret formula
dN
ds
= τB− κT
where κ is the curvature of the path and τ is the torsion of the path.
Hint: Differentiate B×T with respect to arclength.
Page 2 of 5 pages
MAST20009 Semester 1, 2019
Question 4 (15 marks)
(a) Let f : R3 → R and g : R3 → R be C1 non-zero scalar functions. Prove the vector identity
∇
(
f
g
)
=
g∇f − f∇g
g2
.
(b) Consider the vector field G given by
G(x, y, z) = 2y2zi− 3xz2j+ 4xy4k.
(i) Show that G is an incompressible vector field.
(ii) Give a physical interpretation for an incompressible vector field.
(iii) Determine a vector field
F(x, y, z) = F1(x, y, z)i+ F2(x, y, z)j
such that
G =∇× F.
Question 5 (11 marks)
Consider the double integral∫ 2
0
∫ 1+x
2
x
2
(2x− 1)5(2y − x)2 cosh[(2y − x)3] dydx.
(a) Sketch the region of integration, clearly labelling any vertices.
(b) Evaluate the double integral by making the substitutions u = 2x and v = 2y − x.
Question 6 (7 marks)
Let V be the solid region bounded by the spheres x2 + y2 + z2 = 1 and x2 + y2 + z2 = 2.
Find the total mass of V if the mass density is µ = (x2 + y2 + z2)
5
2 grams per unit volume.
Question 7 (11 marks)
Let S be the cone
z = 1 +
√
x2 + y2 for 1 ≤ z ≤ 5.
(a) Sketch S.
(b) Write down a parametrisation for S based on cylindrical coordinates.
(c) Using part (b), find an outward normal vector to S.
(d) Determine the cartesian equation of the tangent plane to S at (1, 0, 2).
Page 3 of 5 pages
MAST20009 Semester 1, 2019
Question 8 (10 marks)
Let D be the region bounded by the curves y =
√
2− x2 and y = |x|. Let C be the boundary
of D, traversed anticlockwise. Let nˆ be the outward unit normal to C in the x-y plane.
(a) Sketch C, indicating the direction of nˆ on each arc of C.
(b) Let F(x, y) = (2x3y + sin4(2y) + xy2, x5 cos3(2x)− 3x2y2). Evaluate the path integral∫
C
F · nˆ ds.
Question 9 (18 marks)
(a) State Stokes’ theorem. Explain all symbols used and any required conditions.
(b) Let S be the surface of the paraboloid
z = x2 + y2 + 3 for z ≤ 7.
Sketch S.
(c) Let F(x, y, z) = (3y + z)i+ yj+ (z2 − x4)k and S be the surface in part (b).
Evaluate the surface integral ∫∫
S
(∇× F) · dS
using
(i) Stokes’ theorem and a line integral;
(ii) a surface integral over the simplest surface.
Question 10 (10 marks)
(a) Let R be a solid region bounded by an oriented closed surface ∂R. Let f(x, y, z) and
g(x, y, z) be C2 scalar functions. Let nˆ be the unit outward normal to ∂R. Show that∫∫∫
R
∇f ·∇g dV =
∫∫
∂R
f∇g · dS−
∫∫∫
R
f∇2g dV.
(b) Suppose that ∂R is a sphere of radius R0 centred at the origin. Let f(r) = r and g(r) = r
2
where r =
√
x2 + y2 + z2.
(i) Find ∇f ·∇g and ∇2g.
(ii) Using parts (a) and (i), show that∫∫∫
R
4r dV =
∫∫
∂R
r2 dS.
Page 4 of 5 pages
MAST20009 Semester 1, 2019
Question 11 (9 marks)
Define oblate spheroidal coordinates (u, θ, φ) by
x = coshu cos θ cosφ, y = coshu cos θ sinφ, z = sinhu sin θ
where u ≥ 0, −pi
2
< θ <
pi
2
, 0 ≤ φ < 2pi.
(a) Let r = (x, y, z). Find
∂r
∂u
,
∂r
∂θ
and
∂r
∂φ
.
(b) Show that the scale factors are
hu = hθ =
√
sinh2 u+ sin2 θ,
hφ = coshu cos θ.
(c) Find an expression for
∇(u2θ3 + φ4)
in terms of u, θ and φ.
End of Exam—Total Available Marks = 120
Page 5 of 5 pages

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