Student
Number
Semester 2 Assessment, 2018
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 4 pages (including this page)
Authorised Materials
• Mobile phones, smart watches and internet or communication devices are forbidden.
• Calculators, tablet devices or computers must not be used.
• No handwritten or print materials may be brought into the exam venue.
Instructions to Students
• You must NOT remove this question paper at the conclusion of the examination.
• You should attempt all questions.
• Start each question on a new page.
• Clearly label each page with the number of the question you are attempting.
• There is a separate 3 page formula sheet accompanying the examination paper, that you
may use in this examination.
• There are 11 questions with marks as shown. The total number of marks available is 110.
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 2, 2018
Question 1 (10 marks) Consider the following function:
g(x, y) =
{
ye−1/x
2
, for (x, y) with x 6= 0,
y, for (x, y) with x = 0.
(a) Calculate lim
(x,y)→(0,0)
g(x, y).
(b) Determine where g is continuous. Justify your answer, referring to any theorems you use.
(c) Using the definition of the partial derivative, calculate
∂g
∂x
and
∂g
∂y
at (x, y) = (0, 0).
Question 2 (10 marks)
(a) Use a Lagrange multiplier to find the point (x, y, z) closest to the origin on the graph of
the function z = x + y − 3. (Hint: to simplify work, take f(x, y, z) = 12(x2 + y2 + z2) as
the function to be minimised.)
(b) Consider the general problem of finding points (x, y, z) which minimise the distance from
the origin on the graph of a function z = g(x, y). Give a system of equations in terms of
x, y, g,
∂g
∂x
, and
∂g
∂y
whose solutions will give such points.
Question 3 (10 marks) A curve C has the parametric equations:
x = 2t, y = t2, z = log t, for 0 < t <∞.
(a) Find the acceleration a(t) and the unit tangent vector T(t) to C.
(b) Find the curvature of C at the point where t = 1.
(c) Let N(t) be the principal normal vector to C. The acceleration at the point t = 1 can be
written as
a(1) = aTT(1) + aNN(1). (You do not need to prove this fact.)
Show that |a(1)| =
√
a2T + a
2
N .
(d) Calculate aT and aN .
Question 4 (10 marks) Let f be a scalar function of order C2 on R3 and let F be a vector field
of order C2 on R3.
(a) Just using the definitions (i.e. without using the identities on the formula sheet),
prove that curl(grad(f)) = 0.
(b) Just using the definitions (i.e. without using the identities on the formula sheet),
prove that div(curl(F)) = 0.
Page 2 of 4 pages
MAST20009 Semester 2, 2018
Question 5 (10 marks) A triangle has vertices (0, 0, 0), (0, 1,−1) and (0, 1, 1). The plane of the
triangle is rotated about the z-axis, and the moving triangle forms a solid (which is a cylinder
from which two parts of a cone have been removed).
(a) Set up a multiple integral in spherical coordinates to calculate the volume of this solid.
(b) Find the volume.
Question 6 (10 marks) Use multiple integration to find the moment of inertia about its axis of
symmetry for a cylinder of radius a, height h, constant density µ and total mass M . Express
your answer in terms of a and M .
Question 7 (10 marks) Let C be the curve
c(t) = (2 cos t)i+ (2 sin t)j+ tk, for 0 ≤ t ≤ 2pi,
and let
F(x, y, z) = 2x i− 4yz2 j− (4y2z − 1)k.
(a) Find f such that F =∇f .
(b) Calculate the work done by the vector field F to move a particle along the curve C in the
direction of increasing t.
Question 8 (10 marks) Evaluate
∫∫
S
F · dS where S is the surface of the bell
z = (1− x2 − y2)e1−x2−y2 for z ≥ 0,
and
F(x, y, z) = (ey cos z, (x3 + 1)
1
2 sin z, x2 + y2 + 3).
(Hint: Use the divergence theorem.)
Page 3 of 4 pages
MAST20009 Semester 2, 2018
Question 9 (10 marks) Let the surface S be the disk x2 + y2 ≤ 9 in the plane z = 2. The
normal to S is directed upwards. Let F(x, y, z) = zi+ xj+ yk.
(a) Evaluate ∫∫
S
(∇× F) · dS,
without using Stokes’ theorem.
(b) Stokes’ theorem asserts that the value of this surface integral is equal to the value of a
certain line integral. Set up and evaluate the line integral.
Question 10 (10 marks)
(a) Sketch the region enclosed by the curves x2 − y2 = 1, x2 − y2 = 9, y = 0 and 2y = x, if
x > 0.
(b) Use the change of variables
u =
y
x
and v = x2 − y2
to find the area of the region. (Hint: Note that
1
1− u2 =
1
2
( 1
1 + u
+
1
1− u
)
.)
Question 11 (10 marks) Use spherical coordinates for this question.
(a) Compute the scale factors hr, hθ, and hϕ for spherical coordinates.
(b) Show that the coordinate system is orthogonal.
(c) Find an expression for ∇θ in terms of r, θ, ϕ and rˆ, θˆ, ϕˆ.
(d) Find an expression for ∇ · (sin θ θˆ) in terms of r, θ, ϕ and rˆ, θˆ, ϕˆ.
End of Exam—Total Available Marks = 110
Page 4 of 4 pages

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