School of Mathematics and Statistics
MAST20009 Vector Calculus, Semester 2 2019
Assignment 4 and Cover Sheet
Student Name Student Number
Tutor’s Name Tutorial Day/Time
Submit your assignment to your tutor’s MAST20009 assignment box
before 11am on Tuesday 22nd of October.
• This assignment is worth 5% of your final MAST20009 mark.
• Assignments must be neatly handwritten in blue or black pen on A4 paper or typed using Lateχ.
For Lateχ assignments, also email a copy of the source code to [email protected].
• Diagrams can be drawn in colour on grid paper (use ruler and compass). Tikz pictures are acceptable
as long as they include a grid.
• You must complete the plagiarism declaration on the LMS before submitting your assignment.
• Full working must be shown in your solutions.
• Marks will be deducted for incomplete working, insufficient justification of steps, incorrect
mathematical notation and for messy presentation of solutions.
1. Consider the domain D ⊂ R2 obtained by removing the unit circle from the square centred at the
origin with side length 4. Equip D with the positive orientation.
(a) Draw D and indicate clearly in your drawing the orientation of D and the compatible orientation on
each of the boundary components.
(b) Let now ~F be the vector field
~F
[
x
y
]
=
[
x− y
xy
]
on R2. Verify Green’s theorem for ~F on D.
2. Let T be the torus, parametrized as in the notes, with R = 3. Consider the function
f(x, y, z) = 2x− y
on R3.
(a) Write down the parametrization of T from the notes.
(b) Produce a high quality drawing (following specifications) of the torus T and the grid you obtain
when fixing values of ϕ or θ, depicting at least five lines in either direction of the grid.
(c) Compute the tangent vectors ~Tϕ and ~Tθ and the outward normal vector ~n.
(d) Compute the surface integral ∫∫
T
f dS.
3. Let ~F be the vector field
~F
[
x
y
]
=
[−y
y
]
on R2, and let c be the semi-circle of radius 1 in the x − y-plane centred at (2, 0) and moving in anti-
clockwise direction from (2,−1) to (2, 1).
(a) Determine the work done by ~F to move a particle along c.
(b) Compute the curl and the divergence of ~F .