The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH2021: Vector Calculus and Differential Equations Semester 1, 2025
Lecturer: Peter Kim
This individual assignment is due by 11:59pm Thursday 3 April 2025, via Canvas. Late assignments
will receive a penalty of 5% of the maximum mark for each calendar day after the due date. After ten
calendar days late, a mark of zero will be awarded.
A single PDF copy of your answers must be uploaded in Canvas. Please make sure you review your
submission carefully. What you see is exactly how the marker will see your assignment.
To ensure compliance with our anonymous marking obligations, please do not under any circumstances
include your name in any area of your assignment; only your SID should be present.
The School of Mathematics and Statistics encourages some collaboration between students when working
on problems, but students must write up and submit their own version of the solutions.
This assignment is worth 6% of your final assessment for this course. Your answers should be well written,
neat, thoughtful, mathematically concise, and a pleasure to read. Please cite any resources used and show all
working. Present your arguments clearly using words or explanations and diagrams where relevant. After all,
mathematics is about communicating your ideas. This is a worthwhile skill which takes time and effort to
master.
Copyright © 2025 The University of Sydney 1
1. Let (, ) = ( + 2 cos + 32, + 2 sin + 43) .
(a) Show that curl() = 0.
(b) Find a function : R2 → R such that = ∇ .
(c) Now, let
(, ) = (, ) − + .
Hence, or otherwise, find the value of the integral∫

· ,
where is the curve parameterised by () = (2 cos , 2 sin ) with ∈ [0, 2].
2. Let be the ellipsoid in R3 defined by
2 +
2
9
+
2
4
≤ 1.
Evaluate ∭

2 .
3. (a) Let be the curve parameterised by
() = (cos3 , sin3 ), : 0 → 2.
Let
(, ) = (, (1 + ))
for a constant ∈ R. Show that the integral∫

·
is the same for all .
(b) Use Green’s Theorem to calculate the area enclosed by the curve given above. Show
your reasoning and work. (Note: There are multiple ways to calculate this area, and they
essentially do the same thing, but please do it using Green’s Theorem.)
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