The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH2021: Vector Calculus and Differential Equations Semester 1, 2026
Lecturers: Peter Kim & Zhou Zhang
This individual assignment is due by 11:59pm Sunday 5 April 2026, 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 © 2026 The University of Sydney 1
1. Let be the curve parameterised by
() =
(
3 cos 4, 3 sin 4
)
, ∈ [0, 2] .
Find the parameterisation of by arc length, i.e., find the parameterisation () of , such that
| |′() | | = 1 for all ≥ 0.
2. Use Green’s Theorem to find the area enclosed by the curve parameterised by
= cos() + 1
6
cos(6),
= sin() + 1
6
sin(6),
∈ [0, 2). This curve is called a hypocycloid. (I am sure there are other ways to calculate the
area, but for this assignment, use Green’s Theorem to do so.)
3. Let be the solid region defined by
=
{ (, , ) ∈ R3 2 + 2 + 2 ≤ 5, 92 + 92 ≤ ( + 5)2} .
Calculate the volune of .
2

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