The University of Sydney
School of Mathematics and Statistics
Assignment 1
MATH2021: Vector Calculus and Differential Equations
Semester 1, 2024
Lecturer: Pieter Roffelsen and Haotian Wu
This individual assignment is due by 11:59pm Thursday 28 March 2024, 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.
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include your name in any area of your assignment; only your SID should be present.
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on problems, but students must write up and submit their own version of the solutions.
This assignment is worth 10% 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.
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1
1. Let F(x, y, z) = (y + 2xyz3, x + x 2 z3, 1 + 3x 2 yz2 ).
(a) Show that ∇ × F = 0.
(b) Find a function f : R3 → R such that F = ∇ f .
(c) Hence, or otherwise, find the value of the integral
∫
F · d s,
C
where C is the part of the curve with parametrisation γ(t) = (2 cos t, 3 sin t, 4t/π) with
t ∈ [0, π].
√
2. Let C be the part of the graph of y = x 2 − 1 between x = 2 and x = 3.
(a) Write down a parametrisation of C, and find the velocity vector of your parametrisation.
(b) Find the line integral
∫
4xy ds.
C
3.
(a) You are given the iterated integral
∫ 1 ∫ √x
f (x, y) dydx.
x3
0
Sketch the domain of integration of this iterated integral (your sketch should be drawn
by yourself not by some software) and rewrite it as an iterated integral in the reverse
order (that is, with inner integral with respect to x and outer integral with respect to y).
(b) Let P be the parallelogram in R2 with vertices (0, 0), (1, 2), (5, −2), and (6, 0). Calculate
the double integral
∬
(y − 2x) dA.
P
4. Calculate the area enclosed by the curve
γ(t) = (6 cos t − cos(6t), 6 sin t − sin(6t)),
t : 0 → 2π.
A sketch of the curve is shown below.
y
6
4
2
-6
-4
2
-2
-2
-4
-6
2
4
6
x