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MATH1023: Multivariable Calculus and Modelling

创建日期：2024-05-27 16:29:56

所属分类：数学mathematics

标签： MATH1023

Multivariable Calculus and Modelling

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Assignment 2

MATH1023: Multivariable Calculus and Modelling Semester 1, 2023
Lecturer: Clio Cresswell
This individual assignment is due by 11:59pm Wednesday 03 May 2023, via
Canvas. Late assignments will receive a penalty of 5% per day until the closing date.
A single PDF copy of your answers must be uploaded in Canvas at https://canvas.
sydney.edu.au/courses/48841. It should include your SID. Please make sure you
review your submission carefully. What you see is exactly how the marker will see your
assignment. Submissions can be overwritten until the due date. 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. If you have technical difficulties with your submission,
see the University of Sydney Canvas Guide, available from the Help section of Canvas.
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 of explanation
and diagrams where relevant. After all, mathematics is about communicating your ideas. This
is a worthwhile skill which takes time and effort to master. The marker will give you feedback
and allocate an overall mark to your assignment using the following criteria:
Copyright c© 2023 The University of Sydney 1
2
1. Consider the following differential equation for y(x) and F (x)
d2y
dx2
+ 2a
dy
dx
+ 9a2y = F (x) (1)
where a is some arbitrary non-zero real constant and y(0) = 3 and y′(0) = a.
(a) For which values of a does y(x) exhibit oscillations?
(b) Solve equation (1) where F (x) = 0.
(c) Solve equation (1) where F (x) = e−ax. Check that your final solution indeed
satisfies y(0) = 3 and y′(0) = a. If not, you have made a mistake!
(d) Solve equation (1) where F (x) = e−ax sin(2
√
2ax + R), for some real constant R.
(e) Part (d) yields more than one particular solution. Is your final solution to part
(d) unique? Discuss briefly (no more than 6 lines). There is no need to prove your
result.
2. Consider f(x, y) =
y2 − x2
2
and g(u, v) = uv.
(a) Draw contour lines for f(x, y) and g(u, v) in a single xy-plane and single uv-
plane respectively. These should correspond to f = 0, 1,−1, 2,−2, 3,−3 and g =
0, 1,−1, 2,−2, 3,−3 and any axes-intercepts should be clearly marked.
(b) At what points and for which c, do the lines v = u and v = −u cross the contour
g = c? At what distance from the origin do these intersections occur?
(c) At what points and for which c, do the lines x = 0 and y = 0 cross the contour
f = c? At what distance from the origin do these intersections occur?
(d) The above work suggests a geometric transformation will transform the graph of
f to the graph of g. You could use a 3D grapher, such as that available from
the MATH1023 Canvas site, to convince yourself further. Say the transformation
is u = T1(x, y) and v = T2(x, y). Find functions T1 and T2 that honour this
transformation where to preserve symmetry require the distances from the origin to
be equal. In other words, require
√
x2 + y2 =
√
u2 + v2, or simply x2+y2 = u2+v2.

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