MAST20029 Engineering Mathematics
Engineering Mathematics
MAST20029
Engineering Mathematics
STUDENT NAME:
This compilation has been made in accordance with the provisions of Part VB of the copyright
act for the teaching purposes of the University.
This booklet is for the use of students of the University of Melbourne enrolled in the subject
MAST20029 Engineering Mathematics.
MAST20029 Engineering Mathematics i
MAST20029 Engineering Mathematics
Semester 2 2022
Subject Organisation
MAST20029 Engineering Mathematics is a core mathematics subject that prepares students
for further studies in all branches of Engineering.
This subject is intended only for students pursuing an Engineering Systems major or who
are enrolled in a Master of Engineering degree, who do not wish to take any further study
in Mathematics and Statistics or Physics.
Students who want to supplement their Engineering Systems major with further study
in Mathematics and Statistics or Physics, should seek course advice before enrolling in
MAST20029. In particular, students who want to specialise in Applied Mathematics within
a Mathematics and Statistics Major, should take MAST20009 Vector Calculus, MAST20026
Real Analysis and MAST20030 Differential Equations instead of MAST20029 Engineering
Mathematics.
Syllabus
This subject introduces important mathematical methods required in engineering such as
manipulating vector differential operators, computing multiple integrals and using integral
theorems. A range of ordinary and partial differential equations are solved by a variety of
methods and their solution behaviour is interpreted. The subject also introduces sequences
and series including the concepts of convergence and divergence.
Topics include: Vector calculus, including Gauss’ and Stokes’ Theorems; sequences and
series; Fourier series, Laplace transforms, including convolution; systems of homogeneous
ordinary differential equations, including phase plane and linearisation for nonlinear systems;
second order partial differential equations by separation of variables.
At the completion of this subject, students should be able to
• manipulate vector differential operators
• determine convergence and divergence of sequences and series
• solve ordinary differential equations using Laplace transforms
• sketch phase plane portraits for linear and nonlinear systems of ordinary differential
equations
• represent suitable functions using Fourier series
• solve second order partial differential equations using separation of variables
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ii Course Information 2022
Pre-requisites
One of
• MAST10006 Calculus 2
• MAST10009 Accelerated Mathematics 2
• MAST10019 Calculus Extension Studies
• MAST10021 Calculus 2: Advanced
and one of
• MAST10007 Linear Algebra
• MAST10008 Accelerated Mathematics 1
• MAST10013 UMEP Maths for High Achieving Students
• MAST10018 Linear Algebra Extension Studies
• MAST10022 Linear Algebra: Advanced
Or
• Enrolment in the Master of Engineering
Credit Exclusions
• Students who have completed MAST20009 Vector Calculus or MAST20030 Differential
Equations may not enrol in MAST20029 Engineering Mathematics for credit.
• Concurrent enrolment in MAST20029 Engineering Mathematics and MAST20009 Vec-
tor Calculus is not permitted.
• Concurrent enrolment in MAST20029 Engineering Mathematics and MAST20030 Dif-
ferential Equations is not permitted.
Classes
The subject MAST20029 Engineering Mathematics has
• three one hour lectures per week;
• a one hour practice class per week.
Lectures and practice classes start on the first day of semester. Details of your lecture times
and practice class time are given on your personal timetable in the Student Portal.
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MAST20029 Engineering Mathematics iii
Lectures
Students are expected to attend the lectures.
• Tuesday at 11am in JH Mitchell Theatre, Peter Hall Building;
• Wednesday at 3.15pm in JH Mitchell Theatre, Peter Hall Building;
• Thursday at 11am in JH Mitchell Theatre, Peter Hall Building.
Lecturer: Associate Professor Marcus Brazil (Subject Coordinator), Room 5.11, Electrical
and Electronic Engineering.
The lectures will include live polls. To participate in these you will need to have a mobile
phone or some other internet-enabled device. You will be able to take part in the polls at
the URL: pollev/marcusbrazil098
Subject Resources
Textbook
The recommended textbook for MAST20029 is:
• E Kreyszig, Advanced Engineering Mathematics, 10th Edition, Wiley, 2011.
The textbook is recommended for extra reading and problems but is not compulsory. Any
earlier editions of the textbook are also suitable. The textbook can be borrowed from the Eastern
Resource Centre (ERC) library and the University Library also has an electronic copy of the
book.
The Kreyszig textbook covers all topics in MAST20029 except for sequences and series.
There are many first year calculus textbooks in the ERC library that can be used as a
reference for the sequence and series section of Engineering Mathematics.
Lecture Notes
All students are required to have a copy of the MAST20029 Engineering Mathematics Lecture
Notes, which can be downloaded from the MAST20029 website.
These notes contain the theory, diagrams, and statement of the questions to be covered in
lectures. Students are expected to bring these partial lecture notes to all lectures, and fill in
the working of examples in the gaps provided. This can be done on a tablet/ipad or on a
printed version of the notes.
Practice Class Sheets
A practice class question sheet to be worked on during the practice class will be issued before
each practice class. Students are expected to attempt the questions during the class. Full
solutions to the questions will be provided sometime after the practice class.
The practice class in the first week of semester covers revision material from first year math-
ematics subjects that is essential pre-requisite knowledge for MAST20029. From week 2
onwards, the practice class will be based on the previous week of lectures.
You should aim to complete all the questions on the practice class sheets during semester.
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iv Course Information 2022
Problem Sheets
All students are required to have a copy of the MAST20029 Engineering Mathematics Prob-
lem Sheet Booklet, which can be downloaded from the MAST20029 website.
This problem booklet is for you to work on during your private study time to learn and
practice key concepts, prepare for your practice classes, and to revise for the mid-semester
test and final exam. There are six problem sheets with answers in this booklet corresponding
to the six major topics covered in lectures. At the end of each week you will be advised on
the LMS which questions should be attempted before attending your next practice class.
Whilst working through the problem sheets, make a list of any concepts or topics you are
having difficulty with and ask for help during the individual consultation sessions. Tutors
will not be discussing the specific questions in the problem booklet in the practice classes.
You should aim to complete all the questions in the problem booklet during semester.
Website
All material to do with the assignments, mid-semester test, exam, practice classes, consul-
tation roster and other announcements will be available from the subject LMS (Learning
Management System), at the address:
canvas.lms.unimelb.edu.au
Expectations
In MAST20029 Engineering Mathematics you are expected to:
• Attend all lectures (either in person or watching the video of the lecture), and take
notes and participate in class activities during lectures.
• Attend all practice classes (either in person or online via Zoom), participate in group
work in practice classes, and complete all practice class exercises.
• Work through the problem booklet outside of class in your own time. You should try
to keep up-to-date with the problem booklet questions, and attempt all questions from
the problem booklet before the exam.
• Read the weekly modules, which will be posted on the LMS every Friday afternoon
during the semester.
• Check the announcements on the LMS at least twice per week to make sure you do
not miss any important subject information.
• Complete all assignments on time.
• Seek help when you need it during consultation sessions.
In total, you are expected to dedicate around 170 hours to this subject, including classes.
This equates to an average of about 9 hours of additional study, outside of class, per week
over 14 weeks.
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MAST20029 Engineering Mathematics v
MATLAB
Students are expected to use the software package MATLAB throughout the subject Engi-
neering Mathematics to complete questions on problem sheets and assignments.
Detailed information about MATLAB is provided on pages xiv-xxiv of this booklet.
Assessment
The assessment is composed of three major components:
• A three-hour exam worth 70% at the end of the semester.
• A mid semester test worth 15% on about Tuesday 6th September (exact date and time to
be confirmed closer to the date).
• Three assignments worth 5% each, due as follows:
(1) 4.00pm on Monday 22nd August;
(2) 4.00pm on Monday 19th September;
(3) 4.00pm on Monday 17th October.
The mid semester test and exam will be conducted online, via Zoom. Full details will be
posted on the LMS, closer to the dates.
Hurdle requirement: Students must pass the assessment during semester to pass the sub-
ject. That is, students must obtain a mark of at least 15% out of 30% for the combined mid
semester test and assignments to pass the subject.
Assignments
• The assignments will be posted on the LMS one week before the due date.
• Your assignment must be handwritten, but this includes digitally handwritten documents
using an ipad or a tablet and stylus, which have then been saved as a pdf. Typed or typeset
assignments will not be accepted (unless you have received permission beforehand due to a
medical reason). The only exceptions to this are the answers to the MATLAB questions,
which should be screenshots of your MATLAB session, as explained on the cover sheet.
• Your assignment must be scanned or photographed and compiled into a single pdf document
to be uploaded to the LMS website.
• Extensions
Students with medical certificates or other appropriate supporting documentation can apply
to Marcus Brazil for an extension of up to 3 days after the assignment deadline by emailing
her a scanned copy of the documentation. The assignment should be submitted via the LMS.
Do not use the Student Portal to apply for special consideration for the assignments; this
online application is for the final Engineering Maths exam only.
• Late assignments
The following applies to an assignment submitted late, where no extension has been sought
and granted.
Between one day late and three days late: A mark penalty of 20% of the assignment total
will be deducted from the student’s result for each day the assignment is submitted late. For
example, if the assignment deadline is 4.00pm on Monday, then an assignment submitted
after 4.00pm on Monday and before 4.00pm on Tuesday is one day late, so 20% of the
assignment total will be deducted.
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vi Course Information 2022
More than three days late: Assignment is not accepted and a mark of zero is awarded for the
assignment.
Mid-Semester Test
Special consideration
Students with medical certificates or other appropriate supporting documentation can apply
to Marcus Brazil for special consideration for the mid-semester test. You must request special
consideration no later than 3 days after the date of the mid-semester test. You then have a
further 5 days in which to provide a medical certificate or other requested documentation.
Applications lodged after these time limits will not be accepted.
Do not use the Student Portal to apply for special consideration for the mid-semester test;
this online application is for the final Engineering Maths exam only.
Special consideration outcomes
The following applies to students who have applied for special consideration or who have
missed the mid-semester test for any reason.
• Students who sat the test but were sick during the test will be allowed to sit the alterna-
tive sitting of the mid-semester test. Any student who sits the original test as well as the
alternative test will be given the mark from the alternative test, even if it is lower than their
mark in the original test.
• Students unable to sit the test due to illness or any other acceptable reason will be allowed
to sit the alternative sitting of the mid-semester test.
• Students who did not sit the test and did not give any reasonable excuse will receive a mark
of zero for the mid-semester test.
Special Consideration for Exam and Whole Subject
If something major goes wrong during semester or you are sick during the examination
period, you should consider applying for Special Consideration through the Student Portal.
You must submit your online special consideration application no later than 4 days after
the date of the final exam in MAST20029 Engineering Mathematics. You will also need to
submit the completed Health Professional Report (HPR) Form with your online application.
The HPR Form can only be completed by the professional using the form provided.
For more details see the Special Consideration menu item on the website:
http://ask.unimelb.edu.au/app/home
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MAST20029 Engineering Mathematics vii
Calculators, Formula Sheets and Dictionaries
Students are not permitted to use calculators, dictionaries, computers, or any electronic
resources in the end of semester exam.
The formula sheet on pages x to xiv of this booklet will be provided in the end of semester
exam, as part of the exam paper.
Assessment in this subject concentrates on the testing of concepts and the ability to conduct
procedures in simple cases. There is no formal requirement to possess a calculator for this
subject. Nonetheless, there are some questions on the problem sheets for which calculator
usage is appropriate. If you have a calculator or an equivalent app on your phone or laptop,
then you will find it useful occasionally.
Getting Help
Lecturers and tutors have consultation hours when they will help you on an individual basis
with questions from the MAST20029 Engineering Mathematics lecture notes, problem sheets
and practice class sheets. Attendance is on a voluntary basis. Details will be provided on
the MAST20029 LMS web site.
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viii Course Information 2022
Lecture Outline
This lecture outline is a guide only. The material to be covered in each lecture may vary
slightly from the following table.
Vector Calculus
1. Vector fields, div and curl operators.
2. Double integrals over general regions, change of order of integration.
3. Double integrals - change of variables, polar coordinates.
4. Triple integrals - change of variables, cylindrical coordinates.
5. Triple integrals in cylindrical coordinates and spherical coordinates.
6. Parametrisation of paths. Line integrals.
7. Work integrals. Conservative fields.
8. Integrals of scalar functions over surfaces, surface area, mass.
9. Integrals of vector functions over surfaces, flux.
10. Gauss’ divergence theorem.
11. Stokes’ theorem.
Systems of First Order Ordinary Differential Equations
12. Systems of linear homogenous ODEs. Solve using eigenvalues and eigenvectors.
13. 2× 2 systems examples, phase space, critical points, phase portraits.
14. Phase portraits of linear systems.
15. Phase portraits of linear systems. Non-linear coupled first order ODEs.
16. Non-linear coupled first order ODEs, linearisation, phase portraits.
Laplace Transforms
17. Laplace transforms. Table of transforms.
18. Inversion of transforms using tables. Solution of ODEs (single and systems).
19. Mid-Semester test (no lecture on that day)
20. S-shifting theorem. Step functions.
21. T-shifting theorem. Impulse and Dirac delta functions.
22. Convolution theorems. Solution of integral equations.
Series
23. Infinite series - partial sums, geometric series, harmonic series.
24. Integral test. Comparison test. Ratio test.
25. Leibniz test. Power series - radius and interval of convergence.
26. Power series (continued). Taylor polynomials.
27. Taylor series, errors.
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MAST20029 Engineering Mathematics ix
Fourier Series
28. Periodic functions, Fourier series, Euler’s formulae, energy density,
Parseval’s identity.
29. Fourier series for odd and even functions, periodic extensions.
30. Application of Fourier series to ODEs.
31. Fourier integrals, odd/even functions, applications to ODEs.
Second Order Partial Differential Equations
32. Examples and classification of second order PDEs.
33. Separation of variables for Laplace’s equation.
34. Separation of variables for the wave equation.
35. Separation of variables for the diffusion equation.
36. Revision lecture
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