WB50 All

About
the HELM Project

HELM (Helping Engineers Learn Mathematics) materials were the outcome of a three-‐year curriculum
development project undertaken by a consortium of five English universities led by Loughborough University,
funded by the Higher Education Funding Council for England under the Fund for the Development of Teaching
and Learning for the period October 2002 – September 2005, with additional transferability funding October
2005 – September 2006.
HELM aims to enhance the mathematical education of engineering undergraduates through flexible learning
resources, mainly these Workbooks.
HELM learning resources were produced primarily by teams of writers at six universities: Hull, Loughborough,
Manchester, Newcastle, Reading, Sunderland.
HELM gratefully acknowledges the valuable support of colleagues at the following universities and colleges
involved in the critical reading, trialling, enhancement and revision of the learning materials:
Aston, Bournemouth & Poole College, Cambridge, City, Glamorgan, Glasgow, Glasgow Caledonian, Glenrothes
Institute of Applied Technology, Harper Adams, Hertfordshire, Leicester, Liverpool, London Metropolitan,
Moray College, Northumbria, Nottingham, Nottingham Trent, Oxford Brookes, Plymouth, Portsmouth,
Queens Belfast, Robert Gordon, Royal Forest of Dean College, Salford, Sligo Institute of Technology,
Southampton, Southampton Institute, Surrey, Teesside, Ulster, University of Wales Institute Cardiff, West
Kingsway College (London), West Notts College.

HELM Contacts:
Post: HELM, Mathematics Education Centre, Loughborough University, Loughborough, LE11 3TU.
Email: helm@lboro.ac.uk Web: http://helm.lboro.ac.uk

HELM Workbooks List

1 Basic Algebra 26 Functions of a Complex Variable
2 Basic Functions 27 Multiple Integration
3 Equations, Inequalities & Partial Fractions 28 Differential Vector Calculus
4 Trigonometry 29 Integral Vector Calculus
5 Functions and Modelling 30 Introduction to Numerical Methods
6 Exponential and Logarithmic Functions 31 Numerical Methods of Approximation
7 Matrices 32 Numerical Initial Value Problems
8 Matrix Solution of Equations 33 Numerical Boundary Value Problems
9 Vectors 34 Modelling Motion
10 Complex Numbers 35 Sets and Probability
11 Differentiation 36 Descriptive Statistics
12 Applications of Differentiation 37 Discrete Probability Distributions
13 Integration 38 Continuous Probability Distributions
14 Applications of Integration 1 39 The Normal Distribution
15 Applications of Integration 2 40 Sampling Distributions and Estimation
16 Sequences and Series 41 Hypothesis Testing
17 Conics and Polar Coordinates 42 Goodness of Fit and Contingency Tables
18 Functions of Several Variables 43 Regression and Correlation
19 Differential Equations 44 Analysis of Variance
20 Laplace Transforms 45 Non-‐parametric Statistics
21 z-‐Transforms 46 Reliability and Quality Control
22 Eigenvalues and Eigenvectors 47 Mathematics and Physics Miscellany
23 Fourier Series 48 Engineering Case Study
24 Fourier Transforms 49 Student’s Guide
25 Partial Differential Equations 50 Tutor’s Guide

© Copyright Loughborough University, 2015

Production of this 2015 edition, containing corrections and minor
revisions of the 2008 edition, was funded by the sigma Network.

Contents 50
Tutor's Guide
50.1 Introduction to HELM 1
50.2 HELM Consortium, Triallist Institutions and Individual Contributors 4
50.3 HELM Transferability Project (HELMet) 7
50.4 HELM Workbook Structure and Notation 10
50.5 Issues and Notes for Tutors 13
50.6 Workbook Layout 18
50.7 List of Sections in Workbooks 1 to 48 22
50.8 Commentaries on Workbooks 1 to 48 29
50.9 Index of Engineering Contexts in Workbooks 1 to 48 66

®

Introduction to HELM 50.1
The HELM project (Helping Engineers Learn Mathematics) was supported by a £250,000 HEFCE
FDTL4 grant for the period Oct 2002-Sept 2005. A HEFCE - funded Transferability Study was
undertaken October 2005-September 2006 encouraging the wider uptake of the use of the HELM
materials.
1. The HELM project

The HELM team comprised staff at Loughborough University and four consortium partners in other
English universities: Hull, Manchester, Reading and Sunderland. The project’s aims were to consid-
erably enhance, extend and thoroughly test Loughborough’s original Open Learning materials. These
were to be achieved mainly by the writing of additional Workbooks and incorporating engineering
examples and case studies closely related to the mathematics presented, enhancing the question data-
banks, upgrading the Interactive Learning segments and adding some more for basic mathematics
topics, and promoting widespread trialling.
The HELM project’s output consisted of Workbooks, Interactive Learning segments, a Computer
Aided Assessment regime which is used to help ‘drive the student learning’ and a report on possible
modes of usage of this flexible material.
The Workbooks may be integrated into existing engineering degree programmes either by selecting
isolated stand-alone units to complement other materials or by creating a complete scheme of work
for a semester or year or two years by selecting from the large set of Workbooks available. These
may be used to support lectures or for independent learning.
HELM’s emphasis is on flexibility - the work can be undertaken as private study, distance learning or
can be teacher-led, or a combination, according to the learning style and competence of the student
and the approach of the particular lecturer.
HELM (2015): 1
Section 50.1: Introduction to HELM
2. HELM project Workbooks
50 Workbooks are available which comprise:
• 46 Student Workbooks (listed in 50.4) written specifically with the typical engineering student
in mind containing mathematical and statistical topics, worked examples, tasks and related
engineering examples.
• A Workbook containing an introduction to dimensional analysis, supplementary mathematical

topics and physics case studies.
• A Workbook containing Engineering Case Studies ranging over many engineering disciplines.
• A Students’ Guide
• A Tutor’s Guide (this document)
The main project materials are the Workbooks which are subdivided into manageable Sections. As
far as possible, each Section is designed to be a self-contained piece of work that can be attempted
by the student in a few hours. In general, a whole Workbook typically represents 2 to 3 weeks’ work.
Each Workbook Section begins with statements of pre-requisites and the desired learning outcomes.
The Workbooks include (a) worked examples, (b) tasks for students to undertake with space for
students to attempt the questions, and, often, intermediate results provided to guide them through
problems in stages, and (c) exercises where normally only the answer is given.
It is often possible for the lecturer to select certain Sections from a Workbook and omit other Sections,
possibly reducing the reproduction costs and, more importantly, better tailoring the materials to the
needs of a specific group.
With funding from sigma the workbooks were updated during 2014 and republished Spring 2015,
and are now available to all Higher Education Institutes worldwide.
3. HELM project Interactive Learning Segments

These are now outdated and unavailable.
2 HELM (2015):
Workbook 50: Tutor’s Guide
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4. HELM project Assessment Regime

In formal educational environments assessment is normally an integral part of learning, and this was
recognised by the HELM project. Students need encouragement and confirmation that progress is
being made. The HELM assessment strategy was based on using Computer-Aided Assessment (CAA)
to encourage self-assessment, which many students neglect, to verify that the appropriate skills have
been learned. The project’s philosophy was that assessment should be at the heart of any learning
and teaching strategy and Loughborough University’s own implementation of HELM makes extensive
use of CAA to drive the students’ learning.
In the past HELM provided an integrated web-delivered CAA regime based on Questionmark Percep-
tion for both self-testing and formal assessment, with around 5000 questions; most having a page of
specific feedback. These are now largely superseded and so no longer available to other institutions.
HELM (2015): 3
Section 50.1: Introduction to HELM
HELM Consortium,
Triallist Institutions
and Individual
Contributors 50.2
HELM learning resources were produced primarily by a consortium of writers and developers at five
universities:
Hull, Loughborough, Manchester, Reading, Sunderland.
The HELM consortium gratefully acknowledges the valuable support of many colleagues at their
institutions and at the following institutions involved in additional writing, critical reading, trialling
and revising of the learning materials:
Universites Other HE/FE Institutions

Aston Bournemouth & Poole College
Cambridge Glenrothes Institute of Applied Technology
City Harper Adams University College
Glamorgan Moray College
Glasgow Royal Forest of Dean College
Glasgow Caledonian Sligo Institute of Technology
Hertfordshire Westminster Kingsway College
Leicester West Notts College
Liverpool
London Metropolitan
Newcastle
Northumbria
Newcastle
Nottingham
Nottingham Trent
Oxford Brookes
Plymouth
Queen’s Belfast
Robert Gordon
Southampton
Southampton Solent
Surrey
Teesside
Ulster
University of Wales Institute Cardiff
4 HELM (2015):
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Individual Consortium Contributors

Loughborough University Finance
Writing team Dr David Green
Dr Tony Croft University of Hull
Dr David Green Writing team
Dr Rhian Green Prof Keith Attenborough
Dr Martin Harrison Dr Mary Attenborough
Stephen Lee Dr Patrice Boulanger
Dr Peter Lewis Dr John Killingbeck
Dr Leslie Mustoe Don Maskell
David Pidcock
University of Manchester
Prof Chris Rielly
Writing team
Dr Joe Ward
Mr Max Caley
Advisors Dr Paul JW Bolton
Dr Nigel Beacham Dr Alison M Durham
Dr Sarah Carpenter Mr Kamlesh Ghodasara
Dr Adam Crawford Dr Colin Steele
Myles Danson
Advisors
Lesley Davis
Dr David Apsley
Dr Aruna Palipana
Dr Adrian Bell
Clare Trott
Dr Shuguang Li
Dr Andy Wilson
Dr John Parkinson
Administration and Production
Prof. Stephen Williamson
Dr David Green
University of Reading
Dr Martin Harrison
Writing team
Louise Kitching
Dr Peter Chamberlain
Sophie McAloon
Dr David Stirling
Dr Aruna Palipana
David Pidcock University of Sunderland
Helen Sherwood Writing team
Dr Joe Ward Dr Malcolm Farrow (later Newcastle Univ)
Clare Wright Walter Middleton
HELM (2015): 5
Section 50.2: HELM Consortium, Triallist Institutions and Individual Contributors
Contributors from other institutions
The following is a list of all those at other institutions who have helped in the development of
the HELM materials by contributing examples and text, critically reading, providing feedback from
trialling, pointing out errors, and offering general advice and guidance.
Felix Ampofof Brian Lowe

Philip Aston Brian Malyan
Rob Beale David Malvern
Kevin Brace John Martin
Pargat Calay Sye Murray
Richard Cameron Shepard Ndlovu
Abbi Coman Ricketts Chris O’Hagan
Grace Corcoran Paul Oman
Bob Crane Steve Outram
Gavin Cutler Norman Parkes
Mansel Davies Sarah Parsons
Ian Drumm Lynn Pevy
Mark Gerrard Alexei Piunovsky
David Graham Margaret Platt
Jim Grimbleby Julie Renton
Christopher Haines Mick Scrivens
Stephen Hibberd Elaine Smith
Alison Holmes Alan Stevens
Ken Houston Richard Steward
Peter Hudson Ian Taylor
Sudhir Jain Stephen Thorns
Sanowar Khan James Vickers
Joseph Kyle Juliette White
David Livie James Wisdom
Peter Long Hilary Wood
David Lowe Lindsay Wood
6 HELM (2015):
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HELM Transferability
Project (HELMet) 50.3
1. Introduction
During the course of the HELM Project some 30 institutions implemented one or more of the three
aspects of the HELM Resources (Workbooks, Interactive Learning Segments, CAA questions) into
at least one of their programmes.
As CAL and CAA are no longer supported, discussions here focus on the HELM Workbooks alone.
2. Use of Workbooks
2.1 Workbooks as lecture notes (hard copy format)
In the cases where HELM Workbooks were used to replace existing lecture notes (20% of implemen-
tations), students were issued hard copies of the Workbooks relevant to their modules. Workbooks
were either issued in full at the beginning of each topic or subsections were issued lecture by lecture.
The lecturers choosing to issue relevant sections lecture by lecture felt that this gave them the ad-
vantage of controlling what the students had before them in the class and allowing the lecturer to
give very focussed direction to the students on what they should be doing during a particular time
period. However, issuing complete Workbooks eased any complications in reproduction and simplified
the situation when students had been absent, as they simply knew which Workbook to collect rather
than having to identify particular subsections. In all cases, the materials were made available prior to
the lecturer beginning the topic, allowing students to preview the material in advance of the lectures.
Lecturers, using the Workbooks as their core notes essentially employed one of two approaches to
their lectures.
Some staff, particularly those teaching the material for the first time or for the first time to a particular
group, developed lectures, often Powerpoint based, which reflected the content of the Workbooks
exactly, using the same or very similar examples. Students found this reassuring; they felt safe that
they did not have to take notes, therefore being able to focus on understanding, and they knew
exactly where they were in the material enabling them to easily follow the lecture, and could make
annotations to their Workbooks in the appropriate places. However, some students did feel that this
made the lecture have less value as they could simply study for themselves at home and that in such
lectures they did not have the opportunity to work through the examples themselves to reinforce their
learning.
HELM (2015): 7
Section 50.3: HELM Transferability Project (HELMet)
Other staff lectured in parallel to the content of the Workbook, using their own examples and
developing some of the theory using their own existing notes. Students were kept appraised of which
sections of the Workbooks were being addressed and were directed to work through specific examples
and exercises to follow up the material covered within the lectures. Students appreciated that this
allowed them to see more examples than just those contained in the Workbooks, but they did tend to
feel that in these circumstances they needed to make more extensive notes. It is helpful to students
that when this second approach is chosen that notation used by the lecturer is kept consistent with
that in the Workbooks even where this is not the lecturer’s preferred method - lecturers are more
able to adapt than students struggling to comprehend new mathematical concepts.
2.2 Workbooks as supplementary linked material for whole modules
The largest subgroup (50% of implementations) used the Workbooks as supplementary notes, linking
them explicitly to the content of the lectures.
In 60% of these cases, hard copies of the Workbooks were issued to students at lectures or tutorials
thus providing an incentive to attend. In the other 40% of these cases, electronic links to the
Workbooks sited on the institutions’ websites were given.
Students were directed to use the material to help them understand the content of their lectures
and to provide additional examples and exercises to use during tutorials and for private study. It
was found that few students made the effort to work through the HELM material if they viewed it
as an optional (albeit very useful) extra. Some students also expressed a lack of conviction about
working through the Workbooks in class, being content to attend tutorials to collect the Workbooks
and taking them home unopened.
Some success has been achieved using a peer tutoring scheme where the Workbooks are used as the
subject matter. This provided a focus for the sessions and excellent support for students struggling
with their mathematics and strong backup for their mentors.
Nevertheless, interviewed students did, in all cases, appreciate having the materials and felt that
when they were preparing for examinations they would be very useful.
2.3 Workbooks as supplementary linked material for parts of modules
Of those institutions using the Workbooks as supplementary materials, around 20% restricted use of
the materials to a small subset of the module content. They tended to use the most fundamental
Workbooks (1: Basic Algebra, 2: Basic Functions, 3: Equations, Inequalities & Partial Fractions) as
they wanted to establish the foundations with what would be regarded as fairly weak mathematicians.
For more advanced mathematical topics they continued with their existing notes. They found that
using the materials for this group allowed the students to be able to spend time honing their skills
before moving on to more challenging topics.
8 HELM (2015):
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2.4 Workbooks as unlinked supplementary material
Some 30% of implementations used the Workbooks as supplementary materials and made the Work-
books available to students but did not specifically attach them to particular modules or courses. In
most cases the Workbooks were made available online for students to download and (if required)
print off themselves. Links to the Workbooks were provided by lecturers and/or through support
centres.
In the support centres, hard copies were sometimes obtainable on request or at least there were
reference copies available. It is difficult to measure the rate of uptake in these circumstances.
However, having this range of materials available as part of a support service for students is seen as
valuable by students and staff alike.
It was noticeable that mature students and those with special needs (typically dyslexics) were
very appreciative of the Workbooks and did make extensive use of them, no matter which usage
mode was employed, citing being able to work at their own pace and being confident that they had
a complete set of notes as large positives.
3 Tutorial assistants
Many institutions use postgraduates for tutoring and the HELM Workbooks provide a sure foundation
for them. (It is well documented that staff often fail to keep their tutorial assistants adequately
informed of their lecture notes and chosen methods, and students seeking help often have woefully
inaccurate notes.)
HELM (2015): 9
Section 50.3: HELM Transferability Project (HELMet)
HELM Workbook
Structure and Notation 50.4

1. List of Workbooks
No. Title Pages

1 Basic Algebra 89
2 Basic Functions 75
3 Equations, Inequalities & Partial Fractions 71
4 Trigonometry 77
5 Functions and Modelling 49
6 Exponential and Logarithmic Functions 73
7 Matrices 50
8 Matrix Solution of Equations 32
9 Vectors 66
10 Complex Numbers 34
11 Differentiation 58
12 Applications of Differentiation 63
13 Integration 62
14 Applications of Integration 1 34
15 Applications of Integration 2 31
16 Sequences and Series 51
17 Conics and Polar Coordinates 43
18 Functions of Several Variables 40
19 Differential Equations 70
20 Laplace Transforms 72
21 z-Transforms 96
22 Eigenvalues and Eigenvectors 53
23 Fourier Series 73
24 Fourier Transforms 37
25 Partial Differential Equations 42
26 Functions of a Complex Variable 58
27 Multiple Integration 83
28 Differential Vector Calculus 53
29 Integral Vector Calculus 80
30 Introduction to Numerical Methods 64
31 Numerical Methods of Approximation 86
32 Numerical Initial Value Problems 80
33 Numerical Boundary Value Problems 36
34 Modelling Motion 63
10 HELM (2015):
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No. Title Pages

35 Sets and Probability 53
36 Descriptive Statistics 51
37 Discrete Probability Distributions 60
38 Continuous Probability Distributions 27
39 The Normal Distribution 40
40 Sampling Distributions and Estimation 22
41 Hypothesis Testing 42
42 Goodness of Fit and Contingency Tables 24
43 Regression and Correlation 32
44 Analysis of Variance 57
45 Non-parametric Statistics 36
46 Reliability and Quality Control 38
47 Mathematics and Physics Miscellany 69
48 Engineering Case Studies 97
49 Student’s Guide 31
50 Tutor’s Guide 75
2. Nature of the Workbooks

The 50 HELM Workbooks comprise:
• 33 Mathematics Workbooks (1 to 33) written specifically with the typical engineering student in
mind containing mathematical topics, worked examples, tasks, exercises and related engineering
examples.
• 1 Workbook (34) emphasising the mathematical modelling of motion, with worked examples,
tasks, exercises and related engineering examples.
• 12 Statistics Workbooks (35 to 46) written specifically with the typical engineering student in
mind containing statistical topics, worked examples with an emphasis on engineering contexts,
tasks and exercises.
• 1 Workbook (47) containing Sections on Dimensional analysis, Mathematical explorations and

11 Physics case studies.
• 1 Workbook (48) containing 20 Engineering Case Studies ranging over many engineering dis-
ciplines.
• 1 Student’s Guide (49) containing helpful advice, various indexes and extensive facts and
formulae sheets for mathematics and mechanics.
• 1 Tutor’s Guide (50) relating success stories and challenges and encapsulating good practice
derived from trialling in a variety of institutions with their individual contexts and cultures.
The Workbooks are subdivided into manageable Sections. As far as possible, each Section is designed
to be a self-contained piece of work that can be attempted by the student in a few hours. In general,
HELM (2015): 11
Section 50.4: HELM Workbook Structure and Notation
a whole Workbook represents about 2 to 3 weeks’ work. Each Section begins with statements of
prerequisites and desired learning outcomes.
The Tasks include space for students to attempt the questions; many Tasks guide the student through
problems in stages.
It is often feasible for the lecturer to select certain Sections from a Workbook and omit others,
reducing the reproduction costs if using hardcopy and better tailoring the materials to the needs of
a specific group.
3. Notation Used
Fonts
In general HELM uses italic serif font letters to represent functions, variables and constants. However,
as exceptions HELM Workbooks use the following non-italic sans-serif letters:
Mathematics
e for the exponential constant and for the exponential function (primarily use in introductory
Workbook 6, elsewhere e is often used)
i where i2 = −1
ln for natural logarithm
Statistics
E for Expectation P for Probability V for Variance M for Median
Complex numbers
p
HELM uses i rather than j to represent (−1) so i2 = −1, although there are one or two exceptions
to this (in Workbook 48: Engineering Case Studies).
Vectors
HELM uses underlining of vectors rather than using bold e.g. a
HELM uses n̂ for the unit normal vector but does not put the ˆ on the basic unit vectors in the x, y
and z directions which have the standard symbols i, j, k.
Identities
Although HELM introduces and uses the identity symbol ‘≡’ extensively in Workbook 1: Basic
Algebra and in Workbook 4: Trigonometry it is not normally used elsewhere and the more normal
‘=’ is used except where emphasis seems advisable. (HELM is therefore not consistent.)
12 HELM (2015):
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Issues and Notes
for Tutors 50.5
1. The use of units in applied problems
Problems in engineering almost invariably involve physical quantities, for example distance, mass,
time, current, measured in a variety of units: metres, miles, amps, litres, etc. When using mathe-
matics to solve applied problems we often are a little slipshod in our approach to the way unknown
variables and associated units are introduced and used.
We may ask for the solution of the equation sin x = 0.5 where x, measured in radians, is in the
interval 0 ≤ x < 2π . But should we say
(a) “Solve the equation sin x = 0.5 where x, measured in degrees, is in the interval 0 ≤ x < 360”,
or
(b) “Solve the equation sin x◦ = 0.5 where x is in the interval 0 ≤ x < 360”, or
(c) “Solve the equation sin x = 0.5 where x is in the interval 0◦ ≤ x < 360◦ ”?
Purists may prefer (b) of these options.
Now consider the following problem.
A train is travelling at 80 km h1 and is 2 km from the next station when it starts to brake so that it
comes to a halt at the station. What is the deceleration, assuming it to be uniform during the time
that the brakes were applied?
Here is the ‘correct’ solution.
Let the deceleration be a km h−2 .
In 2 km the speed of the train will have decreased by 80 km h−1 .
We use the formula v 2 = u2 − 2as, where u km h−1 is the initial speed,
v km h−1 is the final speed and s km is the distance travelled.
u2 − v 2 80 × 80 − 0 × 0 80 × 80
Hence = = = = 1600.
2s 2×2 2×2
The (uniform) deceleration is therefore 1600 km h−2 .
In this approach, a represents the magnitude of the deceleration and is a pure number.
HELM (2015): 13
Section 50.5: Issues and Notes for Tutors
A less ‘professional’ solution now follows.
Let the deceleration be a.
In 2 km the speed of the train will have decreased by 80 km h−1 .
We use the formula v 2 = u2 − 2as where u is the initial speed, v is the final speed and s is the
distance travelled.
u2 − v 2 80 × 80
Hence a = = = 1600 km h2 .
2s 2×2
In this second approach, a represents the deceleration itself and so has units. It seems simpler and
80 × 80
maybe more natural. However, we then get the inconsistency that the fraction , a number,
2×2
is equated to a deceleration a, which is nonsense.
In the HELM Workbooks we have not always given rigorous solutions or taken the greatest care over
the use of units, taking instead a pragmatic line and trying to keep solutions simple.
In truth, in the HELM Workbooks we have not been consistent in our approach, adopting whatever
seemed appropriate to the situation.
We hope that the lecturer will understand what is required for a rigorous solution and forgive us these
lapses. (Any reports of errors or suggestions for improvement of any HELM resources will always be
welcome.)
2. Confidence
We want engineering students to be confident in their mathematics. However, this needs to be
justified confidence. Some informative research by Armstrong and Croft (1999) included a diagnostic
test of mathematics for engineering undergraduates and a confidence test. The results across 28
different mathematical topics (ranging from arithmetic to integration) were illuminating: There was
not a single topic for which the proportion of students expressing a lack of confidence matched or
exceeded the proportion of students answering wrongly. In other words there was overconfidence
across all topics.
It can be concluded that in general any student seeking help needs it and so will many others who
may not admit to it! A student telling his tutor that he has no problems should not be assumed to
be correct - some gentle probing is always advisable.
Reference: Armstrong P.K. & Croft, A.C., (1999) Identifying the Learning Needs in Mathematics of
Entrants to Undergraduate Engineering Programmes in an English University, European Journal of
Engineering Education, Vol.24, No.1, 59-71.
14 HELM (2015):
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3. HELM Workbooks and Students with Special Learning

Difficulties
The HELM project sought expert advice on the style of the Workbooks. This included some interviews
with selected students. In most cases HELM was able to respond to make the Workbooks (based on
the earlier Open Learning Project) more accessible to all students.
When tested with various formats, most students with learning difficulties such as dyslexia preferred a
sans serif font version and this has been adopted. However, a serif font is used for the mathematical
notation. It seems, from past experience that this is preferable since the sans serif mathematical
notation is unfamiliar and ambiguous giving, for example a straight x instead of curved, easily confused
with a times sign, and the digit one looking like the upper case letter “i” and lower case letter “l”
and also like the modulus sign.
Although there is an occasional use of italicised words, the preference is to use emboldening, which
is clearer.
The main suggestion which was not followed concerns text justification. The recommendation is to
use ‘ragged right’ text. Although fully justified text can cause problems for some students it was
decided that HELM’s use of very short paragraphs made that a less important consideration and full
justification improved the general appearance of the page and emphasised the blocks of text.
4. Student Difficulties
On the next two pages are reproduced two actual engineering students’ attempts at an algebra
problem and a calculus problem. They demonstrate a range of errors.
These examples can be used as a basis of discussion with tutorial assistants, new staff and others
as to how to diagnose and remedy problems experienced by students challenged by mathematical
problems.
You will be able to amass your own collection of such examples . . .
An Engineering Student’s Errors in Algebra
Below is the work of a first year undergraduate Manufacturing Engineering student.
1. Study the student’s work.
2. Note down the different kinds of mistakes made.
3. How would you help this student?
HELM (2015): 15
1 3 3
1. 144 2 + 144 2 = 144 4
2. (a)
(x + 6)(x + 5) = (6 + x)(5 + x)
= 6(5 + x) + x(5 + x)
= 30 + 6x + 5x + x2
= x2 + 11x + 30
(b)
(x + 1)(x + 2)(x + 3) = (x2 + 2x + x + 2)x + 3
= (x2 + 3x + 2)x + 3
= x3 + 3x2 + 2x + 3x2 + 9x + 6
= x3 + 6x2 + 11x + 6
3. (a)
11x + 7 = 0
11x + 7 − 7 = −7
11x −7
=
11 11
−7
x =
x
(b)
x2 + 6x − 7 = 0
x2 + 6x − 7 + 7 = 7
6x 7
x2 + =
6 6
7
x2 + x =
6
7
x3 =
6
7
x =
18
(c)
x2 + x − 8 = 0
x2 + x − 8 + 8 = 8
x3 = 8
8
x =
3
16 HELM (2015):
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An Engineering Student’s Solution to a Calculus Problem
Below is the work of a first year undergraduate Electrical Engineering student.
1. Study the student’s work.
2. How many errors can you see?
3. How would you classify each of them?
4. How could you help this student?
Problem
Given: 31
x+7 3
y= , x>
4x − 3 4
find the gradient at x = 1.
Student’s Solution
1
ln(x + 7) 3
ln y = 1
ln(4x − 3) 3
1
3
ln(x + 7)
ln y = 1
3
ln(4x − 3)
ln(4x − 3) ln(x + 7)
1 dy −
= x + 7 4x − 3
y dx (ln(4x − 3))2
 
ln(4x − 3) ln(x + 7)
dy
1
x+7 3  −
= × x+7 4x − 3  
dx 4x − 3  (ln(4x − 3)2 
ln 1 ln 8
 
dy
13
8 −
When x = 1 = × 8 1 

dx 1 (ln 1)2 
0−0
= 2×
0
dy
Therefore = 0 when x = 1
dx
HELM (2015): 17
Description of HELM
Workbook layout 50.6
On the following three pages are reproduced from the Student’s Guide explanatory pages concerning
Workbook Layout.
18 HELM (2015):
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HELM (2015): 19
Section 50.6: Description of HELM Workbook layout
20 HELM (2015):
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HELM (2015): 21
Section 50.6: Description of HELM Workbook layout
List of Sections in
Workbooks 1 to 48 50.7

Workbook 1 - Basic Algebra (89 pages)

1.1 Mathematical Notation and Symbols
1.2 Indices
1.3 Simplification and Factorisation
1.4 Arithmetic of Algebraic Fractions
1.5 Formulae and Transposition
Workbook 2 - Basic Functions (75 pages)

2.1 Basic Concepts of Functions
2.2 Graphs of Functions and Parametric Form
2.3 One-to-one and Inverse Functions
2.4 Characterising Functions
2.5 The Straight Line
2.6 The Circle
2.7 Some Common Functions
Workbook 3 - Equations, Inequalities and Partial Fractions (71 pages)

3.1 Solving Linear Equations
3.2 Solving Quadratic Equations
3.3 Solving Polynomial Equations
3.4 Solving Simultaneous Linear Equations
3.5 Solving Inequalities
3.6 Partial Fractions
Workbook 4 - Trigonometry (77 pages)

4.1 Right-angled Triangles
4.2 Trigonometric Functions
4.3 Trigonometric Identities
4.4 Applications of Trigonometry to Triangles
4.5 Applications of Trigonometry to Waves
Workbook 5 - Functions and Modelling (49 pages)

5.1 The Modelling Cycle and Functions
5.2 Quadratic Functions and Modelling
5.3 Oscillating Functions and Modelling
5.4 Inverse Square Law Modelling
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Workbook 6 - Exponential and Logarithmic Functions (73 pages)

6.1 The Exponential Function
6.2 The Hyperbolic Functions
6.3 Logarithms
6.4 The Logarithmic Function
6.5 Modelling Exercises
6.6 Log-linear Graphs
Workbook 7 - Matrices (50 pages)

7.1 Introduction to Matrices
7.2 Matrix Multiplication
7.3 Determinants
7.4 The Inverse of a Matrix
Workbook 8 - Matrix Solution of Equations (32 pages)

8.1 Solution by Cramer’s Rule
8.2 Solution by Inverse Matrix Method
8.3 Solution by Gauss Elimination
Workbook 9 - Vectors (66 pages)

9.1 Basic Concepts of Vectors
9.2 Cartesian Components of Vectors
9.3 The Scalar Product
9.4 The Vector Product
9.5 Lines and Planes
Workbook 10 - Complex numbers (34 pages)

10.1 Complex Arithmetic
10.2 Argand Diagrams and the Polar Form
10.3 The Exponential Form of a Complex Number
10.4 De Moivre’s Theorem
Workbook 11 - Differentiation (58 pages)

11.1 Introducing Differentiation
11.2 Using a Table of Derivatives
11.3 Higher Derivatives
11.4 Differentiating Products and Quotients
11.5 The Chain Rule
11.6 Parametric Differentiation
11.7 Implicit Differentiation
HELM (2015): 23
Section 50.7: List of Sections in Workbooks 1 to 48
Workbook 12 - Applications of Differentiation (63 pages)
12.1 Tangents and Normals
12.2 Maxima and Minima
12.3 The Newton-Raphson Method
12.4 Curvature
12.5 Differentiation of Vectors
12.6 Case Study: Complex Impedance
Workbook 13 - Integration (62 pages)

13.1 Basic Concepts of Integration
13.2 Definite Integrals
13.3 The Area Bounded by a Curve
13.4 Integration by Parts
13.5 Integration by Substitution and Using Partial Fractions
13.6 Integration of Trigonometric Functions
Workbook 14 - Applications of Integration 1 (34 pages)

14.1 Integration as the Limit of a Sum
14.2 The Mean Value and the Root-Mean-Square Value
14.3 Volumes of Revolution
14.4 Lengths of Curves and Surfaces of Revolution
Workbook 15 - Applications of Integration 2 (31 pages)

15.1 Integration of Vectors
15.2 Calculating Centres of Mass
15.3 Moment of Inertia
Workbook 16 - Sequences and Series (51 pages)

16.1 Sequences and Series
16.2 Infinite Series
16.3 The Binomial Series
16.4 Power Series
16.5 Maclaurin and Taylor Series
Workbook 17 - Conics and Polar Coordinates (43 pages)

17.1 Conic Sections
17.2 Polar Coordinates
17.3 Parametric Curves
Workbook 18 - Functions of Several Variables (40 pages)

18.1 Functions of Several Variables
18.2 Partial Derivatives
18.3 Stationary Points
18.4 Errors and Percentage Change
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Workbook 19 - Differential Equations (70 pages)

19.1 Modelling with Differential Equations
19.2 First Order Differential Equations
19.3 Second Order Differential Equations
19.4 Applications of Differential Equations
Workbook 20 - Laplace Transforms (72 pages)

20.1 Causal Functions
20.2 The Transform and its Inverse
20.3 Further Laplace Transforms
20.4 Solving Differential Equations
20.5 The Convolution Theorem
20.6 Transfer Functions
Workbook 21 - z-Transforms (96 pages)

21.1 z-Transforms
21.2 Basics of z-Transform Theory
21.3 z-Transforms and Difference Equations
21.4 Engineering Applications of z-Transforms
21.5 Sampled Functions
Workbook 22 - Eigenvalues and Eigenvectors (53 pages)

22.1 Basic Concepts
22.2 Applications of Eigenvalues and Eigenvectors
22.3 Repeated Eigenvalues and Symmetric Matrices
22.4 Numerical Determination of Eigenvalues and Eigenvectors
Workbook 23 - Fourier Series (73 pages)

23.1 Periodic Functions
23.2 Representing Periodic Functions by Fourier Series
23.3 Even and Odd Functions
23.4 Convergence
23.5 Half-range Series
23.6 The Complex Form
23.7 An Application of Fourier Series
Workbook 24 - Fourier Transforms (37 pages)

24.1 The Fourier Transform
24.2 Properties of the Fourier Transform
24.3 Some Special Fourier Transform Pairs
HELM (2015): 25
Workbook 25 - Partial Differential Equations (42 pages)
25.1 Partial Differential Equations
25.2 Applications of PDEs
25.3 Solution using Separation of Variables
25.4 Solutions using Fourier Series
Workbook 26 - Functions of a Complex Variable (58 pages)

26.1 Complex Functions
26.2 Cauchy-Riemann Equations and Conformal Mappings
26.3 Standard Complex Functions
26.4 Basic Complex Integration
26.5 Cauchy’s Theorem
26.6 Singularities and Residues
Workbook 27 - Multiple Integration (83 pages)

27.1 Introduction to Surface Integrals
27.2 Multiple Integrals over Non-rectangular Regions
27.3 Volume Integrals
27.4 Changing Coordinates
Workbook 28 - Differential Vector Calculus (53 pages)

28.1 Background to Vector Calculus
28.2 Differential Vector Calculus
28.3 Orthogonal Curvilinear Coordinates
Workbook 29 - Integral Vector Calculus (80 pages)

29.1 Line Integrals Involving Vectors
29.2 Surface and Volume Integrals
29.3 Integral Vector Theorems
Workbook 30 - Introduction to Numerical Methods (64 pages)

30.1 Rounding Error and Conditioning
30.2 Gaussian Elimination
30.3 LU Decomposition
30.4 Matrix Norms
30.5 Iterative Methods for Systems of Equations
Workbook 31 - Numerical Methods of Approximation (86 pages)

31.1 Polynomial Approximations
31.2 Numerical Integration
31.3 Numerical Differentiation
31.4 Nonlinear Equations
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Workbook 32 - Numerical Initial Value Problems (80 pages)

32.1 Initial Value Problems
32.2 Linear Multistep Methods
32.3 Predictor-Corrector Methods
32.4 Parabolic PDEs
32.5 Hyperbolic PDEs
Workbook 33 - Numerical Boundary Value Problems (36 pages)

33.1 Two-point Boundary Value Problems
33.2 Elliptic PDEs
Workbook 34 - Modelling Motion (63 pages)

34.1 Projectiles
34.2 Forces in More Than One Dimension
34.3 Resisted Motion
Workbook 35 - Sets and Probability (53 pages)

35.1 Sets
35.2 Elementary Probability
35.3 Addition and Multiplication Laws of Probability
35.4 Total Probability and Bayes’ Theorem
Workbook 36 - Descriptive Statistics (45 pages)

36.1 Describing Data
36.2 Exploring Data
Workbook 37 - Discrete Probability Distributions (60 pages)

37.1 Discrete Probability Distributions
37.2 The Binomial Distribution
37.3 The Poisson Distribution
37.4 The Hypergeometric Distribution
Workbook 38 - Continuous Probability Distributions (27 pages)

38.1 Continuous Probability Distributions
38.2 The Uniform Distribution
38.3 The Exponential Distribution
Workbook 39 - The Normal Distribution (40 pages)

39.1 The Normal Distribution
39.2 The Normal Approximation to the Binomial Distribution
39.3 Sums and Differences of Random Variables
HELM (2015): 27
Workbook 40 - Sampling Distributions and Estimation (22 pages)
40.1 Sampling Distributions
40.2 Interval Estimation for the Variance
Workbook 41 - Hypothesis Testing (42 pages)

41.1 Statistics Testing
41.2 Tests Concerning a Single Sample
41.3 Tests Concerning Two Samples
Workbook 42 - Goodness of Fit and Contingency Tables (24 pages)

42.1 Goodness of Fit
42.2 Contingency Tables
Workbook 43 - Regression and Correlation (32 pages)

43.1 Regression
43.2 Correlation
Workbook 44 - Analysis of Variance (57 pages)

44.1 One-Way Analysis of Variance
44.2 Two-Way Analysis of Variance
44.3 Experimental Design
Workbook 45 - Non-parametric Statistics (36 pages)

45.1 Non-parametric Tests for a Single Sample
45.2 Non-parametric Tests for Two Samples
Workbook 46 - Reliability and Quality Control (38 pages)

46.1 Reliability
46.2 Quality Control
Workbook 47 - Mathematics and Physics Miscellany (70 pages)

47.1 Dimensional Analysis in Engineering
47.2 Mathematical Explorations
47.3 Physics Case Studies
Workbook 48 - Engineering Case Studies (97 pages)

Engineering Case Studies 1 to 20
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Commentaries on
Workbook 1: Basic Algebra

The lack of fluency with basic algebra among engineering undergraduates is probably the area most
commented upon by those who teach them mathematics and by engineering academics, who find that
this deficiency hampers the student in solving problems in the wider engineering arena. To achieve
the required level of fluency it is necessary to build the student’s confidence by a careful programme
of drill and practice examples. This idea may not be as fashionable as it once was, but effort invested
with this material will reap rewards in later work.
This Workbook is therefore placed first in the series to emphasise the importance of the mathemat-
ics it covers. The five areas that it deals with are: mathematical notation and symbols, indices,
simplification and factorisation (of expressions), algebraic fractions, and formulae and transposition.
Section 1.1 may seem elementary to the majority of students, but experience has shown that many
of them have lacunae in their knowledge and understanding, even at this level. Particular attention
needs to be paid to the understanding of, and correct use of, the modulus sign and the sigma notation.
In the interpretation of index notation, care is needed with negative powers on the denominator of
a fraction and the (obvious) fact that different powers of the same variable (e.g. a2 + a3 ) cannot
be combined together. Correct use of a pocket calculator when fractional powers are involved also
needs attention. In general it must be explained that using a pocket calculator successfully needs a
good grasp of the underlying algebra.
The importance of Section 1.3 cannot be overstated. It is disappointing to find that so many
undergraduates have difficulty in correct manipulation of algebraic expressions. To see in a student’s
attempted solution of a problem in reliability the ‘mathematics’ 1 − (1 − e−t ) ≡ e−t demonstrates
a lack of knowledge of the distributive law. Such errors are all too common and prevent the correct
solution of straightforward problems in engineering. There are no short cuts: the basics must be
thoroughly understood and students must be made aware of this. Factorisation should become
second nature and this, too, demands time and effort.
Handling algebraic fractions is made more difficult these days when school students are not given
a thorough grounding in the arithmetic of number fractions. It would be wise to give students
some revision in dealing with number fractions as a precursor, so that the ‘rules’ for manipulating
algebraic fractions make sense. Students who attempt to apply the rules without understanding
them sometimes come a cropper, and subsequent stages of the attempted solution of an engineering
problem are then incorrect.
The transposition of formulae is a skill whose lack is frequently commented on. To many academics
this skill is too often taken for granted, but many students have at best a shaky grasp of what they
HELM (2015): 29
Section 50.8: Commentaries on Workbooks 1 to 48
are attempting to do and of the ‘rules’ that they are using. Once again, it is practice, which is
required, and examples from their engineering studies should be used as much as possible so that
they see the relevance of what they are doing.
The last point applies quite generally. Although the use of ‘abstract’ examples has merit, every set
of examples should, where possible, include at least one framed in the context of the engineering
1 1 1
discipline relevant to each student. To make R the subject of the formula = + is surely
R R1 R2
more interesting to an electrical engineering undergraduate than being asked to make z the subject
1 1 1
of the formula = + .
z x y
Workbook 2: Basic Functions

Functions are the verbs of mathematics and a firm grasp of their basic definitions and properties are
crucial to later work. Many students are put off by the use of the notation f (x) = x3 and this effect
can be reduced by a gentle, notation-free introduction.
This Workbook covers the areas: the concept of a function, the graph of a function, functions in
parametric form, classifying functions, linear functions, simple standard engineering functions, and
the circle.
For students not versed in the idea of a function the box diagram (input - rule - output) has proved
to be the most useful in discussing the topic, particularly with regard to the inverse of compound
functions. By treating a few very simple functions in this way, it is then time to introduce the concept
of a function as a mapping and the notation f (x). Then compound functions can be studied and
the box diagram approach helps to clarify the domain and range.
When presenting the representation of a function as a graph it is important to stress the word
‘representation’; too many students think that the graph is the function. Now is the time to present
a function as a box diagram, a rule and as represented by a graph in order to show the link between
the three approaches.
At this early stage it is possible to give examples of functions described in parametric form and
demonstrate how to calculate points on their graph. It is then the time to present an example
of a function which would be very difficult, or impossible, to present in the standard f (x) form.
Incidentally, it is a good idea to now consider functions whose input variable is not x, say t .
The inverse of a function, its domain and range can be discussed first by box diagram, then by
rule and then by graph, in that order. The use of the terms ‘one-to-one’ etc can now be safely
mentioned provided that the treatment is gentle. The distinction between functions whose inverse
is itself a function and those for which this is not true can be illustrated by examples such as
f (x) = x + 2, f (x) = 2x, f (x) = x3 and f (x) = x2 . Other examples can be presented via graphs.
With simple examples, and a liberal use of the graphical approach, the concepts of continuity, piece-
wise definition, periodicity, oddness and evenness can extend the portfolio of functions considered.
Illustrating these concepts, however crudely, by examples from engineering is particularly important
here, to prevent the presentation drifting into an apparently aimless academic exercise.
30 HELM (2015):
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When students met the topic of straight lines at school it was almost certainly not as the graphical
representation of a linear function. The advantage, for example, of calculating the exact distance
between two points as opposed to a graphical estimation should be stressed, by asking students
to discuss how they would find the distance between two points in three dimensions and why the
problem is different from the two-dimensional case.
A circle in standard form is an example of an implicit function, but that may be too deep to discuss
at this level. Rather it provides the opportunity to carry out some simple coordinate geometry where
the visual checks are easy to perform. The use of some of these results in the relevant engineering
discipline must be demonstrated.
Polynomial functions are well behaved and can act as a vehicle for looking at concepts of oddness
and evenness, decreasing, increasing and stationary. Simple rational functions, including f (x) = x−1
allow ideas of discontinuity and asymptotes to be introduced. The modulus function and the unit
step function allow students to widen their horizons on the meaning of ‘function’ and can be (and
must be) shown to have practical engineering applications.
Workbook 3: Equations, Inequalities and Partial Fractions

The solution of simple equations underpins much of engineering mathematics. The need to know
whether the equation concerned has any solutions, and if so how many is essential to the satisfactory
solution of problems. Much less well understood are inequalities and a thorough grounding is required
at this level. The concept of the partial fractions of a rational function is useful in so many areas,
for example inverting a Laplace transform.
The areas that this Workbook deals with are: the solution of linear equations, the ‘solution’ of
quadratic equations and polynomial equations in general, simultaneous equations, simple inequalities
and their solution, determination of partial fractions.
There are examples a-plenty of linear equations in engineering and the opportunity should be taken
to introduce the topic via one from the students’ discipline. Linking the solution to the graph of
a straight line crossing an axis is recommended. Are there any cases therefore where no solution
exists? Is the solution unique and why is it important to know this?
The solution of quadratic equations provides an opportunity to bring together several areas of math-
ematics. Looking at graphs of various quadratics suggests the possibility of two distinct solutions, a
unique solution or no solutions; algebra can verify the suggestion. The usefulness of factorisation in
determining the crossing points of the graph on the horizontal axis, and the location of the stationary
point of the function can be stressed, but if no factors exist students may find the method wasteful of
time; honesty is essential. Completing the square is clumsier in determining two crossing points but
readily yields information about the stationary point when there are no solutions of the associated
quadratic equation. The formula method then comes into its own as providing relevant information
in all cases; this is the time to introduce it, but do stress the relevant strengths of all the approaches.
The introduction to higher degree polynomial equations should be gentle and, in the main, concen-
trating on general principles. It is a good moment in the course to emphasise how the complexity of
the problem increases as the degree of the polynomial increases.
HELM (2015): 31
The solution of simultaneous equations allows the opportunity to show the interaction between graph-
ical indication and algebraic determination. Start with the four graphical cases of single intersection
(at almost right-angles), parallel lines, coincident lines, and almost-coincident lines. Then show how
the graphs relate to the algebraic ‘solution’. A simple introduction to ill-conditioned equations can
be given here.
The relationship between a simple inequality and points on the real line helps students to visualise
the meaning of the inequality; its solution algebraically can be related to the picture. The case of a
quadratic inequality is only sensibly dealt with at this level by extensive use of a graphical argument.
The first point to emphasise with expressing a rational function as a sum of partial fractions is
the need to choose suitably-shaped partial fractions; the process of finding the coefficients in the
numerators can then, in principle, follow successfully. It is recommended that a single strategy is
employed: combine the proposed partial fractions into a single fraction with the same denominator as
the original fraction; substitute suitable values of the variable into the numerators to obtain, one by
one, the unknown coefficients; if this does not determine all of them resort to comparing coefficients
of powers of the variable. It is suggested that at least one example is given where one of the unknown
coefficients is zero to warn the students that this can happen with a general strategy and ask whether
they could have foreseen that zero value.
Workbook 4: Trigonometry
There are some colleagues who believe that the sine, cosine and tangent functions should be intro-
duced via right-angled triangles and then extended to their application to oscillatory motion. Others
believe that the application to oscillatory motion should precede their link to triangles. This Work-
book follows the first approach, but with suitable guidance students following the second approach
can also use it successfully.
This Workbook deals with the areas of right-angled triangles and their solution, trigonometric func-
tions of any angle, including their graphs, simple trigonometric identities, the solution of triangles
without a right angle, the application of trigonometry to the study of wave motion.
In this age of the pocket calculator many students are not familiar with the exact values of the
trigonometric ratios for angles such as 30◦ , 45◦ , 60◦ . Does it matter? The view taken here is that it
does. It is also important to note that the inverse trigonometric functions found by pressing calculator
buttons may not provide the answer required. It is worth emphasising that checking whether a triangle
is right-angled before solving it via sine and cosine rules can save much effort.
These days it might be argued that there is little value in knowing, or using, throwbacks such as the
CAST rule; on the contrary, we argue that such knowledge and usage leads to a greater ‘feel’ for the
trigonometric functions than mere blind button-pressing. The ability to sketch graphs of the basic
trigonometric functions adds to their safe application to problem solving.
How far one should delve into trigonometric identities is a matter for debate but surely everyone
should know the identity sin2 θ + cos2 θ ≡ 1 and its close relatives. An awareness of some of the
addition formulae allows the ‘double angle’ identities to be seen as plausible.
A strategy for the solution of triangles in general, using angle sum, sine rule and cosine rule should
32 HELM (2015):
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be presented. This is a good situation in which to explain the idea of adapting the strategy to
different input information. The fact that the information given does not allow a triangle to be
drawn illustrates the principle that ‘if you can draw the triangle then you can calculate the unknown
sides and angles’. The ambiguous case, presented pictorially first, gives emphasis to recognising that
for an angle < 180◦ , knowing its sine does not define it uniquely (except, of course, for 90◦ ).
This Workbook assumes no previous knowledge of the subject and begins with a discussion of the
trigonometry of the right-angled triangle. More general definitions are then given of the trigonometric
(or circular) functions and the main trigonometric identities are obtained.
The sine rule and cosine rule for triangles are introduced and a careful discussion given of which is
the appropriate rule to use in a given situation.
Detailed worked examples and a generous selection of problems (with answers) are provided at all
stages.
As an application a full section is devoted to properties of sinusoidal waves whilst other smaller
optional engineering examples utilising trigonometry (for example in the diffraction of sound, brake
cables, amplitude modulation and projectile motion) are interspersed at appropriate intervals.
Workbook 5: Functions and Modelling

One of the difficulties found frequently by engineering students is the progression from the statement
of a problem in words to a mathematical solution and its interpretation. In this context, a reasonable
proficiency in a range of mathematical techniques is not sufficient. Students need to be able to
abstract salient features from a real problem, make suitable assumptions, assign sensible symbols to
the variables, state the mathematical problem (the mathematical model) and devise a strategy for
solving it. Having solved it, the student needs to be able to interpret the result either in terms of the
predicted behaviour or numerically (using consistent units). If the interpretation is not satisfactory
the student needs to be able to refine the initial model.
This Workbook is intended to introduce this idea of the ‘Modelling Cycle’. The emphasis is on
‘choosing a function for a model’. Since Workbook 5 is early in the HELM series, it deals only with
models that involve linear, quadratic and trigonometric functions. (The last section in Workbook 6
includes modelling examples that use the logarithmic and exponential functions.) An attempt has
been made to use the ‘modelling’ format in all of the engineering examples in the HELM series.
Following a section on the use of linear functions as models we turn our attention to the use of
quadratic functions. An obvious example is that of projectile motion. Given the relative simplicity
of the quadratic function it is worth examining the assumptions made by using it to model such
situations; how accurate are the predictions from the model and does it provide an exact model,
or merely a very good approximation to reality? To avoid confusion in the case of two-dimensional
projectile motion, it is worth spending time with the example of a stone thrown vertically upwards
distinguishing between the graph of the vertical displacement against time and the actual motion of
the stone.
Then we model oscillatory motion by means of the sine and cosine functions, reinforcing the intro-
duction provided in Workbook 4. Finally, we take an example of the inverse square law model; this
HELM (2015): 33
is such a widely-applied model that it needs an early mastery.
The definition of the ‘Modelling Cycle’ and the approach adopted are similar to those used in a series
of Open University courses (TM282 Modelling by Mathematics and MST121 Using Mathematics).
These courses cite some engineering examples but, since the intention was to avoid contexts that
required a lot of prior knowledge, several other contexts are used here. For example, the Open
University materials for TM282 use three contextual themes: Vehicle Safety, Town Planning and
Population Growth. Further illustrations of modelling are to be found in the texts for those courses.
More advanced modelling examples may be found in later HELM Workbooks (particularly 34, 47 and
48) and in the following references:
The Open University MST207 Mathematical Methods and Models,
J. Berry and K. Houston, Mathematical Modelling, publ. Edward Arnold, 1995,
K. Singh, Engineering Mathematics through Applications, Palgrave, 2003.
Workbook 6: Exponential and Logarithmic Functions

Two of the most hard-worked functions in models of engineering problems are the exponential and
logarithmic functions. As with all work-horses, a thorough knowledge of their behaviour is essential
to their efficient use; effort spent now will reap rewards over and over again.
After some examples of functions from the family a2 , the exponential function is introduced by
defining e as the limit of a sequence. A study of the slope of the tangent to curves of the family can
be followed by stating that in the case of the exponential function that slope is equal to the value of
the function at the point of contact. This then allows a generalisation to the slope being proportional
to the value and hence to models of growth and decay, for example Newton’s law of cooling.
The hyperbolic functions can be introduced at this point without too much fuss. Given the contrast
between some of their properties and those of their trigonometric near-namesakes, it might be wise
to hint at their use in parametric coordinates on a hyperbola (hence the trailing h).
The use of the logarithmic function in solving model equations that employ the exponential function
reinforces the concept of an inverse function. The properties of logarithmic functions need treating
with care: the law loga (AB) = loga A + loga B is not true for A and B both negative, for example.
To an extent the material in Section 6.6 on the log-linear transformation duplicates that in earlier
sections of the Workbook. However this section provides an alternative approach and illustrates its
use in various modelling contexts. Indeed the emphasis in Section 6.6 is on the use of exponential
and logarithm functions for modelling.
Workbook 7: Matrices
The algebra of matrices can be a dry topic unless the student sees that matrices have a vital role to
play in the solution of engineering problems. Of course, one obvious application is in the solution of
34 HELM (2015):
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simultaneous linear equations, but there are others, for example in the description of communication
networks, and at least one such example should be introduced at the outset.
Of necessity, some groundwork must be carried out and this can be tedious; where possible, emphasise
the ‘natural’ definitions, for example addition of two matrices of the same shape; then ask how they
would add together say

1 2 3 2 3
and . In new territory nothing can be taken for granted.
3 1 2 1 2
Matrix multiplication is a strange process. The multiplication of two-by-two matrices can be illus-
trated as representing two successive rotations (use simple angles). The generalisation to three-by-
three matrices is a hard one, so it is important to emphasise the scalar product basis of deriving the
two-by-two entries e.g. for the product C = AB, c11 = a11 b11 + a12 b21 etc. And, of course, the
shock to the system of non-commutativity and results such as AB = 0 does not necessarily imply
that either A = 0 or B = 0 (or both) should be exploited. Again it is important to give simple
examples of where multiplication is not possible then give them back their confidence by giving them
simple guidelines on how to negotiate the minefield.
A simple introduction to the concept of the determinant of a square matrix and some simple properties
of determinants should be accompanied by an illustration of how tedious it can be to evaluate a
determinant with even four rows. Please emphasise that there is a different notation for determinants
than that for matrices and that they are not interchangeable.
The inverse of a two-by-two matrix is best introduced by the idea of reversing a rotation. The
fact that even some square matrices do not have an inverse must be justified. That we say “the
inverse matrix, when one exists,” is a useful vehicle to explain the vital importance of existence
and uniqueness theorems. Emphasise that division of two matrices is NOT an operation. Build
up confidence by finding inverses of two-by-two matrices before embarking on the three-by-three
examples; these might be postponed until simultaneous linear equations are introduced. (Workbook
9). The contrast between the Gauss elimination and the determinant method should be drawn.
Whilst students should be made aware of both methods they should be allowed to use the method
which they prefer.
Workbook 8: Matrix Solution of Equations

This Workbook introduces the methods of solution by Cramer’s rule, by inverse matrices and by
Gauss elimination. The relative strengths of these methods should be highlighted, in particular how
they cope with systems of equations that have no unique solution.
Cramer’s rule provides a simple approach to understanding the nature of solution of two equations in
two unknowns and is easily extended to three equations in three unknowns. It should be noted how
much more arithmetic is involved and that extension to larger systems is really impracticable.
The use of the inverse matrix provides an alternative approach, which has useful links to the underlying
theory but is not much help in the cases where no unique solution exists.
In contrast, the method of Gauss elimination can, properly handled, lead to useful information when
HELM (2015): 35
there is no unique solution, especially if there are an infinite number of solutions. These days, students
are not fluent in handling arithmetic fractions and care must be taken to avoid examples, which lead
to anything but the simplest fractions. The method provides a variety of routes to the solution and
this fact may cause students to be wary of using it. A warning could be usefully issued by showing
the exact solution to a system of ill-conditioned equation alongside a Gauss ‘solution’.
Workbook 9: Vectors
In this Workbook we provide a careful introductory account of vectors together with a few basic
applications in science and engineering. The approach is relatively conventional, starting with the
elementary ideas, approached graphically, of vector addition and subtraction and the multiplication of
a vector by a scalar. Applications involving the addition and resolution of forces and a discussion of
the forces on an aeroplane in steady flight are provided. (Some elementary knowledge of trigonometry
is required here and later in the Workbook.) An opportunity can be taken to discuss the contrast
between displacement of an object (vector) and the distance it travels (scalar).
We underline vectors, which better reflects how students and lecturers write them (rather than
emboldening them as is done in textbooks). Note that we write dr/dt (underlining r only) but dr
(underlining the expression dr).
Cartesian representation of vectors, firstly in two and then in three dimensions is covered including
position vectors. The basic unit vectors are denoted by i, j and k . It is important not to rush
through the two-dimensional case: once this is well understood the three-dimensional case can be
sold as a straightforward extension (and the reason for ‘the adding of an extra term’ allowing us to
move into three dimensions can be explained).
A careful coverage of the scalar (or dot) product of two vectors and its properties, together with
some applications in basic electrostatics, is followed by the corresponding vector (or cross) product.
Carefully explain why a × b = −b × a; it is a result to frighten the faint-hearted. In the latter
the determinant form for evaluation in Cartesian coordinates is used and simple applications to the
torque of a force are discussed. Triple products are covered only in exercises. Many students find
difficulty with problems in three-dimensional statics in their engineering modules and the use of the
vector product in tackling these problems can be a good selling-point here (case study or coursework,
perhaps).
The final section involves some geometrical aspects of vectors particularly direction cosines and
direction ratios and vector equations of lines and planes. The simplicity of approach using vectors
can be highlighted by contrasting it with the alternative approach.
Workbook 10: Complex Numbers

Complex numbers are a basic tool in engineering and science and this Workbook gives a gentle
introduction to their properties and uses. A non-rigorous approach is adopted with a complex number
being denoted by z = a+ib in Cartesian coordinates and by z = r(cos θ +i sin θ) in polar coordinates.
The complex exponential form is also covered.
36 HELM (2015):
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The Argand diagram provides a useful means of visualisation of complex numbers but care must be
taken to stress that a point on the diagram represents a complex number and is not the number
itself. The same point can be made when representing a complex number as a vector.
The Workbook initially covers the algebra of complex numbers in Cartesian form and glances briefly
at the solution of polynomial equations with real coefficients. Given the poor grasp of real number
algebra among many students, it is wise not too assume too much algebraic fluency, so take it slowly.
Assuming some knowledge of trigonometric identities allows the introduction of the polar form and its
advantage over the Cartesian form when carrying out multiplication and division (and taking powers).
A simple application to rotations about a point provides a gentle illustration.
Use of the exponential form leads to a discussion of the relations between trigonometric and hyperbolic
functions and identities and the Workbook concludes with De Moivre’s theorem for finding powers
and roots of complex numbers. It is worth remarking that the nth roots of a given complex number ◦
share the same modulus and therefore lie on a circle in the Argand diagram and are spaced 360 n
apart along the circumference.
Workbook 11: Differentiation

At the outset it should be remarked that a sensible ordering of the sections in Workbooks 11 and 12
enhances the presentation of the material to students. After a new technique a reinforcing application
example aids the learning process. There was a time when virtually all students embarking on an
engineering degree course were well versed in the rudiments of differentiation; that is no longer the
case. When the topic was covered in secondary education much time was devoted to it; it is not a
topic that can be skimmed over.
Many of today’s students seem to have difficulty understanding the concept of rate of change and
this needs careful attention. The average rate of change of a function over a given interval should
be introduced and the increment in the independent variable denoted by h: the use of δx can come
dy
later - to make plausible the notation . Linking the instantaneous rate of change to the gradient
dx
of the tangent should be done at the same time.
It is important to define carefully the terms ‘differentiable’, derived function’ and ‘derivative’ and
to distinguish clearly between them. It is also straightforward to give examples where a function is
not differentiable at a point: the modulus function, ramp function and unit step function are good
examples.
Familiarisation with the table of basic derivatives (derived functions) should be encouraged and the
results interpreted on graphs of the appropriate functions. Do stress that angles in this context
MUST be measured in radians.
A graphical explanation of how the derivatives of f (kx), kf (x), f (x + k), f (x) + k are related to that
of f (x) has been found to help. Should it be necessary to explain why the derivative of a constant
function is zero? Probably it is.
The rule for differentiating a linear combination of two functions (and the special cases of sum,
difference and scalar multiple) is reasonably intuitive and need not be overstated. Confidence is the
HELM (2015): 37
keyword here.
The second derivative could be presented in terms of displacement, velocity and acceleration - and a
d2 y
good time to explain that x is not the only notation for an independent variable. The notation 2
dx
etc. does seem clumsy and needs justification.
The remaining sections of the Workbook deal with the techniques of differentiating products and
quotients, the chain rule and parametric and implicit differentiation. The examples chosen must be
simple illustrations; don’t lose the wood of the technique in the trees of awkward differentiation. A
useful approach is to illustrate some applications of one technique before moving onto more advanced
techniques; in particular the last three techniques and especially the last two can be postponed until
earlier techniques have been mastered and time allowed for their absorption.
Workbook 12: Applications of Differentiation

The application of differentiation to the finding of the tangent and (indirectly) the normal to a curve
should be straightforward but it will help if the results are interpreted on a sketch graph of the
relevant function. This suggests the use of simple functions until the student is confident in using
the strategy.
Stationary Points and Points of Inflection
Most students have an imperfect understanding of the definitions of local maximum, local minimum
and point of inflection. Simple graphs can be used to illustrate these features.
Of the following three statements only the first two are known with any certainty by most students:
Given a twice differentiable function f for which f 0 (a) = 0
(1) If f 00 (a) > 0 , then f (x) has a minimum when x = a,
(2) If f 00 (a) < 0, then f (x) has a maximum when x = a,
(3) If f 00 (a) = 0 , then f (x) has minimum or a maximum or a point of inflection when x = a.
Many students think (3) always leads to a point of inflection but the graph of f (x) = x4 clearly
shows this to be untrue when x = 0.
Another misconception is that a point of inflection requires f 0 (a) = 0 . This is not true as can easily
be seen, for example, on the sine curve. This raises another point - for any continuous function there
is always a point of inflection between every local minimum and local maximum. The graph below
highlights these features.
38 HELM (2015):
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y = x4 − 4x3 − 2x2 + 12x + 2

y Maximum
Inflection Inflection
x
Minimum Minimum
Maxima and Minima without Calculus
Students all too readily turn to the calculus when needing to find maxima and minima. There are,
however, cases when alternative approaches are simpler, quicker or more informative:
Example 1
Find the minimum value of f (x) ≡ x2 + 2x + 3.
Completing the square gives f (x) = (x + 1)2 + 2.
This clearly is a minimum when x = −1 and there f (x) has value 2.
Example 2
Find the maximum value of f (x) ≡ 2 sin(x) + 3 cos(x) .
Using the trigonometric identity sin(A + B) ≡ sin A cos B + cos A sin B and utilising the triangle in
the diagram we have
HELM (2015): 39
√
13
3
√

2 3
f (x) = 13 √ sin x + √ cos x
13 13
√
= 13[cos α sin x + sin α cos x]
α
√
= 13 sin(x + α) 2
√ π
This clearly has a maximum value of 13 at x = − α (for example).
2
One point of caution should be stated. When applying differentiation to find for example the di-
mensions of a rectangle of fixed perimeter and maximum area students often get confused between
the graph of the function being maximised and the geometry of the problem. It is also useful to
remember that differentiation can find a local maximum and physical considerations can be brought
in to justify the term “absolute” maximum. Also, in some physical problems, with the knowledge of
the behaviour of the function at the physical extremes of the independent variable the nature of the
single stationary point can be decided without the need to differentiate a second time, or use the
first derivative test.
The Newton-Raphson method is straightforward to derive and relatively easy to apply. Mention should
be made of its proneness to failure under certain circumstances. The presentation of curvature should
be illustrated by simple examples so that the student can relate algebraic results to the graph of each
function. The differentiation of vectors in Cartesian component form follows an intuitive path and
this should not be clouded by too much rigour.
Workbook 13: Integration

The concept of an indefinite integral is introduced in the traditional way, as the reverse of differen-
tiation. It is helpful to illustrate this by integrating the velocity function to obtain the displacement
function and noting that we cannot say where an object is unless we knew where it was at an earlier
time - hence the need for the arbitrary constant. Using the table of derivatives ‘back to front’ and
the rule for linear combinations of functions, which parallels that for derivatives ought not to present
many difficulties.
The
Z 2 extension to definite integrals by itself is not too abstruse but the notation in the statement
2
3x2 dx = x3 1 = 8 − 1 = 7 is quite deep and needs a gentle presentation.
1
Surely one of the great moments in learning mathematics is to realise that the area bounded by the
curve y = f (x) , the x-axis and the ordinates x = a and x = b can be found exactly by evaluating
40 HELM (2015):
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Z b
the definite integral f (x)dx. Make the most of that moment. Simple examples, please. The
a
area bounded by the sine curve over a quarter-period has a satisfyingly simple value.
Further techniques can be postponed until some further simple applications such as those in Workbook
14 have been looked at.
The techniques of integration by parts, by substitution and by partial fractions are best treated in
two stages; first use really simple examples in order to allow students to gain confidence in the basics
of each technique, preferably with illustrative examples from engineering to demonstrate that their
use is not just ‘pure mathematics’, then revisit with more advanced versions of the techniques.
As a final phase, integration of functions involving more complicated trigonometric expressions can
be tackled.
Workbooks 14 & 15: Applications of Integration 1 & 2

These two Workbooks present a selection of the most common applications of definite integration.
Workbook 14 first deals with definite integration as the limit of a sum, then considers mean value
and root-mean-square value of a function, volumes of revolution and the lengths of arcs and the area
of surfaces of revolution. Workbook 15 considers the integration of vectors, finding centres of mass,
and finding moments of inertia.
The importance of recognising a definite integral as the limit of a sum cannot be over-emphasised:
it allows the correct establishment of a definite integral whose value is that of the physical quantity
concerned. The remainder of the solution is the use of techniques of integration met in Workbook
13. The first examples of each application should lead to very simple integrands. The results should
always be examined to see whether they are sensible in the light of the problem.
The mean value of the sine function over a quarter-period, over a half-period and over a full period
can be compared and the need for a more meaningful measure of the ‘average’ of the sine function
suggested. Of course, the root-mean-square value is more tricky to evaluate and the double angle
formula should be provided as a tool to be picked up and used.
When finding volumes of solids of revolution the cylinder, cone and sphere are the obvious examples
and the placing of an axis of symmetry along the x-axis should be highlighted. Careful explanation
of the division of the solid into the union of a set of non-overlapping discs should be followed by
remarking how an approximating sum can be transformed into a definite integral yielding the exact
answer - one of the glories of the calculus.
Approach the problem of finding the length of arc by first using a line parallel to the x-axis, then a line
inclined to the x-axis and thus make plausible the length of a general arc. The resulting formula can
then be shown to yield the same results as in the first two cases. The formulae for this application
and that of surface area only yield simple integrations in a few cases and it is worthwhile pointing out
that other examples may require much more complicated methods of integration, or perhaps need to
employ numerical approximate methods.
The integration of vectors in Cartesian coordinate form is relatively straightforward and can be dealt
HELM (2015): 41
with immediately after differentiating them - use velocity goes to displacement as a suitable vehicle
for discussion. When discussing the centre of mass of a plane uniform lamina the use of symmetry
is a vital ingredient; sensible positioning of the axes may simplify the resulting integrations, not least
finding the value of the area itself. Many students do not have a grasp of the idea of a moment and a
matchstick model is not too elementary a teaching aid. Finally, the calculation of moment of inertia
is not a great leap mathematically but does need a careful foundation of the physical concept. Many
students have only a rudimentary knowledge of mechanical principles and very little can be assumed.
Workbook 16: Sequences and Series

This Workbook gives a comprehensive introduction (adequate for most science and engineering
students) of these topics. Apart from basic algebra there are no significant prerequisites. Even
sigma notation is introduced as a new topic.
In the first section the concept of sequences and their convergence or otherwise is introduced. Arith-
metic and geometrical sequences (progressions) are covered and formulae obtained for the sum of n
terms in each case.
We then move on to more general infinite series. The meaning of convergence is carefully explained
in terms of partial sums. The ratio test for the convergence of series of positive terms is explained
together with a brief mention of other convergence tests and the ratio test is also used to briefly
consider conditional or absolute convergence for more general series.
A brief section follows on the binomial series and the binomial theorem for positive integers. This is
followed by a section on the convergence of power series in which the idea of the radius of convergence
is carefully explained. Various properties of convergent power series are also discussed such as their
differentiation and integration.
The most useful aspect of convergent power series for applications (their ability to represent functions)
is the subject of the final section in which both Maclaurin series and, briefly, Taylor series are discussed.
Workbook 17: Conics and Polar Co-ordinates

This Workbook offers a brief but, for most purposes, adequate introduction to the Co-ordinate
Geometry of the three conic sections - the ellipse, parabola and hyperbola - with the circle also being
introduced as a special case of the ellipse. The basic Cartesian equations of the conics with axes
parallel to one or other of the co-ordinate axes are obtained in the first section and mention made of
more general cases.
In the second section polar co-ordinates are introduced and simple curves whose equations are given
in polars are described while the equations of the standard conics in polars are also explained.
Finally parametric descriptions of curves are covered including those for the standard conic sections.
The prerequisites for this Workbook are relatively modest, with some basic algebra including comple-
tion of the square and algebraic fractions and knowledge of trigonometric functions being required
42 HELM (2015):
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for the first two sections. Knowledge of hyperbolic functions and simple differentiation is required
for the final section but detailed discussion of parametric differentiation is covered in Workbook 12.
The Workbook contains a good number of fully worked examples and many exercises for all of which
answers are provided.
Notes relating to Workbook 17 page 20:
The general conic
Although every conic is of the form
Ax2 + 2Bxy + Cy 2 + 2Dx + 2Ey + F = 0 (with not all A, B, C zero),
not every equation of that form represents a conic. (The equation x2 + 1 = 0 is an obvious counter-
example!)
Meaningful cases are:
Ellipse: AC − B 2 > 0 [eccentricity e < 1]
Possibilities: Ellipse, Circle, Single point.
Parabola: AC − B 2 = 0 [eccentricity e = 1]
Possibilities: Parabola, Two parallel lines, Two coincident lines.
Hyperbola: AC − B 2 < 0 [eccentricity e > 1]
Possibilities: Hyperbola, Rectangular hyperbola, Two intersecting lines.
Definition of conic
A conic is the locus of a point, which moves in a plane so that its distance from a fixed point called
the focus bears a constant ratio called the eccentricity to its distance from a fixed straight line in
the plane called the directrix.
If (p, q) are the coordinates of the focus, e is the eccentricity, and ax + by + c = 0 is the equation
of the directrix then the equation of the conic is
(x − p)2 + (y − q)2 = e2 (ax + by + c)2 /(a2 + b2 )
which can be rearranged into a biquadratic of the form
Ax2 + 2Bxy + Cy 2 + 2Dx + 2Ey + F = 0 (as stated at the beginning) where
HELM (2015): 43
A = a2 + b2 − a2 e2
B = −abe2
C = a2 + b2 − b2 e2
D = −p(a2 + b2 ) − ace2
E = −q(a2 + b2 ) − bce2
F = (a2 + b2 )(p2 + q 2 ) − c2 e2
Workbook 18: Functions of Several Variables

This Workbook is in four sections. We introduce first of all functions of several variables and then
discuss graphical representation (surfaces) for the case of two independent variables x and y.
The heart of the text is the section on partial differentiation (for which, naturally) a basic knowledge
of differentiation in the case of one independent variable is needed. The material is actually quite
straightforward and proceeds only to second partial derivatives including mixed derivatives.
The third section covers stationary points for functions of two variables with computer drawn diagrams
to illustrate the various possibilities. The location of stationary points using partial differentiation is
carefully discussed and the second derivative tests to determine the nature of these points clearly set
out but not proved.
The final section covers errors and percentage changes using partial differentiation. Many worked
examples including applications are provided at appropriate intervals.
Note relating to Workbook 18 page 18:
It is difficult to give a watertight yet straightforward explanation of the tangent surface in 3-D
although the concept is readily understandable. We have therefore not attempted to be rigorous
here.
Workbook 19: Differential Equations

This topic is one of the most useful areas of mathematics for the scientist and engineer in view of
the extensive occurrence of differential equations as mathematical models. This Workbook provides
a sound introduction to first and second order ordinary differential equations (ODEs). No previous
knowledge of the topic is required but a basic competence at differentiation and integration is needed.
The first section introduces ODEs through a model and then discusses the basic concepts of order,
type and general solution together with the need for additional condition(s) to obtain a unique
solution.
44 HELM (2015):
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The section on first order ODEs covers separation of variables and the conversion of linear ODEs
into exact equations of the form
d
(y × f (x)) = g(x)
dx
via the use of an integrating factor.
As would be expected in an introductory treatment, second order ODEs are restricted to the linear
constant coefficient type but these of course have many applications. Both homogeneous and non-
homogeneous types are fully discussed. Knowledge of complex numbers is useful here for dealing
with forcing functions involving sinusoids. A number of engineering application examples from electric
circuit theory are fully worked out.
The final consolidating section of the Workbook returns to the applications aspect with, among other
topics, a reasonably full discussion of mechanical oscillations.
Workbook 20: Laplace Transforms

This Workbook is in many ways a counterpart to Workbook 21 on z-transforms but the two books are
independent. Workbook 20 contains a thorough introduction to the Laplace Transform and its main
properties together with various applications. Only the one-sided Laplace Transform is considered.
A facility with integration is a valuable prerequisite.
The first section introduces causal functions of the form
h(t) = f (t)u(t)
where u(t) is the unit step function. Simple properties of causal functions are studied as are delayed
functions of the form
f (t − a)u(t − a)
and signals of finite duration of the form
f (t)[u(t − a) − u(t − b)]
The definition of the Laplace Transform is used to transform various common functions including
delayed functions. Inversion of transforms is demonstrated using partial fractions (an outline knowl-
edge of which is assumed). Introduction of the shift theorems both in t and the Laplace variable s
allows a wider range of transforms and inversions to be dealt with.
The Laplace Transform of derivatives is covered and this leads on naturally to the solution of ordinary
differential equations and systems of these. Specific electrical and mechanical engineering examples
are demonstrated. A section on convolution integrals and the time convolution theorem is provided
and the concept of transfer or system function for linear systems follows naturally. The negative
HELM (2015): 45
feedback system is investigated. These application topics enhance the value of the Workbook in a
specifically engineering context.
Workbook 21: Z-Transforms

This topic does not appear in older classical texts covering mathematical methods in science and
engineering and hence the detailed coverage from first principles provided in this Workbook should be
of considerable value particularly to those involved in discrete signal processing and digital control.
The Workbook begins with a basic discussion of sequences and simple difference equations before
moving on to a full discussion of the z-transform and derivations of the transforms of important
sequences. The transform is introduced ab initio with no previous knowledge of Laplace transforms
required.
The shift properties of the z-transform are fully discussed and applied to the solution of linear
constant coefficient difference equations. Inversion of z-transforms by partial fractions and by the
use of residues is covered although no detailed knowledge of Complex Variable Theory is needed.
A detailed application of the solution of difference equations to obtain currents in a ladder network
provides a good consolidating example.
A full section is devoted to the concept of transfer (or system) function of discrete systems and the
time convolution theorem is also discussed in some detail.
A final (optional) section briefly extends the theory to sampled functions and the relation between
the z and Laplace transforms is finally obtained.
Workbook 22: Eigenvalues and Eigenvectors

This Workbook assumes a knowledge of basic matrix algebra and an outline knowledge of determi-
nants although brief revision notes are provided on the latter topic. After a brief rsum of the various
situations that can arise in solving systems of linear equations of the form
CX = K
a full discussion is given of the eigenvalue problem
AX = λX
and calculation of the eigenvalues λ and the eigenvectors X is demonstrated both using the charac-
teristic equation and later by numerical methods.
A detailed section on applications considers diagonalisation of matrices with distinct eigenvalues and
this leads on to solving systems of differential equations (such as coupled spring systems) by the
‘decoupling’ method.
46 HELM (2015):
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Further applicable theory deals with the pleasant properties of symmetric matrices in this context
and an outline is given of the situations that can arise with general matrices possessing repeated
eigenvalues.
A student who has mastered the material of this Workbook is in a strong position to study specific
applications such as advanced dynamics and modern control.
Workbook 23: Fourier Series

Fourier series are a well-known tool for analysing engineering and physical systems involving peri-
odic functions (signals). They also arise in solving partial differential equations analytically by the
separation of variables method. The basic idea of course is to represent a periodic signal in terms
of sinusoids and co-sinusoids. Full details as to how to do this are given in a plausible rather than
rigorous fashion in this Workbook. Facility with integration is the main prerequisite of the reader.
An introductory section outlines the basic jargon of the topic (frequency, harmonics etc) and then,
after a discussion of the relevant orthogonality properties, we show how to obtain Fourier series for
functions of period 2π and then of more general period.
The particular form of the Fourier series for functions which are odd or are even is covered and
this leads on, after a brief discussion of convergence, to obtaining Fourier sine or cosine series for
functions defined over a limited interval.
The final section of the Workbook gives a detailed coverage of the complex exponential form of Fourier
series together with Parseval’s theorem and a brief discussion of electrical engineering applications.
Fourier transforms are covered separately in Workbook 24, and the use of Fourier series in connection
with partial differential equations is introduced in Workbook 25.
Workbook 24: Fourier Transforms

In this Workbook a brief but adequate introductory account is given of the complex exponential form
of the Fourier transform. An informal derivation of the transform as a limiting case of Fourier series
is given and mention made of the different ways of writing the transform and its inverse. Our choice
1
is to put the factor in the formula for the inverse and to use the angular frequency ω in the
2π
complex exponential factors.
The Fourier transforms of common signals such as exponential and rectangular are derived from the
definition after which the main properties of the transform are obtained. The time and frequency
shift (or translation) properties, the frequency and time differentiation properties and the form of the
transform for even and for odd functions are among the topics discussed here.
Brief plausible discussions of the Fourier transforms of the unit impulse function (also known as the
Dirac delta function) and of the Heaviside unit step function are given in the final section.
A generous selection of worked examples and of exercises (with outline answers) is provided. There
HELM (2015): 47
is little specific discussion of applications of Fourier transforms but a student who has mastered the
contents of the Workbook would be in a good position to use them in specific fields.
Workbook 25: Partial Differential Equations

This topic is of course a very broad one. This Workbook is restricted to consideration of a few special
PDEs but they are ones which model a wide range of applied problems. No previous knowledge of
PDEs is required but a basic knowledge of partial differentiation and the ability to solve constant
coefficient ordinary differential equations is assumed although brief revision of the latter topic is given.
Numerical methods of solution are not dealt with.
The main PDEs studied - the two-dimensional Laplace equation, the one-dimensional wave equation
and the one-dimensional diffusion (or heat conduction) equation - are fully discussed (although not
derived in detail) so that the student knows how the detailed mathematics to follow may be applied.
Various types of initial and boundary conditions are mentioned.
The separation of variables method of solution for PDEs is the core of the Workbook and this is
developed carefully, initially looking at problems where Fourier Series do not arise. Emphasis is given
to the fact that the method is a logical sequence of steps relevant to a wide range of problems. The
Workbook concludes with problems where Fourier Series are required, the latter being the subject of
Workbook 23 in this series.
Workbook 26: Functions of a Complex Variable

This Workbook builds especially on the work on complex numbers covered in Workbook 10. The
six areas that this Workbook deals with are: the basic ideas of a function of a complex variable
and its derivatives, the Cauchy-Riemann equations, the standard complex functions, basic complex
integration, Cauchy’s theorem, and singularities and residues.
Starting with some simple complex functions and their evaluation, the idea of the limit of a function
as the input variable tends to a special value is best tackled by choosing examples where the limit is
path-dependent, followed by one where it is not; in the latter case the fact that the limit does not
depend on the two or three paths chosen does not imply per se that the limit is path-independent,
and raises the question as to how to be sure that it is path-independent. The derivative of a complex
function should follow, with particular emphasis being placed on the idea of a singular point and on
the similarity between processes of differentiating a complex function and a real function at a regular
point.
The use of the Cauchy-Riemann equations to answer the question of analyticity of a function should
be a demonstration of the elegance of this area of mathematics, and leads nicely into the idea of
conjugate harmonic functions. The explanation of the role of these functions in an application of
Laplace’s equation should be taken. The usefulness of a mapping being conformal will have more
impact if related to an example from the appropriate engineering discipline without the need to
attempt the solution of that example. With what could be a relatively abstract area of mathematics,
the aim should be first to show relevance, in the hope of producing sufficient interest to go through
the necessary algebra.
48 HELM (2015):
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When introducing the standard functions of a complex variable, the similarities to and differences
from the real counterparts need careful treatment. The case of multiple solutions to the equation
ez = 1 contrasts nicely with the unique solution to the equation ex = 1.
Section 4 gives the reader a first glance at complex integration. The message is ‘keep the integrands
simple’. Again, it is useful to give an example where the integral is path-dependent and one where it
appears to be path-independent, and raise the question of how we could verify the path-independence
in any particular case. This area of mathematics is awash with elegant theorems and may appear
daunting to students not weaned on theorems at school; a gentle approach is recommended.
The beauty of the simplicity of Cauchy’s theorem and its wide applicability has the potential to excite
the imagination of even weak students, if sensitively presented. A gentle gradation of examples is
especially called for here. The extension to Cauchy’s Integral Formula should follow immediately to
link the two results at the earliest opportunity.
The classification of singularities of a complex function is a natural consequence of its Laurent series,
which in turn is a neat extension of its Taylor series. Experience shows that Taylor series can be
particularly daunting and the use of simple functions and a relatively low expectation of the depth of
coverage at a first run-through is advisable.
The residue theorem can be introduced by means of simple examples and forms a suitable point at
which to conclude the first venture into functions of a complex variable.
The aim throughout should be to enthuse the student by the elegance and simplicity of the topic and
not go too deeply into each sub-topic. Harder examples are best tackled when the imagination has
been fired and this is done by keeping the examples simple; far better not to tackle harder examples
at this stage than to turn the students off.
Workbook 27: Multiple Integration

Multiple Integration is a topic following on from definite integration and is often taught in the
second year of University courses. The approach taken here has been moulded by modules taught to
Mechanical and Aerospace Engineers in recent years.
The comparison with “single” integration (i.e. volume under a surface plays an analagous role to
area under a curve) is quite deliberate as are the use of the terms “inner” and “outer” integrals
and the demonstration of double integration over a rectangular region (27.1) before going onto non-
rectangular regions (27.2). The majority of the examples given in a rectangular geometry are in terms
of variables x and y, reflecting the fact that most of the applications given involve integration over
an area or volume. However, at least one example (in 27.1) draws attention to the use of alternative
notation in terms of s and t.
Similarly, most examples in a circular geometry make use of r (or ρ ) and θ (or in spherical geometry,
r, θ and φ ). However, the attention drawn to s and t in 27.1 reinforces to the student that there
is nothing ”sacred” about the variable names used. Similarly, while the notation dA or dV is used
to represent an area element or a volume element, it should be appreciated that other notation
could be used. While 27.4 is concerned with the Jacobian and change of variables, it was felt the
transformation to plane polar coordinates was sufficiently universal to allow this special case in 27.2
HELM (2015): 49
and let 27.4 show this case in context and look at other possible transformations.
Note relating to Workbook 27 page 64:
Note that here in discussing Jacobians the notation of two vertical bars is used in close proximity for
two quite different concepts: modulus and determinant.
Workbook 28: Differential Vector Calculus

The main thrust of this Workbook is in 28.2. The introduction of the vector differential operators
div, grad and curl with a necessary background to the subject in terms of scalar and vector fields is
given in 28.1, and 28.3 developes the ideas in alternative coordinate systems. The underline notation
(e.g. u) has been used for vectors and the cartesian unit vectors expressed as i, j and k, giving
consistency with Workbook 9.
However, two sets of notation are used for the vector differential operator i.e. the use of the words
grad f , div F and curl F are sometimes used rather than the ∇ notation ( ∇f, ∇ · F and ∇ × F ).
It was felt important to introduce students to both notational forms as the word notation often puts
the mathematics in more of a physical or engineering context while the operator often simplifies
calculation.
Originally, 28.3 on vector derivatives in orthogonal curvilinear coordinates was intended to be more
general i.e. to mention cylindrical polar coordinates and spherical polar coordinates in the context of
being special cases and to look at a few more coordinate systems and how to calculate relevant vector
derivatives. However, pressure on space did not allow this and the Section was revised to include
only cylindrical and spherical polar coordinate systems. [For these other cases, refer to Handbook of
Mathematical Formulas and Integrals, Alan Jeffrey, Academic Press.]
In 28.3, unit vectors ρ̂, φ̂ and ẑ (cylindrical polar coordinates) and r̂, θ̂ and φ̂ (spherical polar coordi-
nates) are used. Of course, these are equivalent to ρ̂, φ̂ and i, j and k for Cartesian coordinates. In
fact, the vectors ẑ and k are the same vector but the notational form chosen fits the form of the other
vectors under consideration i.e. k for cartesian coordinates and ẑ for cylindrical polar coordinates.
Workbook 29: Integral Vector Calculus

This Workbook uses the same notation for vectors and vector derivatives as Workbook 28. Subsection
1 of 29.1 on Line Integrals was a candidate for being 27.5 (i.e. in the Workbook on Multiple Integrals)
but it was decided to include it here in condensed form (i.e. at the beginning of the Section on Vector
Line Integrals).
A possible challenge when carrying out integrals over a sloping area is of expressing dS correctly in
terms of dx and dy (or dz and one of dx or dy). This point has been emphasized in the text and in
a couple of examples.
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The convention used was that the mathematical expressions were referred to as theorems e.g. Gauss’
theorem while the term law was reserved for the application to electromagnetism (see Engineering
Example).
Clearly, these topics (particularly Green’s theorem) could be mentioned in greater depth.
Workbook 30: Introduction to Numerical Methods

The Workbook assumes familiarity with basic matrix theory. Although more can be said to those
who know more about matrices, what is assumed is the realistic minimum.
Rounding error and Conditioning

The key here is to ensure students have an awareness that errors due to rounding (and indeed to any
other source) can build up, and that injudicious choice of operations can exaggerate the error.
Gaussian elimination and partial pivoting

Solving simultaneous equations needs care if it is to be effective. Although to many Gaussian
elimination is common sense (and the impressive name may cause some not to notice that this is
what you might do if nobody had told you how to solve equations!) there are the issues of errors
building up. Pivoting is a way to tackle this, effectively by avoiding dividing by small numbers.
LU decomposition
In principle one could solve simultaneous equations by inverting the matrices involved, and there is
an explicit algorithm for finding inverses. However, while this is fine in theory, the algorithm involves
many determinants and as the determinant of an n × n matrix can have n! terms, this is not really
practical for even modest values of n. The L-U decomposition is much simpler, and you could if you
wished ask students to estimate the number of operations involved and compare it with inverting.
Matrix norms
The condition number gives an estimate of the accuracy of the solution process of a matrix equation,
in that, very roughly you might expect the errors in the solution to be the size of those on the right
hand side multiplied by the condition number. For square symmetric real matrices, the choice of
norm which gives the smallest condition number will make it the ratio of the eigenvalues of largest
and smallest modulus: this will make sense only to students who know something about eigenvalues.
Iterative methods
Jacobi and Gauss Seidel iteration: see Engineering Example 1 for a case where these take a long time
to converge.
Workbook 31: Numerical Methods of Approximation

Polynomial approximations
Approximation by polynomials is common sense in that we want to approximate the unknown func-
tions by something simple, and polynomials are simple functions. Lagrange interpolation produces
an approximation that is exact at certain points. It may be worth exploring with the students just
how certain an engineer is going to be about the data that are being interpolated, and to look at the
effects of changes, especially if two of the x-values are close.
HELM (2015): 51
For students whose background is suitable one can mention approximation by other types of functions
(e.g. trigonometric polynomials), or functions, which share properties which the solution is known to
have. Least squares approximation of straight lines is another approach: a polynomial of degree 1 is
being made to fit the data as well as possible. One could use the best least squares approximation of
N points by polynomials of some chosen degree, to make the answer less susceptible to changes in
the data. Exploring the effects here with a real set of data relevant to the students might be useful.
Numerical integration
Consider the size of the errors for the composite trapezium (ask the students to estimate it). In
many cases this is small enough that nothing more sophisticated is needed, although there are more
accurate ways of doing numerical integration: they are also more complicated.
Nonlinear equations
The bisection methods, although slow, will always work. Newton-Raphson is much more efficient, if
the starting point is “close enough”, but requires the derivative to be calculated. It can be shown (and
Figure 11 almost does this) that Newton-Raphson will work provided the second derivative does not
change sign in the area considered. There are more complicated ways of guaranteeing convergence
if the derivative does change sign.
For students and teachers willing to investigate this, try Newton-Raphson for solving x3 − x = 0 ,
with various starting points. Which root does it converge to? This is quite sensitive near 0, where
the second derivative does change sign.
The last word . . .

Student comment: “I am amazed that you can base a whole module around guessing and still make
it sound credible.”
Workbook 32: Numerical Initial Value Problems

Much of this Workbook deals with PDEs and, although most Engineering students will not have
encountered this, the Workbook reflects the mathematical difficulties involved in analysing PDEs. It
is important that students appreciate that stability and convergence are important, even though the
analysis may be hard.
General linear multistep methods

It’s worth making the analogy with solving ordinary equations, that some of the apparently more
awkward formulations may be easy to use as iterative methods.
Runge-Kutta method, predictor corrector methods

This gives a way of finding a reasonable initial estimate of the solution, which can be used in the
corrector method: this effectively turns an implicit method into an explicit iteration. As that can
be iterated, the result may be used to obtain good accuracy, provided we use something to get it
started.
The sections on parabolic and hyperbolic PDEs, both of which usually arise in the form of initial-value
problems, illustrate the care needed to make the methods work on particular PDEs, and that the
same method will not necessarily work on other types of PDE.
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Workbook 33: Numerical Boundary Value Problems

Two point boundary value problems
These link well with some of the earlier work on Workbook 30, and it is worth emphasising that the
numerical schemes, which are reasonably intuitive, give rise to potentially large sets of simultaneous
equations, with all the difficulties discussed in Workbook 30. The methods there, however, will allow
solutions to be found.
Elliptic PDEs
These normally arise in the form of boundary-value problems, and the Workbook gives methods for
Laplace’s and Poisson’s equations, and that it is known that these converge. The matrices involved
are all “sparse” in that they have a structure and most of the entries are zero. Finding ways of
exploiting the structure of the matrices can lead to improved numerical methods for their solution.
Again the techniques of Workbook 30 are important, and the students should be able to see that
further progress may need more attention to the methods of solving set of equations, or of matrix
iteration.
Workbook 34: Modelling Motion

This Workbook relates to the modelling ideas in Workbook 5 and Workbook 6 Section 6.6. Workbook
34 would be best used to consolidate the material in Workbooks 9 to 15 (i.e. vectors and differential
and integral calculus). It does so in the context of the dynamics of rigid body dynamics (projectiles,
friction forces, motion on an inclined plane, motion in a circle, 3D examples of motion and resisted
motion). The level is most appropriate to first year engineering mathematics. However it might form
the basis for further trips around the modelling cycle in later years. Further developments might
include the numerical solution of the differential equation for resisted motion that is not linear in
velocity, and more realistic modelling of vehicle motion on bends.
Workbooks 35-46: Probability and Statistics

Probability and statistics deal with uncertainty and variation. Engineering students do not always
easily feel comfortable with these ideas, being perhaps more at home with the apparent certainties
of physical laws. In the world in which engineers work, of course, variation does exist, in materials,
in the environment, in the wear and tear to which products are subjected and so on. Therefore there
is also uncertainty.
Engineering students often need help in coming to terms with these concepts and with the idea
that the “answer” to questions is often a statement, of one type or another, about the remaining
uncertainty in a situation rather than a fixed value for some physical quantity. Physical analogy
or “feel” does not always come so easily with such ideas. In students’ efforts to deal with this,
motivation is often of great importance and motivation is easier when the student sees the material
as relevant. This is one reason why we have attempted to provide examples and explanations, which
relate to engineering. It would, of course, be impractical to provide examples of every topic drawn
from every specialised branch of engineering but students should be able to be less narrow than this
and tutors may be able to provide additional, more specialised, examples by changing the “stories”
in some of ours. The second reason for trying to use relevant engineering examples and illustrations
HELM (2015): 53
is to help students to attain that “feel” which will help them to grasp what is going on. This is
made easier if they can relate probability statements, for example, to frequencies of events of familiar
types. So, for example, it may be easier at first to think of 100 light bulbs, or whatever, of which
“on average” one in twenty, that is 5, might be expected to fail before a certain time, rather than
talk more abstractly of a probability of 0.05. Similarly, it might help to think of a product with two
components, A and B, where one in ten products “on average” has a defect in component A and
one in five products “on average” has a defect in component B, regardless of the state of component
A. Then, out of 100 products, we might expect on average ‘one in five of one in ten products’, that
is one in 50, to have defects in both components.
There is more than one school of thought in probability and statistics but presenting both of the major
approaches in introductory material such as this would risk causing confusion. Although the Bayesian
approach has made major advances in recent years, it is probably still true that the approach which
engineering graduates will be expected to have learned is the more traditional frequentist approach.
Therefore probability is presented in terms of limiting relative frequency and statistical inference is
presented in terms of the familiar ideas of estimators, confidence intervals and significance tests.
Thus properties can be explained in terms of long-run frequencies. This is not too difficult in the
context of a mass-production industry, for example, although care is still required in explaining exactly
what is meant by a confidence interval or the interpretation of a non-significant result in a test. In
branches of engineering concerned with one-off constructions, the application of relative-frequency
ideas requires more careful thought and abstraction although, even here, it is often the case that, for
example, large numbers of similar components are used.
Students should be encouraged at an early stage to distinguish between a population and a sample,
between the (usually unknown) value of a population parameter and its estimate and, of course,
between an estimator, that is a random function, and an estimate, that is a particular value. This
becomes particularly important from Workbook 40 onwards. Experience shows that students do not
always find this easy and often fall into the trap of believing that the data tell us that µ = 1.63 or
that the null hypothesis is (definitely) false.
This is, of course, an introductory service course in probability and statistics and it is neither possible
nor desirable to attempt to achieve either the breadth or depth that would be given in a course for
specialist statistics students. Inevitably there are topics which we do not cover but which may be
encountered by engineering students at some time. We feel that the important thing to be achieved
in a course such as this is a grasp of the basic ideas and concepts. Once these are understood,
coping with new distributions and procedures, perhaps with the aid of suitable computer software
and, where appropriate, specialist advice, can often be relatively straightforward.
In the modern world, statistical calculations are usually done with a computer and we expect that
students will learn how to do some calculations this way. However it is also useful from a learning
point of view to do some, relatively simple, calculations without the aid of a computer program.
When a computer is used, it is best to use special statistical software. There are many packages
available and many of these are easy to use for beginners, for example Minitab. Some excellent
software, e.g. R, is available free of charge on the Web though non-specialist students might not find
this quite as convenient to use. Having said this, many students will have access to a spreadsheet
program and may prefer to use this. Most of the calculations described in the HELM Workbooks are
possible without too much difficulty using popular spreadsheet software. Some engineering students
may well be using, other software such as Matlab in other parts of their courses. Matlab also provides
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statistical and graphics facilities and, if the students are using it anyway, it may be a sensible option.
Few engineering students should have difficulty with the mathematics in the statistics Workbooks
(especially if they have the benefit of the HELM mathematics Workbooks!). Experience suggests
that difficulty is more common with the underlying statistical and probabilistic concepts.
Workbook 35: Sets and Probability

This Workbook serves to underpin all of the probability and statistics covered in the HELM series
of Workbooks 35-46. For this reason it is essential that the students fully understand and become
conversant with the notation used and the concepts explored. The notation used is standard set
notation as found in a myriad of standard textbooks on the subject. Particular attention should be
paid to Section 4 - Total Probability and Bayes’ Theorem. It is the authors’ experience that effort put
in to learning the concepts described and fully studying the worked Examples and then completing
all of the Tasks and Exercises will pay handsome dividends when later Workbooks concerned with
probability and statistics are studied.
Some of the Figures and Examples are in terms of electrical items such as switches and relays wired
together in series and/or parallel. Even though the diagrams are straightforward, care should be
exercised to ensure that the student is familiar with and understands these diagrams. No knowledge
of circuit theory as such is assumed but it is possible that some students may not understand the
ideas involved in series and parallel circuit wiring, for example. It is hoped that this will not be a
problem for engineering students!
Workbook 36: Descriptive Statistics

The first part of this Workbook, entitled “Describing Data”, looks at the basics of calculating means
and standard deviations from tabulated data. Many students feel initially that in the age of electronic
calculators and computers this is unnecessary. However, it should be born in mind that a familiarity
with these basic formulae will pay dividends when later studying Workbook 37: Discrete Probability
Distributions and Workbook 38: Continuous Probability Distributions. The first section of Workbook
36 lays the necessary ground for a proper understanding of the formulae and methods used to calculate
means and variances for discrete and continuous distributions.
The second section of Workbook 36 is aptly titled “Exploring Data”. This is important. The
material enables the student to avoid the twin pitfalls of simply taking data either at their face value
or accepting them simply as a mass of meaningless figures. By using the techniques outlined in this
section of Workbook 36, the student should begin to develop the ability to “look into” a data set
and draw tentative conclusions prior to any attempt to analyse the data. This also has the benefit of
allowing the student to comment on the possible validity of any conclusions reached using statistical
analysis.
Workbook 37: Discrete Probability Distributions

This Workbook looks at the basics of discrete distributions in general and then turns its attention
to three distributions in very common use. The binomial distribution is followed by work concerning
HELM (2015): 55
the Poisson distribution. This latter distribution is looked at in two distinct ways, firstly as a viable
approximation to the binomial distribution, a role in which it can save the user a large amount of
tedious arithmetic normally without significantly reducing the accuracy of the answers obtained. The
second way in which the Poisson distribution is considered is as a distribution in its own right used
to describe the occurrence of events by the so-called Poisson process.
The Workbook concludes with a short section concerning the hypergeometric distribution. This has
applications in, for example, acceptance sampling where, because we are sampling without replace-
ment from a finite batch, the results of individual trials, e.g. of whether an item is defective, are
not independent. This is in contrast to the usual assumptions underlying the binomial distribution.
The distribution of the numbers of defective and acceptable items in a sample from a batch may
seem to be of little direct interest. However, when we turn the question round from “Given certain
numbers within the batch, what is the probability of observing x?” to the inferential question, “Given
observation x, what can we learn about the rest of the batch?” then the hypergeometric distribution
turns out to be very relevant. It will also reappear in Workbook 42 in the context of the test for no
association in contingency tables.
The notations E for expectation and V for variance are introduced.
Workbook 38: Continuous Probability Distributions

This Workbook follows the same general format as Workbook 37 “Discrete Probability Distributions”.
Firstly, a look at the basics of continuous distributions in general confirms that the same general
pattern may be discerned in the development and use of, for example, the formulae used to calculate
means and variances.
The Workbook then turns its attention to two distributions in very common use, the uniform distri-
bution and the exponential distribution. In introducing the exponential distribution, we also begin
the development of applications to reliability and lifetimes. The statistical analysis of the nature of
the expected lifetime of a product has huge relevance to the industrial applications of statistics and
for this reason the exponential distribution is worthy of close study. The concept of a distribution
function can be explained in terms of a lifetime distribution and this can then lead to the idea of a
probability density function. Analogy with physical mass density can also be helpful.
Workbook 39: The Normal Distribution

The normal distribution is extremely important to all students of statistics, and since a very large
number of practical experiments result in measurements, which are closely modelled by this distribu-
tion, it is certainly essential that engineers fully understand and can apply this distribution to practical
situations. A great many problems can be formulated and solved by using the so-called standard nor-
mal distribution. This Workbook will guide the student through a number of applications sufficient
to lay a firm basis for future competence in problem solving using the normal distribution. The usual
notation of N (µ, σ 2 ) for a normal distribution with mean µ and variance σ 2 is used.
The second section of the Workbook builds on the foundations laid in Workbook 37 concerning
the binomial distribution. Here we look at how the normal distribution can by used to approximate
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the binomial distribution provided that the conditions are appropriate. This application can save an
enormous amount of arithmetic without loss of accuracy when applied properly.
Finally, this Workbook looks at practical situations whose description involves sums and differences of
random variables. Without a working knowledge of the behaviour of these quantities, unacceptable
restrictions have to be placed on the range of applications, which an engineer (or anyone else!) is
able to consider.
Workbook 40: Sampling Distributions and Estimation

Having looked at probability distributions in Workbooks 37-39, we now move on to inferential statis-
tics. The approach taken to inference is the traditional frequentist approach. Experience shows that
attempting to teach both frequentist inference and the alternative Bayesian inference in one intro-
ductory statistic course can lead to confusion and, whether we like it or not, frequentist inference still
predominates in much of industry and graduates will be expected to know about significance tests
and confidence intervals.
Concepts of inference can often be difficult for students to grasp at first since students are often
unused to dealing with uncertainty and questions where the answer is a statement about something
which we do not know for certain. For this reason it is important to be careful with the distinctions
between, for example, population parameters and sample statistics.
As the title of this Workbook implies, it is concerned with two crucially important aspects of statistics,
namely samples, their properties and their use in estimating population parameters. The student
should realise that it is often impractical or impossible to deal with a complete population. It is
usually the case that the parameters of a population - the mean and variance for example - have
to be estimated by using information which is available from a sample or samples taken from the
population. Properties of estimators: bias, consistency and efficiency, are introduced. It is important
at this stage to make sure that students understand the distinction between population and sample
and between estimator and estimate and the concept of a sampling distribution.
It is reasonable to enquire as to the degree of accuracy likely to be involved when a population

parameter is estimated. This question is addressed in the Workbook by the construction of confidence
intervals for the mean of a population and the variance of a population. The idea of a confidence
interval is subtle and not always well understood. Again the concept of properties in repeated
sampling is important.
Constructing a confidence interval for the variance of a normal population involves the use of the Chi-
squared distribution. An introduction in sufficient detail is given to enable the student to gain a good
level of understanding of this distribution and its application to finding confidence intervals involving
estimates of the variance of a normal population. Given that a great many practical situations involve
statistical estimation it will be appreciated that this Workbook deserves the very close attention of
the student.
HELM (2015): 57
Workbook 41: Hypothesis Testing
A solid understanding of the theory and techniques contained in this Workbook is essential to any
student wishing to apply statistics to real-world engineering problems. Many students find the material
challenging, one of the reasons being that work on hypothesis testing is often attempted on the basis
of half-understood and half-remembered theory and techniques which necessarily underpin the subject.
The student should already be familiar with the material in the HELM statistics series concerning
the following topics:
• sampling;
• the normal distribution;
• the binomial distribution;
• the chi-squared distribution.
In this Workbook it is assumed that students have some understanding and a working familiarity with
all the above topics. Remember that no attempt is made in this Workbook to teach those topics
and that coverage of them will be found elsewhere in the HELM statistics series.
Student’s t-distribution and the F -distribution are introduced in this Workbook.
Since many students find some difficulty with this topic, it is strongly suggested that the worked
examples are given particular attention and that all of the student Tasks are fully worked. The
students should not allow themselves to fall into the trap of simply looking up the solution to a
problem if they cannot successfully complete it. Suggest that the students keep a record of their
attempts to solve problems in order that any lack of understanding may be identified and rectified.
Even published scientific literature often contains suggestions of misinterpretation of the results of
significance tests. It should be emphasised that a significance test measures (in a particular sense)
the evidence against a null hypothesis. A significant result means that we have “strong” evidence
against a null hypothesis but not usually certainty. A non-significant result simply means that we do
not have such strong evidence. This may be because the null hypothesis is true or it may be that we
have just not obtained enough data.
Workbook 42: Goodness of Fit and Contingency Tables

In order that the student understands the first section of this Workbook (“Goodness of Fit”) it is
essential that the principles of hypothesis testing contained in Workbook 41 are properly understood.
In particular, the student should be recommended to revise, if necessary, any work done previously
concerning the Chi-squared distribution.
Essentially, the first section of the Workbook is concerned with making decisions as to whether or
not a given set of data follow a given distribution (say normal or Poisson) sufficiently closely to be
regarded as a sample taken from such a population. If an underlying distribution can be identified,
then clearly certain assumptions follow. For example, if a data set is normally distributed then we
know that in general we may calculate parameters such as the mean and variance by using certain
standard formulae. If a data set follows the Poisson distribution then we may assume that the
population mean and variance are numerically identical.
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The second section of the Workbook (“Contingency Tables”) is concerned with situations in which
members of samples drawn from a population can be classified by more than one method, for example
the failure of electronic components in a system installed in a machine and the positions in a machine
in which they are mounted. The Workbook discusses how such information may be presented and
follows this by a discussion involving hypothesis tests to decide (using the example outlined above)
whether or not there is sufficient evidence to conclude that failure of a component is related to
position in the machine.
Workbook 43: Correlation and Regression

Section 43.1 introduces the study of regression analysis, that is, the study of possible predictive
or explanatory relationships between variables. The student should note that the real work of this
Section deals with linear regression on one variable only. In the real world, regression may be multiple
(i.e. with several explanatory variables) and may be non-linear. A very short introduction to non-
linear regression is given at the end of 43.1 and is introduced for the sake of completeness only -
simply to signal to the student that non-linear regression does exist and that the real world is often
anything but linear!
Modern computer software makes the calculations for multiple regressions and many nonlinear re-
gressions readily available but an understanding of the basic ideas, gained through looking at simple
cases, is essential to interpret the results.
The main technique used to study linear regression is the method of least squares. In order that
this topic be fully understood, it is necessary to use some multi-variable calculus. The Workbook
deliberately ignores this and simply presents the equations resulting from the use of this technique.
Section 43.2 is concerned with the topic of correlation. Two common methods of measuring the
degree of a possible relationship between two random variables are considered, these are Pearson’s r
and Spearman’s R. The methods of calculation given will result in a numerical value being obtained.
The strength of evidence for the existence of a relationship may be measured by performing a
significance test. The importance of the normality assumption for the usual test of Pearson’s r
should be emphasised. In cases where this assumption cannot be made or where a relationship may
be monotonic but not linear Spearman’s R should be used. It is worth noting that significance tests
involving both measures of correlation involve the use of Student’s t-distribution and that it may be
worth revising this topic before a study of correlation is attempted.
Workbook 44: Analysis of Variance

This Workbook covers three topics, one-way analysis of variance, two-way analysis of variance and an
introduction to experimental design. In order to obtain the maximum benefit from this Workbook,
the students should ensure that they are familiar with the general principles of hypothesis testing.
In particular, a working knowledge of the F -distribution is essential. This groundwork is covered in
Workbook HELM 41-Hypothesis Testing.
The Workbook starts with one-way ANOVA and the introduction makes clear the advantages accruing
from a technique enabling us to simultaneously compare several means. On reading the introduction,
HELM (2015): 59
it should be clear to the student that not only is the amount of work needed to compare several
means drastically reduced but also that use of ANOVA gives the required significance level whereas
use of a collection of pairwise tests would distort the significance level. When working through the
learning material, students should ensure that they clearly understand the difference between the
phrases “variance between sample” and “variance within samples”.
While one-way ANOVA considers the effect of only one factor on the values taken by a variable,
two-way ANOVA considers the simultaneous effect of two or more factors on a variable. Two distinct
cases are considered, firstly when possible interaction between the factors themselves is ignored
and secondly when such interaction is taken into account. The Workbook concludes with a short
introduction to experimental design. This section is included to encourage students to consider the
design of any engineering based experimental work that they may be required to undertake and to
appreciate that the application of statistical methods to engineering begins with the design of an
experiment and not with the analysis of the data arising from the experiment.
Workbook 45: Non-parametric Statistics

This Workbook looks at hypothesis testing under conditions, which are such that we do not, or
cannot, because of a lack of evidence, make the claim that the distributions we are required to deal
with follow a specific distribution. Commonly, we may assume that data follow a normal distribution
and apply say, a t-test under appropriate conditions. Such assumptions are not always possible and
we have to use distribution-free tests. Such tests are usually referred to as non-parametric tests
since they do not refer to distribution parameters, for example the mean of a normal distribution.
This Workbook assumes a familiarity with the techniques of hypothesis testing covered in Workbook
HELM 41, in particular the t-test. A working knowledge of the binomial distribution (HELM 37) is
also essential background reading. The Workbook considers the application of non-parametric testing
to situations involving single samples and situations involving two samples.
A common error is to suppose that there are such things as “parametric data” and “non-parametric
data”. It is not the data, which are “parametric” or “non-parametric”, of course, but the procedures,
which are applied, and the assumptions underlying them. It is hoped that this Workbook will convey
an understanding of the different assumptions which are made, just what it is that is being tested
and when it is appropriate to use a nonparametric test. The tests which are specifically addressed,
the sign test, the Wilcoxon signed-rank test and the Wilcoxon rank-sum test (which is equivalent to
the Mann-Whitney U -test) are straightforward in themselves but the assumptions underlying them
and their interpretation are not always well grasped.
Workbook 46: Reliability and Quality Control

This Workbook is split into two distinct sections, the first covering the topic of reliability and the sec-
ond giving an introduction to the crucially important topic of quality control. Necessary background
reading for this Workbook consists of general probability (HELM 35) and continuous probability
distributions (HELM 38). In Workbook 46 the study of reliability begins with a look at lifetime
distributions and progresses to work concerning system reliability with some surprising results. For
example while it may be taken to be intuitively obvious that a set of electronic components wired in
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series will be as reliable as the least reliable member, this turns out not to be the case - the set turns
out to be less reliable than its least reliable member!
The section concerning quality control starts with a very brief history of the subject and introduces
the student to some essential elementary control techniques including simple quality control charts,
R-charts and Pareto charts. Elementary trend detection is considered via the use of “standard”
checks, which the student may apply to, given data sets.
The material in this Workbook will have a clear relevance for many engineering students. Tutors
might like to consider using some of the material earlier in the sequence of Workbooks if this seems
appropriate to maintain interest and motivation.
Workbook 47: Mathematics and Physics Miscellany

Section 47.1 “Dimensional Analysis in Engineering” sets out several examples, which show the use of
dimensional analysis in the analysis of physical systems. The standard examples such as the simple
pendulum are treated in a detailed manner, so as to emphasize that dimensional analysis has several
“blind spots”; in particular the unknown “constant” which appears in standard simple treatments
must actually be regarded as a dimensionless quantity and so could be an arbitrary function of
an angle, of a Mach number, etc. In several cases the usual simple approach is compared with
the Buckingham approach, which makes the dimensionless quantities the dominant quantities in the
description of a physical system. The examples treated gradually increase in complexity, with the
theory of fluids providing examples for which the set of relevant physical variables is sufficiently rich
to provide two different dimensionless quantities in the Buckingham approach.
[See the references at the end of this Section in the Workbook, which includes suggested Google
search keywords.]
Section 47.2 “Mathematical Explorations” provides an introduction to a range of techniques, which

provide useful alternatives to those of the other Workbooks. One theme, which appears repeatedly,
is that of the use of power series methods in tackling problems, which are often treated, by the use
of ordinary algebra. The tutor might note that there is an implicit underlying theme here which
is worth emphasizing in discussions: when we find the Maclaurin series for tan(x) by studying the
equation dy/dx = 1 + y 2 what we are actually doing is finding a power series solution of a differential
equation. For functions, which are already familiar, this aspect is not obtrusive but these “known”
examples can be used to motivate an approach to the study of power series solutions of more general
differential equations, especially for new cases in which we do not know the solution before we start.
For example, the power series approach is a useful tool in finding the energy levels associated with
the Schrödinger equation in quantum mechanics, particularly for cases in which the equation cannot
be solved by the simple analytic methods set out in simple textbooks.
Section 47.3 “Physics Case Studies” has eleven items. Many engineering problems are based upon
fundamental physics and require mathematical modelling for their solution. This Section contains
a compendium of case studies involving physics (or related topics) as an additional teaching and
learning resource beyond those included in the previous HELM Workbooks.
Each case study will involve several mathematical topics; the relevant HELM Workbooks are stated
at the beginning of each case study.
HELM (2015): 61
Table 1: Physics topics, related Mathematical topics and Workbooks
Physics Case Study Title Mathematical Topics Related

Workbooks
1 Black body radiation 1 Logarithms and Exponentials; 6, 31
Numerical integration
Numerical solution of equations
Differentiation
4 Black body radiation 4 Logarithms and Exponentials; 6, 13, 31
Integration; Numerical integration
5 Amplitude of a monochromatic Trigonometric functions; 4, 10, 16
optical wave passing through a Complex numbers;
glass plate Sum of geometric series
6 Intensity of the interference field Trigonometric functions; 4, 10
due to a glass plate Complex numbers
7 Propagation time difference be- Trigonometric functions 4
tween two light rays
8 Fraunhofer diffraction through Trigonometric functions; Complex 4, 10, 12
an infinitely long slit numbers; Maxima and minima
9 Fraunhofer diffraction through Trigonometric functions; Exponential 4, 6, 10, 12,
an array of parallel infinitely long functions; Complex numbers; Maxima 16, 24
slits and minima; Geometric series; Fourier
transform
10 Interference fringes due to two Trigonometric functions; Complex 4, 10, 12, 16
parallel infinitely long slits numbers; Maxima and minima;
Maclaurin series
11 Acceleration in polar coordinates Vectors; Polar coordinates 9, 17
Time allocation
To work through the whole Workbook will require at least twenty hours of independent study. However
it would be more normal (and preferable) to use it to ‘dip-in-and-out’, and to follow up additional
examples of modelling using particular techniques for an hour or two at a time.
Format
The eleven Physics Case Studies in Workbook 47 have a common format (with rare minor variations),
This is the same as that used for the Engineering Examples in Workbooks 1 to 34. The section
headings are:
Introduction
This consists of a paragraph or two of background information, setting the context and stating
essential engineering information, definitions and fundamental concepts.
62 HELM (2015):
®
Problem in words
This consists of a statement of the problem in words including the purpose of the model.
Mathematical statement of problem

Here the problem is expressed in mathematical form including notation, assumptions and strategy.
Mathematical analysis
This part gives the solution to the problem or explains how it can be solved using techniques in the
indicated Workbooks.
Interpretation
This interprets the mathematical result in engineering terms and includes at least a statement of the
result in words. If appropriate, there are comments about whether results make sense, mathematical
points, indications of further extensions or applications and implications.
Workbook 48: Engineering Case Studies

This Workbook offers a compendium of twenty ‘Engineering Case Studies’ providing additional teach-
ing and learning resources to the sixty-seven ‘Engineering Examples’ included in Workbooks 1 to 34.
Table 2 below summarises the engineering contexts, the mathematical topics and the relevant HELM
mathematics Workbooks for these Case Studies. It should be possible to use this Workbook to
reinforce notions of modelling using a wide cross section of mathematical techniques. The more
elementary mathematical topics and relevant Workbooks are not usually mentioned - for example
basic functions (Workbook 2) pervades every case Study and so is omitted. However, where there is
significant algebraic manipulation (Workbook 1) or equation rearrangement (Workbook 3) this has
been reflected in the table.
The Case Studies have been grouped together by broad engineering topic and not by mathematical
topic or difficulty or length. (The shortest and most straightforward is 11.)
HELM (2015): 63
Table 2: Engineering topics, related Mathematical topics and Workbooks
Engineering Topic Mathematical Topics Related

Workbooks
1 Adding sound waves Trigonometry; Second order ODEs 4, 19
2 Complex representations of sound Complex numbers; Trigonometry 4, 10
waves and sound reflection
3 Sensitivity of microphones Definite integrals (function of a 13, 14
function)
4 Refraction Algebra; Equations; Trigonometry; 1, 3, 4, 11
Differentiation
5 Beam deformation Algebra; Equations; Inequalities; 1, 3, 4, 13, 19
Trigonometry; Definite integrals;
First order ODEs
6 Deflection of a beam Equations; Definite integrals; 3, 13, 19, 20
ODEs; Laplace transforms
7 Buckling of columns Trigonometric identities; Deter- 4, 7, 8, 13, 19
minants; Matrices; Integration;
Fourth order ODEs
8 Maximum bending moment for a Algebra; Equations; Inequalities; 1, 3, 12
multiple structure Maxima and minima (differentia-
tion)
9 Equation of the curve of a cable Trigonometric identities; Hyper- 4, 6, 11, 13,
fixed at two endpoints bolic functions; Differentiation; 19
Integration; First order ODEs
10 Critical water height in an open Algebra; Equations; Derivative of 1, 3, 11, 12,
channel polynomials; Maxima and minima 13
(differentiation); Integration
11 Simple pendulum Algebra - rearranging formulae 1
12 Motion of a pendulum Trigonometric functions; Differ- 4, 11, 16, 18,
entiation; Maclaurin’s expansion; 19
Contour plotting for a function of
two variables; Second order ODEs
13 The falling snowflake Integration; First order ODEs; Sec- 13, 19, 28
ond order ODEs; Vector differenti-
ation
14 Satellite motion Trigonometry; Equations of conics 4, 17, 19, 28
in polar coordinates; Second order
ODEs; Vector differential calculus;
Polar coordinate system
15 Satellite orbits Trigonometry; Equations of conics 4, 17, 19, 28
in polar coordinates; Second order
ODEs; Vector differential calculus;
Polar coordinate system
64 HELM (2015):
®
16 Underground railway signals lo- Trigonometry; Differentiation of in- 4, 11, 17, 18

cation verse trigonometrical functions; Ge-
ometry (arc length); Partial differen-
tiation
17 Heat conduction through a wine Trigonometry; First order ODEs; Sec- 4, 19, 23, 25
cellar roof ond order ODEs; Fourier series; Partial
differential equations
18 Two-dimensional flow past a Algebra - rearranging formulae; 1, 4, 17
cylindrical obstacle Trigonometry; Polar coordinates
19 Two-dimensional flow of a vis- Trigonometry; Integration; Polar coor- 4, 13, 17, 18,
cous liquid on an inclined plate dinates; Partial differentiation; Partial 19, 28, 47
differential equations; Vector differen-
tial calculus; Dimensional analysis
20 Force on a cylinder due to a Algebra; Equations; Trigonometry; 1, 3, 4, 6, 13,
two-dimensional streaming and Logarithmic functions; Integration; 17, 18, 28, 29
swirling flow Orthogonality relations of trigonomet-
ric functions; Partial differentiation;
Polar coordinates; Surface integrals
Time allocation
To work through the whole Workbook will require at least fifty hours of independent study. However
it would be more normal (and preferable) to use it to ‘dip-in-and-out’, and to follow up additional
examples of modelling using particular techniques for an hour or two at a time.
Format
The twenty Engineering Case Studies in Workbook 48 have a common format (with rare minor
variations). This is the same as that used for the Engineering Examples in Workbooks 1 to 34. The
section headings are:
Introduction
This consists of a paragraph or two of background information, setting the context and stating
essential engineering information, definitions and fundamental concepts.
Problem in words
This consists of a statement of the problem in words including the purpose of the model.
Mathematical statement of problem

Here the problem is expressed in mathematical form including notation, assumptions and strategy.
Mathematical analysis This part gives the solution to the problem or explains how it can be solved
using techniques in the indicated Workbooks.
Interpretation
This interprets the mathematical result in engineering terms and includes at least a statement of the
result in words. If appropriate, there are comments about whether results make sense, mathematical
points, indications of further extensions or applications and implications.
Where possible there are comments on the sensitivity of the analysis to input data and the necessary
numerical accuracy of the outputs.
HELM (2015): 65
Index of Engineering
Contexts in

Engineering Topic Workbook Page Number

Acceleration in polar coordinates Wbk 47 67
Admittance of an electronic circuit Wbk 3 69
Aerofoil Wbk 26 14, 19
Aircraft Wbk 9 13
Aircraft wings Wbk 42 4
Aircraft wings Wbk 45 26
Airline booking Wbk 39 29
Alloy impurities Wbk 41 18
Alloy spacers Wbk 44 4, 16
Alloy stretching Wbk 45 27
Alloy-twisting resistance Wbk 45 26
Aluminium alloy tensile strength Wbk 44 45
Aluminium sheet faults Wbk 42 4
Amplifier Wbk 10 26
Amplitude Wbk 3 67, 74
Amplitude modulation Wbk 4 47
Amusement rides Wbk 34 6, 43-50
An LC circuit with sinusoidal input Wbk 19 48
An RC circuit with single frequency input Wbk 19 26
Angular velocity of Earth Wbk 34 40
Anti-lock brakes Wbk 45 11
Arrhenius’ law Wbk 6 32
Assembly machines Wbk 44 33
Asteroid Wbk 17 22
Atomic theory Wbk 47 13, 14
Ball bearing diameters Wbk 40 19

Banked tracks Wbk 34 49
Basketball Wbk 34 26
Battery lifetime Wbk 41 15
Beam Wbk 19 65, 67
Beam Wbk 20 52
Beam deflection Wbk 48 20
Beam deformation Wbk 48 15
Beats Wbk 19 64
66 HELM (2015):
®
Bending moment for a multiple structure Wbk 48 35

Bending moment of beam Wbk 19 65
Bending moment of beam Wbk 43 18
Bicycle Wbk 34 41
Black body radiation Wbk 47 38, 41, 43, 46
Bolt hole diameters Wbk 40 20
Bottle design Wbk 31 52-54
Brake Wbk 4 14
Buckling of a strut Wkb 12 44
Buckling of columns Wbk 48 26
Buffer Wbk 20 39
Cable Wbk 15 21
Cable Wbk 43 7, 12
Cable breaking strength Wbk 45 31
Cable suspended Wbk 48 40
Calculator battery life Wbk 41 15
Capacitor Wbk 20 49
Car accessories Wbk 35 18
Cartons for powder Wbk 41 13
Castings Wbk 41 9
Catalysts Wbk 44 49
CD player output Wbk 42 7
Centre of mass Wbk 27 55-65
Centre of pressure Wbk 27 15
Chain alloy Wbk 45 27
Charge Wbk 9 40
Charge on a capacitor Wbk 20 49
Chemical process Wbk 44 49
Chemical reaction Wbk 6 32
Circle cutting machine Wbk 17 10
Circular motion Wbk 34 35
Coconut shy Wbk 34 25
Columns buckling Wbk 48 26
Communication network Wbk 1 52
Communication network Wbk 7 27
Component lifetime Wbk 38 24, 25
Component variation Wbk 39 17, 18-20
Compressive strength of blocks Wbk 45 4, 14
Compressive strength of concrete Wbk 44 43
Concrete compressive strength Wbk 44 43
Conductor coating Wbk 38 21
Conservation of energy Wbk 34 28
Control charts Wbk 46 24-38
Cornering of vehicle Wbk 34 36-39, 51
Crank mechanism Wbk 4 62
Crank used to drive a piston Wbk 12 33
HELM (2015): 67
Section 50.9: Index of Engineering Contexts in Workbooks 1 to 48
Current Wbk 11 21
Current associated with magnetic field Wbk 28 28
Current in line Wbk 29 26, 67
Current in loop Wbk 29 27
Currents in a ladder network Wbk 21 60
Currents in three loops Wbk 8 30
Currents in two loops Wbk 8 16
Cutting steel quality Wbk 45 9
Dam Wbk 27 3, 15, 36
Defective components Wbk 35 48
Defects (in components and products) Wbk 37 8, 19, 20, 23, 24,
40, 42, 54-58
Deflection of a beam Wbk 48 20
Deflection of a uniformly loaded beam Wbk 19 67
Deflection of a uniformly loaded beam Wbk 20 52
Demodulation Wbk 4 40
Detecting a train on a track Wbk 30 62
Diffraction Wbk 4 6
Diffusion equation Wbk 25 8, 14
Dimensional analysis Wbk 47 2-23
Diode Wbk 31 20
Divergence of a magnetic field Wbk 28 43
Drag Wbk 34 56
Drag Wbk 47 15, 23
Dynamometer Wbk 14 16
Earth horizon Wbk 4 8

Elastic behaviour Wbk 13 19
Electric circuit Wbk 12 26
Electric circuit Wbk 18 38
Electric circuit Wbk 20 36. 44. 49
Electric current Wbk 29 46
Electric current Wbk 35 33, 39, 40
Electric current Wbk 38 20
Electric current to screen Wbk 41 24, 28, 35
Electric fan Wbk 38 25
Electric field Wbk 9 39-44
Electric field Wbk 11 16
Electric field Wbk 13 11
Electric field Wbk 29 19, 63, 67, 68
Electric meters Wbk 39 28
Electric motor Wbk 29 27
Electric potential Wbk 28 50
Electric wire Wbk 33 10
Electrodynamic meters Wbk 14 16
68 HELM (2015):
®
Electromagnetic power Wbk 6 50

Electromotive force Wbk 11 21
Electron Wbk 47 9
Electronic circuits Wbk 3 69
Electronic component failure Wbk 42 19
Electronic component lifetime Wbk 46 6
Electronic filters Wbk 12 2, 60
Electronic monitoring components Wbk 42 5, 6
Electrostatic charge Wbk 13 11
Electrostatic potential Wbk 11 16
Electrostatics Wbk 9 39-44
Electrostatics Wbk 47 13
Energy Wbk 14 13
Energy Wbk 34 10, 28
Energy Wbk 47 18
Engine power Wbk 41 22
Equipotential curves Wbk 26 14
Error in power to a load resistance Wbk 18 38
Estimating the mass of a pipe Wbk 3 27
Exponential decay of sound intensity Wbk 6 46
Extension of spring Wbk 43 21
Feedback applied to an amplifier Wbk 10 26

Feedback convolution Wbk 21 75
Field due to point charges Wbk 9 40
Field strength around a charged line Wbk 29 67
Field strength on a cylinder Wbk 29 68
Flight overbooking Wbk 39 29
Fluid flow Wbk 26 36-37
Fluid flow Wbk 48 80, 86, 91
Fluid power transmission Wbk 12 31
Fluid theory Wbk 47 14, 20
Force on a loop from an electric field Wbk 29 27
Fraunhofer diffraction Wbk 47 56, 60
Fuel injection system efficiency Wbk 45 18
Fuel injection systems Wbk 44 10
Fun ride - rollercoaster Wbk 34 44
Fun ride - ’Rotor’ Wbk 34 46
Fun ride - ’Yankee Flyer’ Wbk 34 47
Gain Wbk 10 26
Gauss’ law Wbk 29 63, 65
Harmonic oscillator Wbk 23 69

Heat conduction Wbk 48 76
HELM (2015): 69
Heat conduction equation Wbk 25 8, 14
Heat conduction equation Wbk 32 48
Heat conduction through a furnace wall Wbk 25 32
Heat flow in an insulated metal plate Wbk 1 85
Height of building Wbk 18 34
Helmholtz’s equation Wbk 25 18
High frequency line equation Wbk 25 16
Hooke’s law Wbk 43 21
Hooke’s law Wbk 47 6
Horizon distance Wbk 4 8
Hydraulic brakes Wbk 12 31
Hypertension and noise Wbk 43 8
Ideal gas equation Wbk 47 18
Ideal gas law Wbk 18 13, 18
Ideal gas law and Redlich-Kwong equation Wbk 18 18
Impedance Wbk 12 60-63
Instant coffee production Wbk 46 30, 32, 35
Insulating blocks Wbk 45 4, 14, 29
Interference field Wbk 47 51
Interference fringes Wbk 47 64
Interference fringes Wbk 4 31
Inverse square law decay of electromagnetic power Wbk 6 50
Joukowski transformation Wbk 26 19
Kepler’s laws Wbk 47 12

Kinetic energy Wbk 1 79
Kinetic energy Wbk 6 32
Kinetic energy Wbk 34 10, 28
Kirchhoff’s law Wbk 3 10
Kirchhoff’s law Wbk 8 28-30
Kirchhoff’s law Wbk 20 49
Ladder network Wbk 21 60

Laplace’s equation Wbk 25 7, 17, 25, 36
Laplace’s equation Wbk 26 11
Laplace’s equation Wbk 33 19, 27, 30, 34
Lifetime Wbk 38 11, 13, 24, 25
Lift Wbk 47 15
Light bulb lifetime Wbk 46 5
Light bulbs Wbk 38 11
Light ray propagation Wbk 47 53
Light rays Wbk 12 29
Light rays Wbk 17 16
Light waves Wbk 4 31
70 HELM (2015):
®
Lightning strike Wbk 29 46

Lorentz force Wbk 29 27
Mach number Wbk 47 16

Magnetic field Wbk 11 21
Magnetic field Wbk 28 28, 43, 45
Magnetic field from a current line Wbk 29 29
Magnetic flux Wbk 13 51
Magnetic flux Wbk 29 43
Magnets Wbk 39 28
Manufacturing components Wbk 35 48
Masses on spring Wbk 20 47
Maximum height of projectile Wbk 34 12
Maximum range of projectile Wbk 34 14
Measuring the height of a building Wbk 18 34
Metal bar temperature Wbk 32 53-57, 60-64
Microphones Wbk 48 10
Mixture - pressure in Wbk 31 79-81
Modelling vibrating systems Wbk 23 68
Models - beetles Wbk 5 18
Models - carton Wbk 5 33
Models - falling rock Wbk 5 6-10, 26-30
Models - ferry Wbk 5 18
Models - profit Wbk 5 18, 30
Models - rain Wbk 5 12
Models - rain level Wbk 5 15
Models - road level Wbk 5 14
Models - rocket Wbk 5 10
Models - satellite Wbk 5 10
Models - snowfall Wbk 5 17
Models - sound Wbk 5 46
Models - supply and demand Wbk 5 21
Models - tide level Wbk 5 39-44
Modulation Wbk 4 40
Network Wbk 1 52
Network Wbk 7 4, 25-28
Newton’s law of cooling Wbk 32 3
Newton’s laws of motion Wbk 47 13
Newton’s second law Wbk 9 13
Noise Wbk 43 8
Noise barriers Wbk 4 6
Noise reduction by sound barriers Wbk 4 6
HELM (2015): 71
Ohm’s law Wbk 3 25
Ohm’s law Wbk 29 47
Optical interference fringes due to glass plate Wbk 4 31
Orbit Wbk 17 22
Orifice plate flow meter Wbk 47 20
Output signal Wbk 20 64
Paint weathering Wbk 44 29

Parabolic mirror Wbk 47 27, 28, 29
Parachute Wbk 6 48
Parachute Wbk 34 58
Parallel design of components Wbk 46 7-9
Pareto charts Wbk 46 35
Pendulum Wbk 47 7, 10-11
Pendulum Wbk 48 50, 51
Pipe Wbk 47 20
Pipe mass Wbk 3 27
Piston ring diameter Wbk 39 17
Planetary motion Wbk 47 12
Plastic bottle design Wbk 31 52
Plastic tube tensile strength Wbk 44 47
Point - scratch resistance Wbk 35 20
Point - shock resistance Wbk 35 20
Poisson’s equation Wbk 25 18
Poisson’s equation Wbk 33 19, 28, 31
Population dynamic models Wbk 32 8-11
Pressure Wbk 9 4
Pressure Wbk 27 3, 15, 36
Pressure Wbk 47 16
Pressure in an ideal multicomponent mixture Wbk 31 79
Pressure of gas Wbk 18 13
Production line data Wbk 46 27, 30, 32, 35, 36
Projectile Wbk 4 47
Projectile - angled launch Wbk 34 12
Projectile - energy Wbk 34 10, 28
Projectile - height Wbk 34 12
Projectile - horizontal launch Wbk 34 9
Projectile - inclined plane Wbk 34 30
Projectile - range Wbk 34 14
Projectile - without air drag Wbk 34 9
Propagation time difference Wbk 47 53
Propellant Wbk 45 6, 7, 16
Pulley belt tension Wbk 14 8
Pumping engine bearing lifetime Wbk 46 12-13
72 HELM (2015):
®
Quadratic resistance Wbk 34 57, 59, 62

Quality control Wbk 37 8
Quality control Wbk 46 21-38
Radiation Wbk 47 38, 41, 43, 46

Radiation emitted by microwave oven Wbk 42 9
Radioactive decay Wbk 27 58
Railway signals location Wbk 48 72
Range of projectile Wbk 34 12
Redlich-Kwong equation Wbk 18 18
Refraction Wbk 12 29
Refraction Wbk 48 13
Relays Wbk 41 10
Reliability in a communication network Wbk 1 52
Reservoir Wbk 27 42, 54
Resistance - linear Wbk 34 56
Resistance - quadratic Wbk 34 57, 59, 62
Resisted motion Wbk 34 56-63
Reverberation Wbk 6 46
Roadholding of car Wbk 44 31
Rocket Wbk 8 31
Rocket fuel shear strength Wbk 45 6, 7, 16
Rollercoaster ride Wbk 34 44
Roundabout Wbk 34 36
Route network Wbk 7 27
Sampling Wbk 21 3, 85-95

Satellite motion Wbk 48 60, 63
Schrödinger’s equation Wbk 25 18
Series design of components Wbk 46 7, 9
Shear force and bending moment of a beam Wbk 19 65
Shear strength Wbk 43 18
Shear stress and strain Wbk 13 19
Shot putting Wbk 34 22
Signal sampling Wbk 21 85-95
Simple harmonic motion Wbk 4 68
Simple harmonic motion Wbk 25 6
Skateboarding Wbk 34 31
Skiing Wbk 34 15
Snowflake falling Wbk 48 56
Solenoid Wbk 13 51
Solid rocket fuel Wbk 45 6, 7, 16
Sonic boom Wbk 14 12
Sound Wbk 48 2, 7, 10
Sound intensity Wbk 5 46
Sound intensity Wbk 6 46
HELM (2015): 73
Sound waves Wbk 4 6
Spot welds Wbk 43 18
Spring Wbk 43 21
Spring Wbk 47 6
Spring - damped Wbk 20 39
Springs Wbk 20 47
Steel alloy corrosion Wbk 44 21
Steel bar Wkb 13 19
Steel cables Wbk 41 25, 29, 37
Stiffness Wbk 13 18
Strain Wbk 8 10
Strain Wbk 13 19
Strain gauge resistance Wbk 39 18-20
Streamlines Wbk 26 14
Stress Wbk 8 10
Stress Wbk 13 19
Stresses and strains on a section of material Wbk 8 10
String Wbk 47 5, 7
Submarine equation Wbk 25 16
Surface tension Wbk 47 14, 17
Suspended cable Wbk 15 21
Suspended cable Wbk 48 40
Switches Wbk 41 10
System reliability Wbk 46 7-9
System response Wbk 20 71
Tank - ellipsoidal Wbk 27 79

Tank - elliptic Wbk 27 37
Telegraph equation Wbk 25 16
Temperature of wire Wbk 33 10
Tensile strength Wbk 41 21
Tensile strength Wbk 44 45, 47
Tension Wbk 14 8
Tension in spring Wbk 47 6
Tension in string Wbk 47 7
Terminal velocity Wbk 6 49
The current continuity equation Wbk 29 46
The web-flange Wbk 17 13
Thermal diffusivity Wbk 32 46
Thermal insulation Wbk 1 85
Tiddly-winks Wbk 34 19
Tolerance limits Wbk 46 24
Torque Wbk 9 52
Torque Wbk 13 19
Torque Wbk 28 6
74 HELM (2015):
®
Torsion Wbk 13 19
Torsion Wbk 17 13
Torsion of mild-steel bar Wbk 13 19
Total energy Wbk 34 28
Traffic flow Wbk 37 11, 46
Train on a track Wbk 30 62
Transmission line equation Wbk 25 16
Transverse vibrations equation Wbk 25 18
Turbochargers Wbk 41 17
Turbulence Wbk 47 16
Two dimensional fluid flow Wbk 26 36
Tyre mileage Wbk 38 13
Undersea cable fault location Wbk 3 25

van der Waals’ equation Wbk 47 18, 19, 23
Velocity of a rocket Wbk 8 31
Velocity on a bend Wbk 34 51
Vibrating system Wbk 20 47
Vibration Wbk 23 69
Vibration of string Wbk 47 5
Vintage car brake pedal mechanism Wbk 4 14
Viscosity Wbk 47 14, 15, 16
Volume of liquid in an ellipsoidal tank Wbk 27 79
Volume of liquid in an elliptic tank Wbk 27 37
Washing machine faults Wbk 42 20

Water flow Wbk 47 20
Water height in an open channel Wbk 48 45
Water wheel efficiency Wbk 12 28
Waterflow Wbk 28 12, 13, 25, 30
Wave equation Wbk 32 70
Waves Wbk 4 40-42
Waves Wbk 47 17
Waves Wbk 48 2, 7, 10
Wear on rollers Wbk 40 11
Weathering of paint Wbk 44 29
Woodscrew size variation Wbk 40 6
Work done moving a charge in an electric field Wbk 29 19
Young’s modulus Wbk 8 10

Young’s modulus Wbk 20 52
HELM (2015): 75
NOTES

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About

the HELM Project

HELM Workbooks List

50.2 HELM Consortium, Triallist Institutions and Individual Contributors 4

50.3 HELM Transferability Project (HELMet) 7

50.4 HELM Workbook Structure and Notation 10

50.5 Issues and Notes for Tutors 13

50.6 Workbook Layout 18

50.7 List of Sections in Workbooks 1 to 48 22

50.8 Commentaries on Workbooks 1 to 48 29

50.9 Index of Engineering Contexts in Workbooks 1 to 48 66

Introduction to HELM 50.1 

1. The HELM project

• A Workbook containing an introduction to dimensional analysis, supplementary mathematical

• A Tutor’s Guide (this document)

3. HELM project Interactive Learning Segments

4. HELM project Assessment Regime

Hull, Loughborough, Manchester, Reading, Sunderland.

Universites Other HE/FE Institutions

Individual Consortium Contributors

Felix Ampofof Brian Lowe

Project (HELMet) 50.3 

2.1 Workbooks as lecture notes (hard copy format)

2.2 Workbooks as supplementary linked material for whole modules

2.3 Workbooks as supplementary linked material for parts of modules

2.4 Workbooks as unlinked supplementary material

Structure and Notation 50.4

No. Title Pages

No. Title Pages

2. Nature of the Workbooks

• 1 Workbook (47) containing Sections on Dimensional analysis, Mathematical explorations and

ln for natural logarithm

E for Expectation P for Probability V for Variance M for Median

HELM uses underlining of vectors rather than using bold e.g. a

Issues and Notes  

for Tutors  50.5 

1. The use of units in applied problems

Purists may prefer (b) of these options.

Now consider the following problem.

Here is the ‘correct’ solution.

Let the deceleration be a km h−2 .

In 2 km the speed of the train will have decreased by 80 km h−1 .

We use the formula v 2 = u2 − 2as, where u km h−1 is the initial speed,

v km h−1 is the final speed and s km is the distance travelled.

Let the deceleration be a.

In 2 km the speed of the train will have decreased by 80 km h−1 .

3. HELM Workbooks and Students with Special Learning

You will be able to amass your own collection of such examples . . .

An Engineering Student’s Errors in Algebra

Below is the work of a first year undergraduate Manufacturing Engineering student.

1. Study the student’s work.

2. Note down the different kinds of mistakes made.

3. How would you help this student?

An Engineering Student’s Solution to a Calculus Problem

Below is the work of a first year undergraduate Electrical Engineering student.

1. Study the student’s work.

2. How many errors can you see?

3. How would you classify each of them?

4. How could you help this student?

Introduction to HELM 50.1

Project (HELMet) 50.3

Issues and Notes

for Tutors 50.5

Workbook layout 50.6