Unit lecturer: Dr D A S Rees Department of Mechanical Engineering. Room 4E 2.54.

Telephone: (01225) 386775 (Office)
E-mail Address: or

QUICK LINKS     (Last updated 10/02/2022 22:10)
Notes, videos, problem sheets:
   Course notes You should already have a copy of these, but here are the originals should they be needed.
   My videos Links to my lecture videos (and the associated slides) will appear here gradually as the semester progresses.
   Problem sheets and solutions These will be unveiled gradually as the semester progresses.
   Other videos Some other videos which I have curated from youtube.
Ancilliary information:
   Syllabus Increasingly detailed lists of what we will be doing.
   Resources Some links to external support.
   Greek letters and miscellaneous info.
   Handouts Some extra information that may be useful.
   Textbooks Some advice on textbooks.
   Past exam papers These cover the last five years. Includes outline solutions and general feedback.
   University calculator These are used when exams take place in the normal manner.


Ordinary Differential Equations (5 lectures)
   Video (42.39) Slides 1: Classification, reduction to first order form. (29/01/2021)
   Video (41.40) Slides 2: Separation of Variables, First Order linear. (29/01/2021)
   Video (42.52) Slides 3: Homogeneous linear constant-coefficient ODEs (05/02/2021)
   Video (30.29) Slides 4: Inhomogeneous linear constant-coefficient ODEs I (05/02/2021)
   Video (20.02) Slides 5: Inhomogeneous linear constant-coefficient ODEs II (05/02/2021)
Matrices (5 lectures)
   Video (41.47) Slides 1: Matrices: definition, multiplication and the zoo (12/02/2021)
   Video1 (20.54)   Video2 (40.23) Slides 2: Matrices: Determinants and Cramer's rule (19/02/2021)
   Video1 (24.54)   Video2 (6.57) Slides 3: Matrices: Gaussian Elimination (19/02/2021)
   Video (47.38) Slides 4: Eigenvalues and eigenvectors 1 (26/02/2021)
   Video 2 (25.29) Slides 5: Eigenvalues and eigenvectors 2 (26/02/2021)
   Video 2b (9.39) Slide 5b: Eigenvalues and eigenvectors 2b Extra example. (26/02/2021)
Laplace Transforms (4 lectures)
   Video (39.25) Slides 1: Introduction, LTs of some functions, some ODE solutions. (02/03/2021)
   Video (43:59) Slides 2: Unit impulse, integrals, LT. ODE solutions with impulsive forcing. (03/03/2021)
   Video (33:41) Slides 3: Unit step function, s-shift and t-shift theorems. (10/03/2021)
   Video (25:00) Slides 4: Convolution: definition and theorem. Solution of systems of ODEs. (11/03/2021)
Numerical Maths (iteration schemes) (2 lectures)
   Video (39:00) Slides 1: Ad hoc iteration schemes for finding roots. (15/03/2021)
   Video (32:56) Slides 2: The Newton-Raphson scheme. (15/03/2021)
Least Squares (1 lecture) Joke
   Video (50:00) Slides: Lots of Least Squares stuff.
Fourier Series (3 lectures but 1 video)
   Video (50:01) Slides: Everything you need and more.


Matrices. A 25 minute set of examples of the some of the uses of matrices.

The problem sheets linked below use an older typesetting. It'll be better to use those at the back of the printed notes.

Howlers from the 2019/2020 S1 exams: Sheet 0 Answers 0 [Available]

ODEs: ODEs Sheet 1 [Available] Solutions [Not Available] on the separation of variables and first order linear equations.
ODEs: ODEs Sheet 2 [Available] Solutions [Not Available] on the solution of linear constant-coefficient equations.
ODEs: ODEs Sheet 2b [Available] Solutions [Not available] (supplementary) on random, miscellaneous and difficult but hopefully interesting ODEs.

Fourier Series: Sheet [Available] Solutions [Not available] on Fourier series and their role in solving ODEs with periodic forcing.

Laplace Transforms: Introductory Sheet [Available] Yes, do have a go at this one before we start the lectures on the topic!
Laplace Transforms: Sheet 1 [Available] Solutions [Not Available] on basic transforms and their application for solving ODEs, and on the unit impulse.
Laplace Transforms: Sheet 2 [Available] Solutions [Not Available] on the two shift theorems and the convolution theorem.

Matrices: Sheet 1 [Available] Solutions [Not Available] on products of matrices.
Matrices: Sheet 2 [Available] Solutions [Not Available] on determinants, Cramer's rule and Gaussian Elimination.
Matrices: Sheet 3 [Available] Solutions [Not Available] on eigenvalues, eigenvectors and the solution of systems of ODEs.

Least Squares: Sheet [Available] Solutions [Not Available] Ounces/grams data for Q2

Numerical Mathematics: Sheet [Available] Solutions [Not Available]


The syllabus for this semester includes

1. What am I doing and when? Answer.
2. A more detailed account of the order in which subtopics appear.
3. The official detailed syllabus is here, but it is not accurate due to curriculum creep.

This unit is assessed 100% by examination only. There is no coursework element.


Main University webpage for students.
Mech Eng info for students (Moodle) (You'll need to type your userid and password).
There is a Maths and Stats Resource Centre at the University of Bath. Called MASH, it operates drop-in sessions at various times.
In addition, there are further online resources at the Mathcentre website.


Greek letters. Don't get your etas, zetas and xis mixed up. Nor your phis and psis. And meet your first discontinuous letter.
If you want hints on pronunciation depending on where you are in the world, then this is the place.
SI prefixes


Note on the solution of homogeneous systems of ODEs.
Handout on the speed of convergence of Fourier Series.


Essentially anything with the words "Engineering Mathematics" in the title is likely to be sound. So Glyn James, Stroud, Kreyszig, Croft and Davidson, Kuldeep Singh are all excellent.

Beware of the word "Advanced", as in "Advanced Engineering Mathematics" because some of these are seriously advanced. So check the contents pages. In fact, in book titles, the word "Elementary" doesn't always accord with the standard dictionary definition, and occasionally "An Introduction to..." requires you to have gained a PhD prior to even contemplating the merest possibility of looking inside the cover. So always check the index. You've been warned.

HOWEVER, my advice is to check out some of these books in the Library prior to purchasing anything. Some people get on with Stroud but hate Croft, for example, whereas others feel precisely the opposite. Always check prices on


There is a scanned version of the exam formula book here. (This is the latest 2019/2020 version).

PLEASE NOTE THE FOLLOWING. Due to syllabus changes over the years Fourier Series was introduced in 18/19.

17/18 paper   Outline solutions   Formal/Informal feedback document
                      [Note: Ignore the question on numerical integration.] [Note: Least Squares should be y=1.06-0.362x] [Note: There is no Fourier Series question.]
18/19 paper   Outline solutions   Formal/Informal feedback document
19/20 paper   Outline solutions (full)   Outline solutions (short)   Formal/Informal feedback document
20/21 paper   Outline solutions   Informal feedback document   Formal feedback document
21/22 paper   Outline solutions   Informal feedback document


The University will be supplying calculators for the Mathematics exams.
Currently (i.e. September 2018) the designated species is the Casio FX-85ES.
Further information on functionality may be found here.