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 6/11/2022)

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 from last year (and the associated slides). These will appear here gradually as the semester progresses..
   Problem sheets and solutions. The problems sheets are at the back of the Course Notes document. The solutions will be unveiled gradually as the semester progresses.

Ancilliary information:
   Syllabus Increasingly detailed lists of what we will be doing.
   Resources Some links to external support.
   Miscellaneous Mathematical links, some important, some useful, some frivolous.
   Handouts Some extra information that may be useful.
   Other videos Links to a few external videos that may be useful. Suggestions for others gratefully received.
   Textbooks Some advice on textbooks.

   Information about exam preparation and this must be read carefully. [This is up to date: 06/11/2022]
   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.


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 things have evolved a little since that document was drafted. It needs a bit of an upgrade and that will be happening for next year, 23/24.

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 and strange units of measurement.

Table of integrals.

History of the trigonometric functions.

Information on the hyperbola including where the name came from.

Why I am smug, what I don't expect in class, and I'm not sure that I believe this.

If it ever seems like this in a lecture, then you need to give me a good kicking!

Maths teaching in the past. Dodgy examples of exam/coursework submissions.

L'Hôpital's rule - the musical. (This isn't NSFW but....turn down the volume first!)

This is why your maths must be correct.


Electronic copies of all the problem sheets and their solutions will eventually appear here. You'll be emailed whenever anything gets added. You are not expected to hand in your completed work.

Curve sketching: Sheet 1 (2 pages) [Available]   Solutions (Q1-7) [Available]   Solutions (Q8,9) [Available]
        Extra: Sheet 1b (1 page) [Available]   Solutions [Available]
Complex numbers: Sheet 2 (2 pages) [Available]   Solutions   [Available]
Differentiation: Sheet 3 (2 pages) [Available]   Solutions   [Available]
Differentiation: Sheet 4 (1 page) [Available]   Solutions   [Available]
Integration: Sheet 5 (1 page)   [Available]   Solutions   [Available]
Integration: Sheet 6 (2 pages)   [Available]   Solutions [Available]  
Integration: Sheet 7 (3 pages)   [Available]   Solutions   [Available]
Series: Sheet 8 (1 page)   [Available]   Solutions   [Available]
Series: Sheet 9 (2 pages)   [Available]   Solutions   [Available]
Vectors: Sheet 10 (2 pages)   [Available]   Solutions   [Available]
Vectors: Sheet 11 (2 pages)   [Available]   Solutions   [Available]
Exam Question 10: Sheet 12 (2 pages)   [Available]   Solutions (Q1-4) The solution to Q5 is to come....   [Available]
Sheet 12 is designed to give an indication of what sort of question Q10 will be on the exam paper. It mixes up elements from different sections of the unit.


These are meant to assist in your understanding of the various topics in the syllabus. If there is anything you didn't understand in the lectures, look it up here, and you won't be hindered by the remnants of a Welsh accent. Alternatively, you may wish to check out what I am about to do in the next lecture. Most of the examples presented in these printed notes are different from the ones given in the lectures so these may also be of some use.

The full set of notes is here.

But if you wish to go to the first page of the different sections, then these are the links:
0. Introductory material
1. Curve sketching
2. Complex numbers
3. Differentiation
4. Integration
5. Series
6. Vectors
Problem Sheets


Extra examples for curve sketching.
Checklist for hyperbolic functions.
Proof of the H=hxxhyy-(hxy)2 test (for interest only - not examinable).


Introduction (1 video).
   Video (6.45)
Curves: (4 videos).
   Video (27.43) Slides 1: Polynomials, moduli, exponentials, hyperbolics.
   Video (36.49) Slides 2: Symmetries, envelopes, square roots and ratios of polynomials.
   Video (4.12) Scan: Extra ratios of polynomials case.
   Video (11:21) Scan: Extra examples of square roots of functions.
Complex numbers: (2 videos).
   Video (42.15) Slides 1: Definition, arithmetic, complex exponentials, de Moivre.
   Video (23.21) Slides 2: Roots of complex numbers. Relation with hyperbolic functions.
Differentiation: (5 videos).
   Video (14.22) Slides 1: Introduction, limit definition, notation.
   Video (39.19) Slides 2: Product, chain and quotient rules.
   Video (19.04) Slides 3: Critical points
   Video (27.24) Slides 4: Intro to partial derivatives
   Video (40.49) Slides 5: Surfaces, critical points and classification
Integration: (6 videos).
   Video (19.00) Slides 1: Introductory bits and pieces.
   Video (32.53) Slides 2: Integration by substitution.
   Video (27.39) Slides 3: Integration by partial fractions.
   Video (36.07) Slides 4: Integration by parts.
   Video (43.34) Slides 5: Applications I.
   Video (27.30) Slides 6: Applications II.
Series: (4 videos).
   Video (64.14) Slides 1: Binomial series
   Video (27.38) Slides 2: Taylor's series.
   Video (38.52) Slides 3: d'Alembert's convergence test
   Video (26.17) Slides 4: l'Hôpital's rule
Vectors: (4 videos).
   Video (35.28) Slides 1: Introduction and scalar product
   Video (30.57) Slides 2: Vector product
   Video (34.07) Slides 3: Areas, points and lines.
   Video (27.13) Slides 4: Points and planes. Some generalities.  


Solving a cubic which often needs complex numbers. Tartaglia, del Ferro, Cardano, Ferrari et al.(36 minutes)
Multiplication and division of complex numbers. (7:01)
Roots of complex numbers despite the title of the video! (16:05)
Euler's identity as expounded by a viola-playing friend of mine. (5:19)
Introduction to partial differentiation (18:21)
Taylor's series. The initial motivation is a little different from mine. (12:42)
Integration (partial fractions). Not so keen on the last example. (50:05)
Integration (polar coordinates). Integrating the Gaussian, but please please please don't expect me to do it this way! (4:46)
Newton-Raphson. There are many interesting things in this video. Probably it is better to watch this after knowing what the method is. (26:05)


Past papers may be obtained from the library webpages although the following links are the originals, rather than library scans.

paper , outline solutions, feedback .
2019/20 paper , outline solutions, feedback .
2020/21 paper , outline solutions, feedback .
2021/22 paper , outline solutions, feedback .
2022/23 paper , outline solutions,

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


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.


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 first. 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

You will generally find that textbooks have little to say about curve-sketching.

Last updated: 6th November 2022.