MATHEMATICS 2

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

Telephone: (01225) 386775 (Office)
E-mail Address: D.A.S.Rees@bath.ac.uk or ensdasr@bath.ac.uk

EXAM NOTES (2022/2023)

   Revision session in UH on Thursday 27th April.    notes      video
   Online session 1. Wednesday May 3rd.    notes      video (Open the notes in Acrobat and you'll need to rotate the image)
   Online session 2. Tuesday May 9th.    notes      video
   Online session 3. Thursday May 18th.    notes      video

My full revision timetable is below (and it includes the timetable for ME20021 Modelling Techniques 2). Please note that, at other times, I may be off-campus or engaged in final year project duties. Of course I am emailable.

You will see that I have some online sessions and some in-person sessions. So do make sure that you are well prepared before seeing me; by this I mean that you need to make sure that all the necessary bits of paper are immediately to hand - those behind you in the queue won't be as patient as I am if you're searching for that bit of paper which you knew for sure was hiding in that folder or was it that bag or back in your room on the table.....

Exam questions will be similar in style to those available online. The following link is to a two-page document on aspects of the Maths 2 exam.

QUICK LINKS     (Last updated 02/05/2023 )
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.
Examinations:
   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.

MY VIDEOS

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.


OTHER VIDEOS

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


PROBLEM SHEETS AND SOLUTIONS
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 2022/2023 S1 exams: Sheet 0 Answers 0 [Available]

ODEs: ODEs Sheet 1 [Available] Solutions on the separation of variables and first order linear equations.
ODEs: ODEs Sheet 2 [Available] Solutions [Available] on the solution of linear constant-coefficient equations.
ODEs: ODEs Sheet 2b [Available] Solutions [Available] (supplementary) on random, miscellaneous and difficult but hopefully interesting ODEs.

Fourier Series: Sheet [Available] Solutions [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 [Available] on basic transforms and their application for solving ODEs, and on the unit impulse.
Laplace Transforms: Sheet 2 [Available] Solutions [Available] on the two shift theorems and the convolution theorem.

Matrices: Sheet 1 [Available] Solutions [Available] on products of matrices.
Matrices: Sheet 2 [Available] Solutions [Available] on determinants, Cramer's rule and Gaussian Elimination.
Matrices: Sheet 3 [Available] Solutions [Available] on eigenvalues, eigenvectors and the solution of systems of ODEs.

Least Squares: Sheet [Available] Solutions [Available] Ounces/grams data for Q2

Numerical Mathematics: Sheet [Available] Solutions [Available]

SYLLABUS

The syllabus for this semester includes

• Ordinary Differential Equations
• Matrices
• Laplace Transforms
• Numerical Mathematics (iteration schemes)
• Fourier Series
• Least Squares Fitting of Data

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.

RESOURCES

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 AND MISCELLANEOUS INFORMATION

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

HANDOUTS

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

TEXTBOOKS.

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 amazon.co.uk.

PAST PAPERS

There is a scanned version of the 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
22/23 paper   Outline solutions   Informal feedback document

UNIVERSITY CALCULATOR

The University will be supplying calculators for the Mathematics exams.
Currently (i.e. April 29th 2023) the designated species is the Casio FX-991EX.
Further information on functionality may be found here.