How does one make sure one's mathematical skills are up to date and relevant? One way would be to aim to do past papers from relevant mathematics courses once a year as they come out. Such a set of papers and resources for study of these papers is provided by the Cambridge University mathematics department. This post collects together a few resources for self study of the Cambridge Mathematical Methods courses for Scientists (NST). I am continually amazed at the generosity of the Cambridge Maths site in offering to the public these resources
The past papers are collated here.
https://www.maths.cam.ac.uk/undergradnst/pastpapers
The syllabus for both years is given here
https://www.maths.cam.ac.uk/undergradnst/files/misc/NSTschedules.pdf
There are two courses one for first year students and one for second year science students who want to study mathematics in more depth.
The recommended text book is Riley Benson and Hobson
https://www.amazon.co.uk/Mathematical-Methods-Physics-Engineering-Comprehensive/dp/0521679710
Which has a solution manual which covers the odd problems in great detail so ideal for self study
Investing in these two books would cover most mathematical methods that a physicist is likely to need in their careers. However it would probably take a good few years to do all the examples in Riley Benson and Hobson so it is more for reference than anything.
Buried in the vaults of the Cambridge Mathematics department are the following resources
Answers to first year past papers can be found from this website
https://www.robinson.cam.ac.uk/iar1/teaching/index.html#nst1a_maths
Unfortunately I have not been able to find any lecture notes or the example sheets. However if you have Riley Bence and Hobson you should be able to find the techniques that you don't know already covered there. However having the answers to the exam questions is obviously helpful
Year II is much better served the lectures occur in three parts
The first part is covered by a set of lectures from Dr Simon Cowley who from his web site seems quite an engaging person
http://www.damtp.cam.ac.uk//user/sjc1/index.html
Lecture notes for the various courses he has taught over the years can be downloaded from here
http://www.damtp.cam.ac.uk//user/sjc1/teaching/
The ones of main interest are the notes for the first part of the Cambridge NST Part IB
http://www.damtp.cam.ac.uk//user/sjc1/teaching/
This covers things such as vector calculus, matrices partial differential equations Fourier Transforms and solution of differential equations, by series including a survey of Legendre Polynomials, Green's Function and a bit on real analysis (for physicists not mathematicians 😅). This is more or less equivalent to MS224 and the mathematical methods part of MST326 but also includes Green's Functions and goes into more detail about the solution of differential equations by Series solution especially the Frobenius Method.
Notes for the second lecture course in Part IB are given by Dr Hunt
https://www.damtp.cam.ac.uk/user/reh10/lectures/
Scroll down to the Mathematical Methods part
He also gives the examples sheets that were part of the course and hints for their solution
This second part covers variational principles, solutions to Poisson's equation and a summary of the basic techqniques of Complex Analysis again taught as Physicists would use them rather than a Pure Mathematician. There is a bit of overlap with the OU course MS327 but neither MST326 or MS327 cover inhomogeneous partial differential equations in any depth.
Finally the third part covers group theory and representation theory for physicists including an analysis of multi-mode oscillations. MS327 covers multi-mode oscillations but do not relate it to group theory.
https://www.damtp.cam.ac.uk/user/examples/N23L.pdf
The examples sheets for the whole year are also given scroll down to the bottom
Ok I aim to have generated a complete set of solutions to the 2019 papers by the end of the year. This is a nice way IMHO to extend ones mathematical ability and keep it topped up. Overall the topics covered here in terms of mathematical knowledge overlap with MS224, MS327 and MST326 but also extend it to cover topics such as
1) Solution by Series of Differential equations including the Frobenius Method
2) Green's Functions and their application to Partial Differential equations
3) Complex Analysis (although to be fair if you do M337 you will do far more than is given here)
4) More detail on Special Functions including Legendre Polynomials
5) Group Theory
Also I suspect the questions will be harder than the OU exam questions
Anyway I hope this post will encourage you to extend and revise your mathematics skills.
