Monday, 14 November 2011

M208 Review Part II

This is the second part of my review of M208 covering both linear Algebra and Analysis

Linear Algebra covers 1 block of 5 units. The first part should be fairly familiar to those who have done MST221 as it covers some coordinate geometry a bit of vector algebra mostly concerned with the vector equation of a straight line and some properties of conics (ellipses, hyperbolae and parabola's) Then there is a bit on linear equations and their solution using matrix techniques and Gaussian elimination this extends work in MS221 to three dimensions. A slightly tedious and tricky aspect is row reduction but if care is taken it is quite straightforward. There is also an introduction to the calculatrion of eigenvalues and eigenvectors of a matrix.

Then there is a brief introduction to what I feel is one of the most fascinating branches of mathematics the extension of Vector techniques to general spaces including functions. This fascinating area leads eventually to the mathematical structure of quantum mechanics (But none of this is mentioned in the unit alas). I have remarked in an early post how linear algebra can be extended to functions with the notion of a scalar product and othogonality being generalised to functions that can be defined as orthogonal by an integral over a suitable range see my earlier post for further details

As I've said before in the early days of the OU there was a whole course devoted to linear analysis M203 which developed the analogy covering both the pure aspects and it's applications to the classical differential equations of classical physics namely the heat, diffusion and wave equation. (And there was no pretending that a second level course is really a third level course as is currently done with a lot of the current third level courses). Anyway for those who want more on this subject as I've recommended before this is the set book on which the course was based is called an Introduction to linear analysis

and has the right combination of pure and Applied maths really worth investing in as a prelude to Functional Analysis. Finally as I've also mentioned those who want to understand the mathematical structure of quantum mechanics Linear Algebra and Analysis is a necessary prerequisite. Essentially the mathematical structure of quantum mechanics is that of a linear vector space over the field of complex numbers. I believe the second part of the OU quantum mechanics covers this. Although obviously it does not cove Functional analysis. So a good block just slightly disappointed that it didn't take things further. I'll have to wait until the MSc to pursue my interest or may be I'll stick with Krieder for now.

Finally we come to the best part of M208 namely Analysis. A lot of people found this difficult and it probably is the hardest part of M208. Analysis is the attempt to make calculus rigorous and it can be truly said to divide Pure mathematicians from Applied mathematicians, Physicists and Engineers. I have spent most of my career ignoring this whilst developing my caclulus skills. Part of the problem is that 30 years ago the books available to explain analysis were a very concise, just presented the theorems with out much motivation and it seemed really difficult to see what was going on. Fortunately times have changed the best introductory book on analysis for those wanting to do M208 is by Brannan and I would strongly recommend those contemplating doing M208 this year to invest in it as quickly as possible

It contains all the analysis covered in M208 but has the convenience of not being split up into units and there are no irritating breaks for video programmes or CD's.

Ok so what does analysis cover it starts off with the real number system and care is taken to define upper and lower bounds of real numbers. It is possible to construct the real numbers from the rational numbers by a procedure due to Dedekind but this is not covered in M208 partly because introduced right at the begining of a course it would be even more off putting than it is already.

Then a study is made of convergence of sequences and series culminating in the first definition of continuity based on convergence of series. It is quite straightforward to master the techniques for testing whether or not a  series convergence or not but remembering the details of each proof is quite tricky but these are never examined. One does have to be careful to introduce all the definitions carefully and examine the conditions of each part but as there are structured answers one (eventually) gets used to what is needed.

In the second part another definition of continuity is introduced the notorious epsilon delta definition of continuity this puts analysis on a really rigorous basis. Fortunately in M208 and Brannan this is introduced gently and with plenty of examples. These culminate in the testing for continuity of pathological functions such as the Blancmange Function which is a function which is everywhere continuous and nowhere differentiable. This proof is quite tricky but well worth reading through once or twice. (You don't have to understand it to be able to answer either the TMA or exam questions but if you understand it you will really have achieved something. I can't claim to understand it fully yet but I do hope to look at it again until I do)

After that then the climax comes when it is explained how to differentiate and integrate rigourously, There is not much extension of techniques that have already been learnt. The focus is on understanding how integration and differentiation can be defined rigourously not on developing techniques of integration.
the final unit develops more on Taylor series which some people might find a bit dry but remember this technique is used in numerical analysis so worth getting to know if you are interested in mathematical modelling.

So Analysis is probably the most challenging part of M208 which is why I recommed anyone thinking of doing M208 to get Brannan as quickly as possible, you will not unless you are really dedicated have time to do justice to the analysis units in 8 weeks and exam revision. Investing in Brannan will buy you time so you can get used to the abstract nature of analysis, the first step to maturity as a Pure mathematician and I only wish I had studied analysis earlier.

In conclusion then M208 is a curates egg of a course, the pressure of TMA deadlines means that a lot of the formal stuff will be bypassed. This could of course be remedied by the OU making the TMA's at least focus more on the conceptual aspects of the subject. Like everything you get as much as you put into it. Technically it is possible to do well by repeating almost by rote the answers given at the back of the books and by annotating the handbook with set answers. If's that what you want fine, but whether you deserve to be called a mathematician by following that course is another matter.

1 comment:

  1. Nice review Chris. I am enjoying M208 at the moment and my confidence is beginning to improve. Can't wait for the course to start proper.