## Sunday, 29 January 2012

### M338 and MS324 progress so far

Just a short post here, I've more or less finished the first TMA for M338. The first question seems apparently straightforward. A skectch of a simple function, specifying the subsets over a specified interval, and then finding the image set of a function which isn't quite an inverse as the function is many valued. Whilst straightforward it's probably a warm up getting us used to the specification of continuous functions in terms of the behavior of the  inverse of open sets of a function. Sort of backwards but seems fundamental in topology.

The second question is a standard proof that  a composite function involving trigonometric functions is continuous.

Then another question involving proving that a function is continuous on the real line using the epsilon delta definition of continuity. This is then combined with the first new stuff namely the extension of continuity to Euclidean space namely a vector space of N dimensions. We live in three dimensional space but it is of course possible to conceive of N dimensional space. The definitions of continuity in N dimensional space involve the Pythagorean distance between two points instead of the modulus function. Anyway much of the same ideas to continuity apply to N dimensional space with this distance function. The questions seem relatively straightforward but as with all pure maths it's important to learn the phraseology and make sure all aspects are specified. It is tricky to get this correct and I find it takes two or three goes to get it right. Sometimes it feels more like writing an essay rather than doing maths anyway it's all part of getting in to how Pure mathematicians think.

So it's a relief to get back to 'real' maths with MS324. So far the website hasn't opened but I've been working my way through unit 0.   This is a revision unit to get one up to speed the first part covers Integration and differential equations. The second part concerns multi valued calculus and a bit of vector calculus leading up to the divergence theorem. So far so good, I've not found the problems too difficult and I think I stand a good chance of 'winging' this one namely just concentrating on the TMA's and past exam papers as a light relief to all the heavy stuff on M338. There are end of section questions and I will see if I can do those first before tackling the main units. I can't wait till I get the TMA's which should be available once the website opens tomorrow.