Sunday, 23 October 2011

Normal service will be resumed in two weeks

Hi I apologise for not been able to post anything of substance over the next two weeks but I'm afraid I'm having to devote a lot of my spare time to work at the minute. The crisis should be resolved in two weeks time so I'll resume normal service. I think this week is this blogs 1st birthday. I would like to thank all who find this blog interesting and for all the new friends I've made. Who says mathematicians/physicists are anti social. It's only that it's difficult to find like minded people I wish more people were interested in maths as  opposed to Soap Operas or Football the world would be a much more interesting place.

Friday, 14 October 2011

General Relativity a reading list

Prompted by a discussion on the Maths Choice Forum I'm afraid I couldn't stop myself so here is a reading list for budding General Relativists. The question was what is a good mathematical approach to General Relativity. One of the contributors seemed a bit sniffy about the first book of the physics course The Relativistic Universe. Ok it's not the most mathematical approach but the basics are there namely introduction to relativity and the main solutions namely the Schwarzschild metric and the Robertson Walker metric. So a good introduction for those wanting more here is a brief survey.  

"Of the books I'm familiar with the most mathematical is Hawking and Elliss The large scale structure of space time but you do need a background in Topology and differential geometry before it becomes even remotely comprehensible. It deals with the singularity theorems that Hawking and Penrose discovered. There is little direct application to astrophysics.
And although there is a chapter on cosmological models you would have to spend a lot of time filling in the gaps

The monster which does justice to both the mathematics and astrophysical applications is Misner Thorne and Wheeler. Although some might find it a bit long winded more for reference than reading through
A reasonable compromise between concrete applications and abstract mathematics is Wald
All of these books are probably a bit dated an uptodate  version but more geared to applications than mathematics is
A book on mathematical physics which includes the background to differential geometry and also functional analysis for those interested in quantum physics is
Finally one of the first books on Geometric Algebra and its application to physics  is
Coauthored by the same Lasenby who wrote General Relativity an Introduction for Physicists.

Insertion added 15th October
Forgot to add these two classics
Chandresekhar on Black Holes

This has the encouraging paragraph at the end of chapter 9

"Every effort has been taken to present the mathematical developments in this chapter in a comprehensible
logical sequence. But the nature of the development simply does not allow a presentation that can be followed in detail, with modest effort:. The reductions that are necessary to go from one step to another are often very elaborate and so on occasion may require as many as ten, twenty or even fifty pages !! (my exclamation marks) "

Doesn't exactly make you want to read it does it.

And Weinberg's First book on General Relativity

All of these books have been sitting on my shelves and I occasionally take a peek sometimes even work through a chapter or two.

I'm in two minds about the usefulness of a purely abstract approach to General Relativity. Undoubtedly the Hawking and Penrose theorems regarding singularities are of great significance but the Topological Approach seems a bit like existence theorems in Pure maths. Yes they have been shown to exist, but what causes them or what are the consequences for that you need the detailed mathematical modelling that Astrophysicists use to construct what happens when stars collapse or work out the implications of a given cosmological model for say elemental abundances. For that the abstract mathematics is an elegant means to an end, be it differential forms or now geometric algebra, namely the components of the Riemann Tensor. Once you have those and solve the Einstein Field equations you end up with differential equations and the standard methods of solving them.
I'm currently trying to understand  the applications of variational methods to the standard solutions to general relativity especially the big bang. This seems a reasonable compromise between physical insight and the abstract methods certainly beats the usual slog of computing 128 Christoffel symbols only to find out most are zero.
I think to get the full picture you need both approaches but whether it is possible for a person to get the full picture in the typical lifetime of an undergraduate or a 1 year MSc course is another question.

Anyway hope to get back to my studies on the Variational method as applied to the Robertson Walker metric sorted by Christmas.

Wednesday, 12 October 2011

M208 and M337 Exams Debriefing

Well what a day I started with M337 I have to say it was a fair exam even though I probably wont get higher than grade 3.
Part 1 has 8 questions the topics were

1) Allegedly simple questions on algebraic manipulation of complex numbers. However the last part was a fiendish raising a number to a complex power. In general for a complex number z
$$z^{\alpha} = exp(\alpha Log z) $$
where Log z = log|z| + iArg z

So it should have been simple to work out but I got bogged down

2) Sketching some sets and their difference and deciding whether or not the sets are regions, compact etc
Straightforward but I froze and for the life of me couldn't after about 2 attempts sketch the difference between the two sets.
It took me about 1/2 an hour to do these questions which should have been a simple warm up was not feeling very happy.

Then the core 3 questions on Complex Integration my favourite topic think I managed to do justice to all three parts and finished these in the next half hour so calmed down and felt a bit better,

Question 6 involved Rouche's theorem which is a method of finding zero's of a complex function within a given interval managed to get the first part out which is quite straightforward but unable to answer the last part as you had to work out how many of the zero's lay in the upper half and I couldn't see how to do it.

Question 7 was a standard one on fluid flows and their complex potentials and sketching the stream lines
I got the complex potential but couldn't sketch the stream line as it involved a complicated circle and I couldn't remember where the centre of a circle of the form

    $$x^2 + y^2 + y = c$$ was

Question 8 was a question on Complex Iteration and determination of whether or not a point lay in the Mandelbrot set got about 3/4 of this question out

So reckon probably about 3/4 of part 1 answered correctly

Part II there were 4 questions
9 Part 1 was on the Cauchy Riemann conditions which I think I managed to answer more or less correctly
  Part II was on curves and the effects of a transformation. Normally the curves are quite straightforward but this time they weren't and so only managed to answer about 1/2 of the second part

Then my last question 10 should have been a straightforward derivation of a Laurent Series for the function

$$\frac{z}{1-cosz} $$

however I couldn't get it out all the usual tricks didn't work and I couldn't see how to proceed

I abandoned this and was able to answer a question on the Complex Integration of the series as they gave the answer. The second part was about the singularities of the function but by this time my brain was frazzled so I gave up.

So about 1/2 of part 2 answered. So not a satisfactory paper. I've done enough to pass probably grade 3
but certainly no more.

M208 This went much better
Part 1 had twelve questions
1 Sketch of a graph.
2 Exercise on proof involving the converse of a statement
3 Question on whether or not two sets were groups
4 Another question on groups and their cosets
5 A question on row reduction
6 A question on basis for a linear transformation I did not answer this as I hadn't revised this so left it
7 A question on the solution to an inequality
8 A question on whether or not two sequences were convergent or not
9 A question on the symmetry groups of a hexagon
10 A question on homomorphisms
11 A question on L'Hopital's Rule
12 A question on showing that an Integral was less than a certain value
    Think I got most of this correct 
So apart from question 6 managed to give reasonably full answers to most of the questions but may have lost one or two marks here and there

Part II had 5 questions of which I answered 2
13 A question on the diagonalisation of a 3x3 matrix. Seemed to come out so confident that I got most of the marks for this question
17 A question on the Taylor polynomial of a function which I got most of

     A question on the epsilon delta definition of continuity. By this time it was the last 15 minutes and I was feeling quite tired. Also as I did not write out a sample problem of this type in my handbook unlike some people I didn't phrase the answer properly so will probably only get a few marks for this question.

Still having answered most of part 1 and 3/4 of part 2 I think I've done enough to get grade 2 and if the examiners are feeling generous might get distinction.

Some people might think I'm being perverse in refusing to annotate the handbook. My argument is that the exam is a test of how much you know without any crutches. Not an ability to spot model answers to a question and then simply copying them out. There was much heated debate on this in the forum. In my opinion annotation should not be allowed however no doubt other people will think differently.

So a bit disappointed by my performance on M337 but reasonably satisfied with my performance on M208
Will try and review both courses in the next week or two and also how I plan to use down time productively.

Monday, 10 October 2011

Tick Tock E Day approaches

Well tomorrow is E day for me I have Complex Analysis in the morning followed by Pure Maths in the afternoon. Having been through two or three past papers for both courses. I seem able to do most of the
questions in Part 1 and about 1 and a half questions in Part 2 in the time available. So distinction is probably not likely and certainly not for Complex Analysis. Anyway I'll be happy with grade 2. I'm not going to do anymore revision. I'm just going to try to relax till tomorrow. It's a bit like a Kid waiting for Christmas Eve to be over so that he can open his presents.

Good luck to all my fellow course mates for tomorrow

I'll report back on Wednesday when I'll either be over the moon or down in the dumps

Till then Bye for now.