## Friday, November 02, 2007

### Complex proof, fireworks and...

... millions of other things too.

It was today as I bumped into Prof. D (my once upon a time calculus lecturer), that I realised how full I was. My head is full of lots of different information and deadlines at the moment, and I can't really sit in peace. If I am then that means that there is something which I am not doing. With this fullness I have realised that my blogging is one thing which I am finding the most difficult to force into my stomach. I can't do anything about that, and believe me when I say that on two separate occasions this week, I have written at least half a post but only to never to finish it. I aim to complete this one though!

One thing that I like about Prof. D is that whenever I bump into him, he always finds time to stop and have a conversation with me. He doesn't teach me anymore and I was quite surprised that he remembered my name from the time I told him it. I talked to him about the 'not to be mentioned' topic, and as always he was very encouraging. I told him that my friends do not see what I see, and that it would be nice to find someone who had similar views. It is unfortunate for him (and possibly not for me!) that he can't go somewhere. Before I had bumped into him I had had Complex Analysis in my head. Whilst talking to him this topic was brought up, and so towards the end I remarked about the gazillions of things floating in my head.

As was rightly said, I must not take my eyes of the ball, that is my studies. I am guilty of doing that at the moment, but until I resolve certain matters I find it hard to let go. I am enjoying it all at the moment, and I feared that I would end up saying this. I am not known as a fool for nothing. My slight reluctance to do anything of this nature was linked to this too. However, in my defence once the car is moving I no longer have to accelerate. Prof D. is one of the most friendliest person that you will find on this planet! I always enjoy talking to him.

It is true that I have been neglecting my studies. It is not that I haven't been going to lectures etc. because I have. What I am finding difficult to do is spend that hour or two to actually study the material, and try to understand it. I have yet to find that right frame of mind for that. One of the possible reasons for this is that I am sort of following the material at the moment\{PDE's and stats}, hence why I am not taking time out to drill home the message. This week has been horrible in the sense I have wasted a unique opportunity to do that drilling. I have made some progress with stats, but that is an embarrassing amount.

Everyone seems to be telling me the same thing. My lecturers, friends and family are echoing that I should ease of all other business for the time being. Some know me too well, but that is indeed from past experiences. All this fullness leads to me becoming very stressed out. It is too soon in the year for that too happen, which is why the alarm bells have been sounding. Yes - I have been a misery guts whilst posting for the past few days, but if anything else is added to my current fullness, I will be sick. I can feel myself going over the edge, but I must remain calm and positive. (Easier said than done).

I felt that my second year might be better than my first because I would know what is expected of me. I felt that I would be able to organise myself better, but this external thing has prevented me from doing such a thing. My worry is that will this thing always have a negative impact on me?

I have been playing around with a site for the past few days, when I shouldn't have. (But that being said, I felt immense pleasure when things fell into place). I shouldn't have because it has nothing to do with my studies, and everything to do with the this damn not to be mentioned topic! Enough of that now.

When I said Complex proof, how many of you guessed that I meant a proof from complex analysis? Well it was actually meant to have a double-ish meaning, so don't worry if you thought I meant a hard proof. ;)

Theorem
If f is differentiable at z_0, then f is continuous at z_0.

In my lecture notes this was done in a rather slick way - too slick for me. I followed it but it seems to be one of them proofs which I will never be able to 'recall' (or maybe memorise). Hence why I decided to see if there was another way to do things. I asked my lecturer to have a look through my version of things, but unfortunately I hadn't written it out "nicely". I will type my nice version out here, and the challenge is to spot the mistake!

Proof
Assume that f is differentiable at z_0, and for contradiction suppose that f is not continuous at z_0.

Then since f is differentiable, we can write:

$\begin{array}{ccc} \displaystyle \lim_{z\to z_0} \frac{f(z)-f(z_0)}{z-z_0} & = & f'(z_0)\\*[4ex] \displaystyle \frac{\displaystyle\lim_{z\to z_0} (f(z) - f(z_0))}{\displaystyle \lim_{z\to z_0}(z - z_0)} & = & f'(z_0) \end{array}$ by the quotient rule provided the limit exists and bottom one is not equal to zero.

That is: $\displaystyle\lim_{z\to z_0} (f(z) - f(z_0))= f'(z_0) \times \displaystyle \lim_{z\to z_0}(z - z_0 ).$

We know that $\displaystyle \lim_{z\to z_0}(z - z_0 )=0$, therefore the above tells us that $\displaystyle \lim_{z\to z_0}(f(z) - f(z_0) )=0$. This means that f(z) is continuous at z_0. However, this is a contradiction for we assumed that f(z) is not continuous at z_0. Therefore, by contradiction we can say that if f is differentiable at z_0, then it is continuous at z_0.
$\blacksquare$

I spotted the error whilst writing the proof out with more details. If you have spotted the error, then is there any way round it? Or how would you prove this?

It is unfortunate that the camera I have requires a dodgy USB wire. It is not the normal ones which is why I never seem to post any more pictures. The normal one I have plenty of, but I only had one of this type. But now I have 0.5 of the dodgy USB for I no longer know where it is! I semi went to a fireworks display and it was good fun. Brought back memories from my youth, when I had these red wellingtons and we would run around in the back garden with sparklers! My dad always had the bucket and after the sparklers he would attend to the fireworks. (Which were quite a magnificent display).

I am reminded about the innocence of children and their naivety - how they see certain things differently. Bear and mini-bear had been with us, and as with all displays you get the odd drunk person walking about, making a mess on the floor whilst maybe smoking. This person kept on going round in circles, and was pointed out by someone. Bear had noticed this person and then said: "She's lost her car, that is why she is walking up and down looking for it. She keeps on walking and will find it soon. Her friends were helping her as well."

Never once as a child did I wish to be a grown up! Milo (and others) thought otherwise, but I loved my childhood. As I said ages ago: one of the best thing about being a child is the lack of responsibility. You never have to mull over a decision more than twice, for they are mostly made for you. [Not that there isn't anything good about being more grown up...]

I am going to have an early night today, so tomorrow morning and Sunday can be dedicated to vector calculus. (And hopefully some Algebra). Before I start talking about the depressing things I have to do, I think I need to mention something else. The amount of conversations I am having with myself has significantly increased! So beware since my limbs seem to respond to my discussions!