Monday, November 05, 2007

Mean, green, Fourier Series fighting machine

Yet I am still forgetting that a_0/2. STILL!

However, now I know why! Having done all the integration and reached a sense of small satisfaction, I don't check my work. That being said, another thing which I haven't been doing is writing out the general Fourier series formula that I posted yesterday. Hopefully in an examination I will have the sense to write that out first, and then go about finding it. Then it will always be there -- a_0/2.

I must apologise before hand for how I may behave today. And please please please - if I dare nod off in the test, wake me up! Waking up at the time I did was indeed difficult. I am sitting here now because I have given up with my notes on the chain rule and what not. My Calculus book made a really nice pillow, which signalled for me to close it.

Discussion has been happening at Gower's Weblog about teaching with examples first. As I look at the horrible definitions for 'total derivatives' and what not, I know that an example first in this instance would be superb! I probably can, after much time and a break, make sense of the definition, but it will take longer. I did try hunting my last years notes, which were beautifully filled with examples, but I have stupidly already taken the required section out before hand and not replaced it.

This is the chain rule for functions of several variables by the way! I have just read and ... well just read that for f(x,y)= z: dx = f_x(x,y) dx + f_y(x,y) dy. This is the total differential(?) My notes then go on and say something about the chain rule to do with this. Not making sense I'm afraid.

I think I know what the problem is (wow)! When I start fiddling with functions of several variables, I start small. I look at and try to understand say the properties and life of y=f(x). Then I want a bigger house and look at and try to understand z=f(x,y). If I am still unsure about how this property ladder works, I may dip my hand in higher variables. However, by the time I have studied z=f(x,y) with plenty of examples, I am ready to 'generalise'! Yes - I then look at f(x_1,......x_n) = y.

In my notes we have started of with the 'general' case and that is proving problematic. The Calculus book which I am using, does what I do. It starts small and goes big, but I don't erm... have that much time to absorb all of it in. I thought about just reading my notes and trying to make sense of them, but I am failing miserably in that department. (Hence the book becoming a pillow).

On a positive note: spherical coordinates are no longer ugly looking so and sos. I still cannot understand how we sketch them and how certain coordinates give certain objects (eg a sphere), but I no longer have to memorise them. I just need to sketch that neat diagram and derive the Cartesian equivalent if and when the time arises. This makes it much nicer for me, and less taxing for my brain (which as we know has a poor memory).

Excuse the nonsensical stuff in this post. If I went to see the doctor, he would ban me from posting (such is the current state of my writing), but he doesn't have to know about the chain rule and total derivatives and other potentially nerve-wrecking things!

I have given up on the Fourier series half-range series, but let me see what I can recall. A function f(t) defined in a limited interval (i.e. not periodic) CAN NOT be represented by a Fourier series. Recall Fourier series can only represent periodic functions. However, then it is up to us to see whether this ugly function can be made periodic. It can actually be made periodic in infinitely many ways. Normally a full period is (0, 2\pi), therefore a half-period is (0, pi).) This limited function is defined in the half-period.

I am wondering now, what if we had a function in the interval (o, 2pi) but it wasn't periodic? We then wouldn't have infinitely many choices of how we can make it periodic, since we could just say f(t + 2pi)= f(t)? Or maybe we could have infinitely many choices if we said f(t + 2l)= f(t), then if we extended the interval to (-2pi, 2pi), we would get 2l= 4pi so l=2pi? [Sorry more about this some other time!] By this way - this paragraph is not the truth: it is just me asking myself questions and failing to answer them.

Back to the half period. If we want to find a sine half series, we create the function F(t) ={ f(t) in (0, pi) and depending on f(t), choose a nice appropriate odd function in the interval (-pi, 0). (For cosine series, choose nice even function). If you don't care about odd or even, then choose anything else in the whole world, but not what you chose for the above two.)

I told you I was the mean, green... OK maybe not, but after doing some questions on half-range sines I should be!

Another reminder to myself. And odd function times an even function is an odd function. If we integrate an odd function over a symmetric interval i.e (-a, a) (a \in R), the integral is always zero. An odd function times an odd function is even, as is even times even. If we then integrate an even function over the same symmetric interval we get:

\int^a_{-a} even function dt = 2 \int^a_0 even function dt.

This should hopefully save me from doing nasty integrals. I knew there was a reason why I liked the interval (-pi, pi) more than (0, 2pi)!

Corrections, and proper formulas etc will be inserted tonight sometime! I need me some weetabix and some chain rule.

PS: Can I say something (well I am going to say it anyway!) The best mathematician in the world has to be Abel. Namely because when we write abelian group, it is not a capital a. I am sick of pressing shift F to write Fourier series!


Jake said...

PS: Can I say something (well I am going to say it anyway!) The best mathematician in the world has to be Abel. Namely because when we write abelian group, it is not a capital a. I am sick of pressing shift F to write Fourier series!

That isn't a good example though as 'abelian' is an adjective; you are describing a property of the group whereas 'Fourier Series' is really a proper noun so would be capitalized anyway.

Беанс said...

I am sure I can think of some way to make 'Foruier' into an adjective! ;)

The red lines under 'fourier' became annoying and I didn't want to write FS everywhere.

Беанс said...

After all my work on trying to spot which integrals are zero, I am ashamed to report that I was unable to spot a_n=0. However, the a_0 was I hope, divided by 2! I actually did look towards my neck during the test, and had a good chuckle to myself whilst I did the 'dividing by 2'.

(There is a reason why I am called mad after all!)