Thursday, April 26, 2007

L N i.e. log_e (and other snippets)

First question: How do you pronounce ln? Do you say 'lun' of L N? If 'lun' then I'm intrigued as to where that comes from! Now I'm going to get to why this post is about ln (pronounced as LN! :D) 'lun' just doesn't sound 'right'!

So what is so special about ln that it deserves a post title. Well you might have noticed that things in maths which 'bug' me- for more than an hour that is- get this special honour. I was also tempted in writing 'integrating inverse trig functions' as the title, but thankfully I resisted. Now please can someone clarify the following:

Hopefully you can confirm that for me, otherwise I'm very worried indeed. I've been told that the modulus sign isn't 'necessary' at this level, since if you have ln then it naturally means that you take the absolute value. The natural question which arises is why have I not written ln|-g| = ln g. I mean that is technically right(?), since g is a number (the evil number!). Now bare with me if you're wondering what I'm blabbering about again. You see I was doing a question and this 'beautiful' question required using the separation of variables method. I did this and then applied the initial conditions which resulted in the blimmin, ln(-g). I left this as it is, did nothing fancy with the minus sign like in the picture, and thankfully got the required answer.

You've probably realised that I didn't get the right answer on the first attempt. I confess- it frustrated the life out of me, but the 'get in there' feeling did come to pass.... well until now. The other problem which I have as a result of this, is the darn constant of integration. Ultimately when you integrate you get a constant. When you use separation of variables you get two constants, but two constants can be combined and called another constant 'A' I guess. That's easy enough to understand so you only have the constant on one side- normally the right hand side. Now I'm going to hazard a guess that I kept on getting 'weird' constants because of the fact that I was messing with the ln(-g).

Actually I have just done the question again in six ways- yep that's right six! I'm lying when I say that I have nothing better to do with my time, but I just had to get it out of my system. The first three ways I had the constant on the right hand side, and the other three it was on the left side. I did with the ln as the following: ln(-g), -ln(g) and as ln(1/g). Now this is one of them 'don't try it at home moments' because frankly I have a mess. The constants are indeed the same (thankfully) after some work. But the answers with the constant initially on the left is different to the one where it is on the right. I'm afraid I haven't 'double checked' my answers since this would require patience above all, and I've lost count on the amount of attempts I've already attempted this. Maybe I have made an error, but what I forgot to mention is that when I use ln(-g) as it is, the answers is the same regardless of where the constant started!

This is telling me that don't mess with logs- they're evil so and sos who enjoy making me suffer. But seriously, unless I'm having a dense moment (again) why is this happening. I mean it ultimately boils down to the picture above, but I should 'technically' be getting the same answer, right?

I'll leave that as it is for now, because thankfully I have 'scored one' for the team. This nagging feeling will annoy me for a while, but I'm going to 'double check' my work on the weekend- I've had enough for now. So I mentioned inverse trig functions above, and the following picture is a 'standard result' which we're never told directly. I confess that integration is not one of my strongest points, namely because I can't remember the standard results! I also have a problem with 'spotting' solutions. Anyway, as I was doing this other question I ultimately became 'stuck' again. One could call this integration question as trivial, and when I tell you that I had it in the standard form, then it definitely is 'trivial'. Drum roll please for the following 'standard integral'....

Now that can possibly be a lot of mumbo jumbo, since I've written it down myself. However the following is what this stems from:
A big thanks to a maths lecturer from the mss building for that! I was having a nightmare trying to check whether I'd integrated something correctly and Google wasn't being nice! (differentiation seemed the obvious thing, but I couldn't remember how to differentiate trig inverse trig functions :/ I am doing a maths degree right?!!). So I acted on impulse and instead decided to ask a lecturer who I see around, but who doesn't know me, for assistance. Thankfully he did help me, and what I'd been doing had been right but I'd forgotten the above which was annoying me since I couldn't understand my work. It seems that I best go over my first semester calculus stuff as well!

Now I'm guessing that you might find the following in the staff room for the maths lecturers:
(Might need clicking) but I guess not everyone finds the fact that I like interacting with lecturers good, and even the staff seem to agree as well! :D Haha, to those who haven't yet met/taught me- beware! Or maybe it should be me who keeps an eye out for things- I mean I'm definitely outnumbered! (Oh, and that's not the real picture- my spies are working on acquiring one. ;) )But even after this I'd still talk to the lecturers any day, which makes it more frustrating for them I suppose. :D (Once again thanks to the lecturer who helped me today).

Well obviously the day hasn't been all gloomy- we had no sequences and series lecture because it was cancelled! Now if I was sent an email saying that the Linear Algebra lecture had been cancelled, then I could have possibly been reduced to tears, but on this hand I was 'happy' although I hope that the lecturer does get better. The linear algebra lecture was good today, I'm glad that we did linear transformations, since although I knew what a linear transformation was (T(u+v) = T(u) + T(v) and T(cu)= cT(u) for all u,v in W and c in R) that was all I knew. The rest had been mumbo jumbo.

Funnily enough, although I'm not 100% sure, Bella was having a nice conversation in today's lecture. Thankfully it didn't bother me today, since I had a good nights sleep yesterday but does it make me a 'bad friend' that at times it does bother me? I mean each to their own right? Some people don't have to listen and are able to later on understand stuff, but like I said I didn't sit next to her today so it was OK! Prof S also commented that someone got -4 in the multiple choice test yesterday! And I have to agree with what he said, that sometimes the 'ugliest' or 'horrible' of things are beautiful in their own way. (However this might make the person who got -4 proud!). I got my result and it was another 'goal scoring' moment. (Although I must confess that I did ponder on the question which I'd missed and had been 99% sure off!). You see yesterday after the test I'd bumped into my personal tutor, and upon meeting him I'm thinking of changing the alarm bell signals to, 'Don't guess the answer if you don't know it, but if you're 99% sure then answer it!'. Well I agree with what he said about not guessing if you're clueless rather than if you have a 'vague' idea. Alas, this makes me sound ungrateful (as Po put it) so I'll compare my mark to -4 and not 18 (as someone got) and be content. :) (I tend to be harsh on myself).

This 'reflection' only hit me afterwards- I am actually quite happy. Pleased enough not to have a rant at multiple choice tests as I had initially planned. :D A word of advice though- the process of elimination is your friend when doing multiple choice tests. Example- you have to find the eigenvectors for a matrix P and have three options. Find one first and check the possible answers, if it can't be any one then cross it out. Find another and once again, from the two options remaining cross another one out. Now you're left with the answer and don't have to work the final eigenvector out. (well that's what I did anyway and it worked!).

Finally, if I may continue. I'm just going to say a couple of words on this and leave the 'deep' stuff to another post. As I sat back and looked towards the 'future' today, as one does, I couldn't but help feel that life isn't a linear function. I mean that's obvious right, but sadly I wish it was. I mean if you have a straight line and I give you an x co-ordinate then you'll be able to find y. If in life I was to give you a time, then wouldn't it be 'beautiful' or 'nice' if at that time I was doing what I'm doing now. Well slightly similar anyway. I know that makes life a 'boring' process, but I mean in terms of maths. 'My Maths' to be precise. I have certain aspirations- each as unlikely as the other- but this linearity would definitely make them possible. Alas I must resign myself to the clouds, where everything is nice and linear and where one doesn't have to worry about what one does not manage to achieve, but rather how one continues to progress. It's hard for me to find consolation in the clouds. But it's hope that keeps man going I guess, so that is what I will now do- hope that one day I can achieve a certain type of linearity in terms of my maths. :)


Anonymous said...

In your first equation box involving logs. The first equality is not correct, ln |-g| is not equal to ln | g^-1 |.

beans said...

Ah indeed! I know what I've been doing wrong. If it's -ln(g) then we can write this as ln(g^{-1}), but not if you have ln(-g)! Thanks a lot, although shockingly as I discussed this with a few friends, none of them spotted this as well!

(Sir Anonymous it is for 24 hours :D until I think of a cool maths name!) Cheers. :)

beans said...

I actually feel quite 'dumb' now! I can't believe that I thought that. :o :o No wonder I kept on gettting the stinking wrong answer!

In fear of me repeating this mistake again, I will not edit this post to hide my embarrasment!:o (although one hopes that this will not happen again!).