Cryptokyo's GYG: Quest for the 1st

Woweee congrats on the 1st in first year! About to start maths at uni so reading this will be so helpful!

Well done on the first! 👍

I'm Looking forward to hearing more about what Maths at uni is like

Hello there! Welcome to my GYG thread as I try to get a 1st in my second year studying mathematics at university. I hope to use this thread mainly as a tracker on my progress over the year and to make me stop procrastinating. I am not sure why I am so good at procrastinating but I need to stop doing it as it will eventually bite me in the arse at some point.

Feel free to ask me questions about maths at university and I may drop in random post about random things I've been working/ thinking about. These can range from maths problems to programming projects I'm working on.

My prior academic achievements in spoiler.


I hope to dip into some books over the course of the year and the ones I have already dipped into are listed below:
[ul]
[li]Introduction to Complex Analysis (Nehari)[/li]
[li]Visual Complex Analysis (Needham)[/li]
[/ul]
Anyway, enjoy! I better get started

Update #1

Preparing for next year
So today has been rather sporadic. I've had confirmation the courses I'll be doing next year, these are: Complex Analysis, Analysis In Many Variables, Numerical Analysis, Algebra, Probability, Number Theory, Geometric Topology and Special Relativity and Electromagnetism.

This is mostly pure modules as I am trying to avoid statistics like the plague. One of the key things I've learnt is probability =/= statistics. Probability is much more like pure maths and the problems are more novel rather have any exact application - well this is my experience so far. And stats, well stats 1 m8. Although, I must confess I avoided doing statistics in the my first year so my opinion lacks experience in some respects although I did 4 Stats exams at A Level so I feel my thoughts on the matter have some conviction.

The two rogue modules in there are geometric topology and special relativity and electromagnetism. The geometric topology module seems to include learning about notes and how to classify them so that seems exciting. However, I really have no clue what I'll be doing in it! I partly chose it because its got a pretty cool name. Special relativity and electromagnetism is now of those things I've always wanted to know but I'm not sure I have any particular interest in Physics as a subject.

Random topics I've been doing
So today really has been a bit of revision of the first year content and playing with it a bit. I got further than I expected and connected topics from my first year linear algebra, calculus and analysis to have a more in depth look at Fourier series. I've written up the stuff I did in LaTeX so I'll probably add it on here in my next update after some additions/edits. The nice thing about the way I approached it is that the maths you need to know to do it is mostly A-Level with a few extra touches and in my notes I streamlined some of the more heavy topics which I find can hinder rather than help the understanding in some respects.

The work I did today revolves around the operators called inner products. The dot product (scalar product) is an example of an inner product. We can find if two vectors are orthogonal by seeing if their dot product is zero. Using inner products we can see if functions are orthogonal and it turns out that by cleverly choosing the inner product we can create families of functions that are mutually orthogonal. And using this principle we can get to Fourier series. Fourier series is a bit like Taylor series in that it is an approximation to a function (and in some/most cases converges to it) however it deals with functions that repeat themselves after a fixed amount - these are called periodic functions i.e. there exists a positive number $p$ such that $f(x)=f(x+kp)$ for all integers $k$.

If you have any questions, feel free to ask!
Tag list

Update #2

Canadian Travels

Sorry for the long time between updates. I have been in Canada for the past week and I am about to embark on an American-esque road trip so it may be a short while before the next update after this. Also, this means the time in the day in which I have to do any maths is quite short. However, it is nice to have some time to relax before my next (most likely hectic) term of university.

From Orthogonal Functions to Functional Analysis

As mentioned in my last post, I have been looking at orthogonal functions and then how they relate to Fourier series. The main issue I encountered was that the maths I had learned prior to looking at this was all based upon the finite-dimensional case of vector spaces but Fourier series uses an infinite-dimensional vector space. And of course infinity causes many problems when it comes to adding stuff up. Luckily, I have Kreyzig's book on Functional Analysis with me that covers just the maths I need to know for further understanding so I am now working my way through this book. I have just finished a section of the book on metric spaces. A metric space is a set with a function that can measure the distance between two points. The real line is a metric space where we measure the distance between to numbers x and y using |x-y|. But there are many more wacky examples.

I have attached some notes on orthogonal functions and Fourier series that I produced just after the last update. But I must stress the caveat in section 1.4. The maths is not rigorous but with a few (truthful) assumptions we can glance into the maths going on from an intuitive approach.

Tag list

Update #2

Canadian Travels

Sorry for the long time between updates. I have been in Canada for the past week and I am about to embark on an American-esque road trip so it may be a short while before the next update after this. Also, this means the time in the day in which I have to do any maths is quite short. However, it is nice to have some time to relax before my next (most likely hectic) term of university.

From Orthogonal Functions to Functional Analysis

As mentioned in my last post, I have been looking at orthogonal functions and then how they relate to Fourier series. The main issue I encountered was that the maths I had learned prior to looking at this was all based upon the finite-dimensional case of vector spaces but Fourier series uses an infinite-dimensional vector space. And of course infinity causes many problems when it comes to adding stuff up. Luckily, I have Kreyzig's book on Functional Analysis with me that covers just the maths I need to know for further understanding so I am now working my way through this book. I have just finished a section of the book on metric spaces. A metric space is a set with a function that can measure the distance between two points. The real line is a metric space where we measure the distance between to numbers x and y using |x-y|. But there are many more wacky examples.

I have attached some notes on orthogonal functions and Fourier series that I produced just after the last update. But I must stress the caveat in section 1.4. The maths is not rigorous but with a few (truthful) assumptions we can glance into the maths going on from an intuitive approach.

Tag list

Another excellent and fascinating update! good luck on your ventures! I shall keenly read said notes

That is one holiday lol

Please comment on the notes if you wish. It's my first time writing these sorts of notes so do just tell me if it all just seems like a load of tosh . Hopefully if this is a success I'll make some more on other topics.



Ahaha yeah. Should be fun