Oh yea haha a family friend gave me a quick 1 hour condensed intro into Lie groups and tangent bundles and their links to Jacobians and stuff like that. Pretty hard to take it in that time though But yeah I love group theory and enjoy analysis, but have little experience with the latter.
So you study these areas im guessing?
My interests lie mainly in PDEs, differential geometry and analysis. I don't usually get to do any abstract algebra apart from Lie groups. You could probably pick up a text on real analysis right now. It'll give you a nice head start into understanding epsilon-delta proofs.
And how many hours do you spend on it? And do you go by step, i.e. Do you go chapter by chapter, in order or pick what you prefer?
You should definitely take a look at "How to Think Like a Mathematician" by Houston, which would answer most of your questions about self-studying uni maths. You can pick out whichever chapter you want from this book, but in general, any kind of rigorous, university level mathematics textbook should be read chapter by chapter, unless you are comfortable with the material. More often than not writers like structuring the chapters in order that would best fit the student that is reading them, a prime example of this is Beardon's textbook on V&M and GT where he starts with an introduction of groups and connect it to points made in linear algebra later on. However, chapters that deal with concepts such as, for example, Möbius transformations in Beardon's "Algebra and Geometry" textbook are only used to show you the applications of what is currently being learnt; and thus some textbooks, whose main purpose is to give the fundamentals of a particular course, could use different examples, or may even omit such applications of the theory as a whole.
I am not sure about others, but I usually self-study for about 5-8 hours, but it heavily fluctuates. Yesterday I managed 8 hours and a half, but the day before that I only did 4 hours. Either way, I would recommend you to take a look at Houston's textbook and manage your time as you wish. Don't look at other people since it solely depends on you, and take it into account that most people do not have any prerequisite knowledge of the material taught at university before October, so you should only decide to self-study if you are really eager to start early. Also, don't think of relaxing during the summer and not doing mathematics as something unproductive, it is important that you do not get 'burnt out' of mathematics before you even start to do 'actual' mathematics.
Yeah basically what insight said. But if you are starting looking at a completely new topic Id recommend reading the chapters in order until you have a good feel for it (especially the first few ones). and timewise probably upto 8 hours . Basically stop if you feel you arent making much progress due to tiredness etc.
I was going to do D&R during summer, but was told it is best to self-study the fundamentals of high-level mathematics first, so I chose to do V&M and Groups during summer. How are you finding D&R so far? I don't think the content itself requires VC, but the example sheets do seem to include divergence and what not.
On a side note, are you planning on taking Maths with Physics at Cambridge?
I haven't been on TSR for a while, how is part IA doing? You are working through Beardon's textbook right?
ah yea I was wondering how you were doing the other day . but yeah its great so far! A bit of bad hay fever seemed to reduce my thinking abilities for the first few days but been much better after I realised waking up early helps for some reason . and yea its the beardon I've been going through, been a bit distracted GT though as its so nice. I skipped the section on spherical geometry though, it seemed like one of those topics that'll be more useful for geometry later on but not too important for the other material in the book but idk.
ah yea I was wondering how you were doing the other day . but yeah its great so far! A bit of bad hay fever seemed to reduce my thinking abilities for the first few days but been much better after I realised waking up early helps for some reason . and yea its the beardon I've been going through, been a bit distracted GT though as its so nice. I skipped the section on spherical geometry though, it seemed like one of those topics that'll be more useful for geometry later on but not too important for the other material in the book but idk.
Yeah, I've been pretty busy lately with maths and social life (finally managed to get one) so I haven't been that active on TSR. Definitely gonna be here for results day though!
I also find Groups pretty nice, I even decided to casually read through (not doing exercises or taking notes) of "A First Course in Abstract Algebra" by Fraleigh, which is a pretty good one on Groups and it even goes in depth with rings, field theory and Galois theory. Definitely recommend it if you are the 'abstract algebra' type. It also has way more exercises than Beardon's after each chapter (one of the reasons I got it), and the only downside to it is that it is not written in LATEX (pretty old one). I am either going to start working through rings and fields from this textbook, or do some Analysis I from Burkill's textbook, after I finish Beardon's. Probably gonna do the former since I am really starting to like abstract algebra, and Analysis I is only in Lent term.
I don't think spherical geometry is that important, you are right. I really liked it, but I had a bit of trouble with some theorems so gonna have to read through my notes in detail at some point. Are you currently on 'Quaternions and Isometries'? What do you think of quaternions? I had so much fun with them!
Yeah Ive been enjoying quaternions, but I prefer doing it all in terms of the previous group theory stuff and then going back to beardon to get the general ideas refreshed
ah yea I was wondering how you were doing the other day . but yeah its great so far! A bit of bad hay fever seemed to reduce my thinking abilities for the first few days but been much better after I realised waking up early helps for some reason . and yea its the beardon I've been going through, been a bit distracted GT though as its so nice. I skipped the section on spherical geometry though, it seemed like one of those topics that'll be more useful for geometry later on but not too important for the other material in the book but idk.
Spherical geoemtry is IB geometery i think. Do not skip, geometry is the best.
Also does anyone know of some Cambridge like 'example sheets' from other unis? Currently Im looking at the the Oxford ones but wandering if theres something more alike.
Also does anyone know of some Cambridge like 'example sheets' from other unis? Currently Im looking at the the Oxford ones but wandering if theres something more alike.
The american uni's often share their materials quite freely, don't have any links though. Oxford ones are fun (if possible could you read ahead and do the second year ones then send me your solutions, it would be greatly appreciated)
The american uni's often share their materials quite freely, don't have any links though. Oxford ones are fun (if possible could you read ahead and do the second year ones then send me your solutions, it would be greatly appreciated)
ah ill have a look then, thanks. but yeah I think the 2nd year stuffs a little way a way yet! I recognise the odd few terms but little more