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A Summer of Maths (ASoM) 2016

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Original post by l1lvink
I've been quietly stalking this thread for a few weeks now, but most of what you guys are talking about is way beyond me :frown:

What would you recommend is a good way to get into some of this stuff?
My background is A level Maths and 1st year Physics at York


What topics are you interested in?


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Original post by l1lvink
I've been quietly stalking this thread for a few weeks now, but most of what you guys are talking about is way beyond me :frown:

What would you recommend is a good way to get into some of this stuff?
My background is A level Maths and 1st year Physics at York


I'm assuming you've met basic single-variable calculus at A-level and multi-variable or vector calculus in your physics course, as well as complex numbers and introductory differential equations.

If you then get comfortable with linear algebra and real analysis, that opens a huge amount of topics for you. Partial differential equations is the obvious one, but related to that is calculus of variations which is used in Hamiltonian mechanics and Fourier analysis which is like everywhere in physics. Differential geometry is the language of general relativity; group theory is used in quantum theory; probability theory in statistical mechanics.
Original post by drandy76
What topics are you interested in?
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Really, I don't know, whatever is somewhat more 'beginner' maybe?

Original post by Alex:
I'm assuming you've met basic single-variable calculus at A-level and multi-variable or vector calculus in your physics course, as well as complex numbers and introductory differential equations.

If you then get comfortable with linear algebra and real analysis, that opens a huge amount of topics for you. Partial differential equations is the obvious one, but related to that is calculus of variations which is used in Hamiltonian mechanics and Fourier analysis which is like everywhere in physics. Differential geometry is the language of general relativity; group theory is used in quantum theory; probability theory in statistical mechanics.


So, where would you recommend I start with linear algebra and real analysis?
(edited 7 years ago)
Original post by l1lvink
Really, I don't know, whatever is somewhat more 'beginner' maybe?



So, where would you recommend I start with linear algebra and real analysis?


Heard numbers and sets is meant to be comparatively easy


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Original post by drandy76
Heard numbers and sets is meant to be comparatively easy


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Alright, are there any good resources that you can recommend for those?
Original post by l1lvink
Alright, are there any good resources that you can recommend for those?


1st page has some Cambridge notes on them, the Dexter ones, haven't gone through them myself so not too certain of the quality however


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Original post by drandy76
Heard numbers and sets is meant to be comparatively easy


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Recommending N&S to a 1st Year Physics student? I think it's better for him to start with something like V&M or Groups which is also comparatively easy, and yet fundamental to both pure and applied mathematics.

Original post by l1lvink
So, where would you recommend I start with linear algebra and real analysis?
Dexter's notes on V&M and Groups:
V&M: https://dec41.user.srcf.net/notes/IA_M/vectors_and_matrices.pdf
Groups: https://dec41.user.srcf.net/notes/IA_M/groups.pdf

There are some mistakes in the lecture notes, if you are really troubled by that, or just prefer to study from a textbook, most of us in this thread (who are working through V&M and Groups) are studying from Beardon's "Algebra and Geometry" which covers both courses.

V&M covers the basics of linear algebra, and Groups gives an introduction into abstract algebra. If you want to get familiar with real analysis, you can work through Dexter's corresponding lecture notes (https://dec41.user.srcf.net/notes/IA_L/analysis_i.pdf) or get the book "A First Course in Mathematical Analysis" by Burkill which is a highly recommended reading for the Analysis I course in IA of the Tripos.
Original post by drandy76
1st page has some Cambridge notes on them, the Dexter ones, haven't gone through them myself so not too certain of the quality however
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They are good but quite a bit of typos/mistakes in them. Unless buying/borrowing a textbook is a problem, I would recommend to work through a textbook over Dexter's lecture notes. I feel like textbooks explain the content better, and they also contain exercises which are fundamental in understanding the material in depth.
Thanks a lot for the suggestions, I'll have a look at them

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Has anyone else noticed how many typos Beardon's textbook has?
Original post by Insight314
Has anyone else noticed how many typos Beardon's textbook has?


yes, they're a few. any particularly bad ones you've noticed?
Reply 471
What should I study if I want to make my life as easy as possible next year?
I'll be going to imp if it matters
Original post by KFazza
What should I study if I want to make my life as easy as possible next year?
I'll be going to imp if it matters


anything you're interested in really. probably number theory is the most basic to start off with.
Anyone know of some good questions on Cayley's theorem? (In particular its implications/ on the theorem "If G G is a group, H H a subgroup of G G , and S S is the set of all right cosets of H H in G G , then there is a homomorphism θ \theta of G G into A(S) A(S) and the kernel of θ \theta is the largest normal subgroup of G G which is contained in H H ".
(edited 7 years ago)
Original post by EnglishMuon
Anyone know of some good questions on Cayley's theorem? (In particular its implications/ on the theorem "If G G is a group, H H a subgroup of G G , and S S is the set of all right cosets of H H in G G , then there is a homomorphism θ \theta of G G into A(S) A(S) and the kernel of θ \theta is the largest normal subgroup of G G which is contained in H H ".


I am not sure if this is what you are looking for, but I have quite a few on isomorphisms (not sure if they include isomorphisms on a group of permutations though).

ImageUploadedByStudent Room1470244480.918540.jpg
ImageUploadedByStudent Room1470244514.061150.jpg

Start from 7.6 since the first ones are a bit too basic for you.

These exercises are from an old textbook so tell me if you don't understand the notation that Fraleigh has used.


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Original post by EnglishMuon
yes, they're a few. any particularly bad ones you've noticed?


Not really, most of them are not that fatal but it is a bit irritating to have typos on definitions and theorems.


Thanks, they are not quite on Cayley's theorem but I like my isomorphisms none the less :smile: and yeah the notation looks fine to me, the first GT book I ever read was from the 50s so most stuff seems modern compared! One annoying thing though is older books seem to write the cycle notation for permutations the opposite way round (i.e. the left cycle is the one you apply first rather than the right one as done in Beardon).
Original post by EnglishMuon
One annoying thing though is older books seem to write the cycle notation for permutations the opposite way round (i.e. the left cycle is the one you apply first rather than the right one as done in Beardon).


I swear they tell you beforehand which way it is computed, or at least Beardon does.
Original post by Insight314
I swear they tell you beforehand which way it is computed, or at least Beardon does.


Yeah they normally do its just a little annoying when i answer some questions doing one way but they are after the other because I forgot :tongue:
Original post by EnglishMuon
Yeah they normally do its just a little annoying when i answer some questions doing one way but they are after the other because I forgot :tongue:



How do you usually compute a product of cycles? I am curious because there are other ways of doing it. I was taught by my maths teacher a month or so ago how to do them using 'branes'. I can't explain that well how you do it, it is a pretty interesting way, but here is the basic idea behind it:

ImageUploadedByStudent Room1470251812.909626.jpg

Edit: I just realised I compute them left to right, as opposed to how Beardon does them haha.
(edited 7 years ago)

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