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
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.
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?
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.
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.
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 Posted from TSR Mobile
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.
Anyone know of some good questions on Cayley's theorem? (In particular its implications/ on the theorem "If G is a group, H a subgroup of G, and S is the set of all right cosets of H in G, then there is a homomorphism θ of G into A(S) and the kernel of θ is the largest normal subgroup of G which is contained in H ".
Anyone know of some good questions on Cayley's theorem? (In particular its implications/ on the theorem "If G is a group, H a subgroup of G, and S is the set of all right cosets of H in G, then there is a homomorphism θ of G into A(S) and the kernel of θ is the largest normal subgroup of G which is contained in 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).
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.
Thanks, they are not quite on Cayley's theorem but I like my isomorphisms none the less 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).
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.
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
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:
Edit: I just realised I compute them left to right, as opposed to how Beardon does them haha.