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Maths and Theoretical Physics Course Thread TR031 TR035
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24-08-2011 5:05pm#1
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\o/ Finally we get a thread. How about including the TPs in the title though? There's a good few of 'em lurking around, even if most of us do eventually transfer to Maths. >_>
Oh, and I'll be happy to answer any questions too.
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Maybe_Memories wrote: »Good idea! I've no idea how to change the title though
Use your Modly powers!
My modly powers are useless in this forum.. :pac: You might be able to do it by editing your post, or perhaps one of the TCD mods would do us a favour?0 -
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PurpleFistMixer wrote: »Hey, not most TPs switch to maths!
SS TP here, in case anyone has any questions... (or questions about doing schols in TP I guess...)
I could have been talking about the lurkers! I'm sure plenty of 'em have switched.. Or will switch!
Actually, surprisingly only 5 people have transferred from my year, so you are of course correct. A lot more have dropped out, and a few (3/4 perhaps) have wandered into physics.
Someone should just tell the first years how to do the Jordan Normal questions now tbh.. >.> Also, if you guys don't like Pete, you're dead to us. :P0 -
I could have been talking about the lurkers! I'm sure plenty of 'em have switched.. Or will switch!
Actually, surprisingly only 5 people have transferred from my year, so you are of course correct. A lot more have dropped out, and a few (3/4 perhaps) have wandered into physics.
Someone should just tell the first years how to do the Jordan Normal questions now tbh.. >.> Also, if you guys don't like Pete, you're dead to us. :P
I'm not convinced anyone knows how to do the Jordan Normal questions :S0 -
Tears in Rain wrote: »I'm not convinced anyone knows how to do the Jordan Normal questions :S
I do! :P
And posting a method for JNF is actually a very good idea, which I might do later on today.
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Since JNF is one of the things most people find difficult and the supplemential exams are next week I thought some people might find this useful..
Suppose we have a linear operator A:V->V where A is the matrix of the operator relative to some basis, presumable the standard orthonormal basis of R^3.
The purpose of finding the Jordan Normal Form is because it makes the matrix appear simpler in a different basis so we can more easily understand what the operator does.
Right, so take the matrix A. Firstly, find its characteristic equation found by
det(A-tI)=0, where I is the identity matrix.
The solutions to the equation are the eigenvalues of our matrix.
Define a new matrix B=A-tI. There will usually only be one or two eigenvalues, I haven't seen a question of Vlad's yet where there was 3 distinct eigenvalues.
For each eigenvalue find rk(B), rk(B^2), ..., rk(B^k-1), rk(B^k)
until rk(B^k-1) = rk(B^k).
Obviously then dimKer(B^k-1) = dimKer(B^k) because of the formula
rk(A) + dimKer(A) = dim(V)
NOTE: CASE 1 AND 2 ASSUME TWO EIGENVALUES
CASE 1: rk(B) = rk(B^2)
Find a basis of ker(B) given by the vector v, Bv=0.
v is our basis vector. Since the kernels of powers stabilise instantly we get a thread of length one, and we have our one basis vector, and we are done.
CASE 2: rk(B) =/= rk(B^2) = rk(B^3) =/= 0
Reduce the vectors which span Ker(B^2) using the vector which spans Ker(B) to get a relative basis vector f, which is the non-zero vector obtained after reducing. Since the kernels of powers don't stabilize for two steps, find Bf and reverse the order to get the basis vectors Bf, f.
CASE 3: If B is nilpotent, that is to say (B^k)=0 for some k. There are certain intricacies to this case.
Firstly, assume two distinct eigenvalues.
Find the vectors that span ker(B^k-1), and our first basis vector is the vector which "makes up for the missing leading 1". If k=2, find Bf and we are done.
Reverse the order to have a basis; Bf, f.
Now assume only one eigenvalue. If k=3, do the same as above, but find f, Bf and (B^2)f. Reverse the order for the basis. (B^2)f, Bf, f.
****** Now suppose only one eigenvalue and k=2. Find f, and Bf, and then reduce ker(B) using Bf to get a vector g.
Basis given by g,Bf,f.
Right, you have your jordan basis. More than likely you will be using two of the cases above, one for each eigenvalue.
So you get a transition matrix C, which transforms A into it's Jordan Normal Form, J.
J = (C^-1)AC
In general, J is given be each of your eigenvalues along the diagonal.
The multiplicity of each factor of eigenvalue in your characteristic equation determines the size of your jordan block. Best way to illustrate is with an example. Suppose the characteristic equation is t(t-3)^2
Eigenvalues are 0 and 3, and 3 has a multiplicity of 2, so the jordan normal form is given by
.... 0 0 0
J = 0 3 1
.... 0 0 3
A one is placed over the intersection of two same eigenvalues, so if your characteristic equation was (t-3)^3, J would be
.... 3 1 0
J = 0 3 1
.... 0 0 3
This is true in all cases except the one marked with ******. In this case you essentially have two jordan blocks of different size, but each has the same eigenvalue, so you'd get
.... 3 0 0
J = 0 3 1
.... 0 0 3
This is a result of the Cayley-Hamilton theorem and minimal polynomials etc.
One last thing, you put the basis vectors in corresponding order of the eigenvalues of your JNF. Example, if your eigenvalues are 2 and 3, and for 2 you get a basis of f, and for 3 you get a basis of g and Bg, and you arrange your J as
.... 2 0 0
J = 0 3 1
.... 0 0 3
then the columns of your transition matrix C is f, Bg, g
So there you have it, Jordan Normal form explained. 0 -
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While it is a good idea to provide explanations etc. for common problems, I would personally advise doing up nice LaTeX documents and putting them on your personal maths webpages - it's more permanent, easier to refer to, and easier to read than a forum post! (not that I object with forum posts, it just makes more sense to LaTeX stuff like that, to me.)
... And if you don't know LaTeX, it's a good reason to start learning.
Now my own question: does anyone have any idea of who will be teaching general relativity this year?
edit: Also some exciting news which TPs may be interested in: this year's Hamilton lecture will be given by Edward Witten! It will be sometime in October on something like "The Quantum Theory of Knots".0 -
Thanks for the two explanations of JNF guys
PurpleFistMixer wrote: »blah blah blah words...your personal maths webpages...
What's the deal with those anyway? How do we get/use them?0 -
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PurpleFistMixer wrote: »Do they not do 061 in first year any more? The incredibly basic computer course where you get your maths account and are encouraged to make a website? (there are probably other elements to the course but I don't remember them) In practice I just scp to my maths account and do everything from there, but if you don't have a maths account I can see a problem.
I went to the first few but they were painfully slow so I gave up on them. That said, there doesn't seem to be anything on Timoney's 061 page explaining how to do it...either way it's a moot point since I just figured it out 0 -
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I feel like Pete will be wasted in linear algebra. Its fairly self-explanatory, very leaving cert-esque. Having said that, I don't know what I'd rather he be teaching, or who I'd put in Linear ahead of him, but I feel he could teach a more complex course very effectively.
Also, is it just me, or are maths students the craziest bunch of people you've ever met for going out and social stuff? Honestly the first semester last year nearly killed me, and bankrupted me in the process.
I had that impression too, that the course, at the start at least, was 'Leaving Cert-esque', but it's something I really only felt for the first half of the year. After Christmas, when we had moved on from computing numerical results to actually reasoning about vector spaces algebraically, proving theorems, and incorporating more abstract ideas into the fold, the course seemed to be a lot more in line with what I felt university level Maths was about. I found the first half of the course incredibly dull, but the second half really interesting. (That said, I couldn't do the second half, but that's a different story).
As for Pete being wasted on it, to be honest, I think to a certain extent, all first year Maths modules are 'beneath' all of the lecturers in the sense that it's so trivial to them, but on the other hand, it's the mark of a good teacher to be passionate and bring new insight into a subject that would ordinarily be considered dull and trivially easy. Who for example, would consider Feynman, a Nobel prize winning physicist, wasted in teaching the introductory Physics classes that went on to be compiled as his Lectures in Physics?
Furthermore, obviously as a student I'm no expert in pedagogy, but I'd imagine that teaching introductory Mathematics modules has additional difficulties and responsibilities, since in some sense you're responsible for stewarding the transition from the blind rule-following of Leaving Cert Maths (and let's face it, no matter how much they try to deny it, or change it, this is what LC Maths is and will be for the foreseeable future) to the world of university Maths which involves developing a sophisticated mathematical intuition where previously there was simple pattern matching from a small palette of possible problems, mathematical rigour where previously arm-waving arguments were acceptable, and working with abstract mathematical structures where previously there was a tacit assumption of working in the Euclidean plane over the field of Reals. This transition can be jarring, and being responsible for introducing it to students is a task I don't feel is 'beneath' anyone.
Also, nights out yeah, can't speak for Maths but got nicely comatose on some TP nights out
edit: Noticed you're going into SS while I've just finished JF, so if anything I said there is a load of bollocks, please correct me 0 -
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dropinthocean wrote: »What's the reading list/ suggested reading for a first year TP/ maths student. I understand that it will be mostly what's in lectures that's important but just if one wanted to get a head start
First of all, sorry if I come across as condescending in this post, when I say something's difficult I'm mostly drawing from my own difficulties and experiences with a subject. Something I hate is when people pretend to be experts and know everything about a subject when it's far more fruitful to acknowledge your own shortcomings. This isn't America's Next Top Model and there's no need to maintain a façade of "OMG I'm the best." /rant (To clarify, this isn't an attitude I've encountered in TP, and as far as I can tell there aren't any big egos in my course. I have encountered it in people from other courses.)
OK, as for books, I'll divide this into subject areas. You already know this but again I emphasise that it's what's covered in lecture's that's important, less so in books, and less so again in certain books than in others. With that caveat:
Linear Algebra: There are two books recommended here, vastly different in tone and scope. The first, Elementary Linear Algebra by Anton and Rorres is an introductory book on the subject. Linear Algebra as a subject is something that I find suffers with regards to the quality of literature available, as it's a subject that's often foisted on to (say) social science majors, medical students, business students etc. i.e, people who aren't interested in the mathematics itself, but are required to do a course to graduate anyway, similar in this respect to Probability and Statistics courses. Because of this, the books tend to be huge wordy tomes full of colourful pictures and lots of "real-world examples" to motivate concepts and maintain interests. Anton/Rorres is exactly this sort of book, and while the verbosity of the material is useful at the start of the course when the concepts are new, using it to review material is an experience akin to wading through mud while wearing weighted boots. Personally, I hated the book, but your experience may be different.
The second book, Lectures in Linear Algebra by I.M. Gel'fand is as different as humanly possible. While Anton/Rorres covers mostly the second half of the course, Gel'fand covers most of the (in my opinion, vastly more interesting) second half. Its tone is completely different, it's completely aimed at the pure mathematician, with scarcely any appeal to geometric intuition, and no 'real world examples'. While Anton/Rorres could be used to stun a charging bull elephant, Gel'fand is A5 in size and approximately 170 pages long. The exposition is very dense, and requires time to work through solutions. This might all sound very forbidding, but personally I loved the book. Unfortunately, the brevity comes at a price, and it's hardly recommendable as an introductory text. Indeed, I only reached a level of competence such that I found it useful in the last few weeks, while studying for repeat exams. That said, when you do reach 'its level', it's a very useful and charming book, and you can review whole swaths of the course in a few pages. But again, I stress, not suitable for getting a head start.
Vlad, who lectured the course the last few years (but won't be lecturing in 2011/2012) has a few further notes on Recommended Reading here.
Analysis: The course here follows somewhat closely Spivak's Calculus. In case it hasn't come across already, I really don't like wordy doorstop-like textbooks. At first sight, Calculus comes across as one of these, but it's really not. It's beautifully typeset and has a leisurely exposition that's neither condescending nor dull, and Spivak's tone is friendly and humorous. The subject matter is fairly easy, and anyone coming from LC Maths should be able to dive straight into it, but the concepts it teaches are the foundations of mathematical analysis that you'll be using throughout the course. (erm...I think at least..remember, I've only just finished first year, so take everything I've written with a boulder sized grain of salt). The Analysis course is probably the first example of the kind of mathematical rigour you'll encounter in the course, so getting a head start by looking at this kind of stuff can't do you any harm.
Mechanics: The course here mirrors almost exactly the first 9 or so chapters of Kleppner and Kolenkow's An Introduction To Mechanics. It's fairly similar to the classical mechanics problems you would have encountered if you did Applied Maths in school. Personally I think mechanics at this level is a little dry and boring, though obviously essential, so take from that what you will. The problems can be fun, and Kleppner and Kolenkow do inject some humor into the book, but I think ultimately Newtonian classical mechanics just really isn't that interesting as a subject, rather it's a stepping stone to more advanced classical mechanics, as well as a tool to develop a physical intuition about the world as well as problem solving skills that can be applied to all Physical problems. That said, this may be my own personal prejudice talking, and while my tone here may be pessimistic, I would stress I did still enjoy this course, just not as much as others (and analysis, which I loved, I found to be the most derided by my classmates).
Physics:
I don't know if you're studying TP or Maths, but assuming the former, you'll also be handed an absolute beast of a book called University Physics. If Elementary Linear Algebra could stop a charging bull elephant, University Physics could take out a Tyrannosaurus. It's....grand I guess. It tries to cover pretty much all of introductory Physics, i.e. electromagnetism, classical mechanics, thermodynamics, special relativity etc. It's one of the books that I looked over to get a head start on the course before starting, and to be honest, I rarely open it these days. The reason it's given out free by the department is because it gives you access to a set of online problems that are graded and contribute to your end of year mark. I don't really know what else to add, other than if there's an area of Physics you're interested in, you're probably better of looking at a book dedicated to the subject, e.g. Griffith's Electrodynamics if you're interested in electromagnetism, than going straight to the Jack of all trades, Master of none approach of University Physics.
Buying Books:
The above raises the question of when and which books are worth actually buying. Realistically, you can get away with not buying any books at all for first year, and if you're doing TP you'll get University Physics for free. If there's any book worth buying, it's probably Kleppner/Kolenkow, since the course so closely mirrors the book. You can probably get it second hand, and pawn it off on someone at the end of year. Personally, I found Calculus worth buying too, though I don't know anyone else who has it. If you're doing Physics, I think Feynman's Lectures on Physics belong on every Physics student's bookshelf. I got these as a birthday present one year, and I'm constantly consulting them to get a different point of view on a topic introduced in class. Anyone familiar with Feynman's popular science writing will recognise his conversational tone (the books are transcribed almost directly from the lectures, with little effort of converting the text to a more standard prose style) as well as his emphasis on gaining an intuitive understanding of the subject.
Finally I would stress, multiple copies of all of the above books are available in the library for loan, it is not necessary to buy any of them, and money would probably be better spent on Fresher's Week naggins.
Good luck, and enjoy Maths/TP 0 -
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Tears in Rain wrote: »This isn't America's Next Top Model and there's no need to maintain a façade of "OMG I'm the best." /rant (To clarify, this isn't an attitude I've encountered in TP, and as far as I can tell there aren't any big egos in my course. I have encountered it in people from other courses.)
Yeah to be honest the vast majority of maths and TP students are very laid back. I mean we all (or nearly all) take our respective courses seriously but there's no crap like "you can't do such and such you're clearly going to fail and I'm so much better." Obviously there is healthy competition and that's good.
I agree with your comment on Gel'Fand's book. When you first read it it's like a kick in the face. Getting used to the style takes a while. But when you do get used to it it's a very interesting and dare I say entertaining book to work through.
Personally I can't wait 'till second semester this year. ODEs with Pete. Gonna be awesome! 0 -
Maybe_Memories wrote: »Yeah to be honest the vast majority of maths and TP students are very laid back. I mean we all (or nearly all) take our respective courses seriously but there's no crap like "you can't do such and such you're clearly going to fail and I'm so much better." Obviously there is healthy competition and that's good.
Personally I can't wait 'till second semester this year. ODEs with Pete. Gonna be awesome!
Yeah, thought there might be people like that being a fairly high points course. Pretty happy everyone turned out to be sound!
Pretty much all of second year looks savage from the looks of things, can't wait to get back...0 -
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Tears in Rain wrote: »edit: Noticed you're going into SS while I've just finished JF, so if anything I said there is a load of bollocks, please correct me
I'm actually just finished it but I knew almost no first years last semester. And no all seemed fine to me!0 -
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Tears in Rain wrote: »I have a nice stash of ebooks too...but in reality I know trying to read them on a computer will turn into a day wasted on facebook, boards, IRC and reddit. That's one advantage that real books have, you can sit in the library surrounded by them and avoid at least a decent proportion of the usual distraction.
The big distraction with the library is the sheer volume of books, and they're all so interesting.. I spend so much time looking through Quantum and Fluid mechanics books. 0 -
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