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Maths and Theoretical Physics Course Thread TR031 TR035
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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
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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!
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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.
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PurpleFistMixer wrote: »That or I'm jealous that Vlad has been going around handing out gifts and I'm missing it.
I briefly thought that too.. >.>
What was so nice about the paper actually? Hope you guys got on well! 0 -
I briefly thought that too.. >.>
What was so nice about the paper actually? Hope you guys got on well!
Jordan Normal question was a 2x2 matrix.
Sylvester's Criterion was on it, and showing a vector is part of a subspace (reducing a 5x5 matrix isn't very fun...).
Question 1 was difficult though.0 -
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Maybe_Memories wrote: »Jordan Normal question was a 2x2 matrix.
Sylvester's Criterion was on it, and showing a vector is part of a subspace (reducing a 5x5 matrix isn't very fun...).
Question 1 was difficult though.
2x2?! Seriously? That was very, very generous of Vlad. You can pretty much pass semester 2 on that question alone like, or you could with the marks assigned to it on our paper last year.
What was question 1, out of nosiness? You guys had separate papers for each this year, right?0 -
2x2?! Seriously? That was very, very generous of Vlad. You can pretty much pass semester 2 on that question alone like, or you could with the marks assigned to it on our paper last year.
What was question 1, out of nosiness? You guys had separate papers for each this year, right?
1. (a) Define the rank of a linear operator.
(b) Consider the vector space V of all 2x2 matricies. Show that for every 2x2 matrix A the mapping L:V->V given by the formula L(X) = AXA - 3X is a linear operator. Compute the rank of this operator for
A = 2 -1
``` 1 2
Showing that L was a linear operator was grand. But getting the rank wasn't. If L is the operator then surely L(X) is the image of some vector X? But that doesn't make sense because then the AXA part of the formula isn't defined...
Yup, we did. 0 -
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Maybe_Memories wrote: »1. (a) Define the rank of a linear operator.
(b) Consider the vector space V of all 2x2 matricies. Show that for every 2x2 matrix A the mapping L:V->V given by the formula L(X) = AXA - 3X is a linear operator. Compute the rank of this operator for
A = 2 -1
``` 1 2
Showing that L was a linear operator was grand. But getting the rank wasn't. If L is the operator then surely L(X) is the image of some vector X? But that doesn't make sense because then the AXA part of the formula isn't defined...
The point to take on board is that the 'vector space V of 2x2 matrices' is a vector space, so the elements of V are the 'vectors'. Thus a 2x2 matrix X is a vector in this 4-dimensional space. Its components are given by the four components, say
X = ( p q )
( r s )
Compute the components of L(X) in terms of p,q,r,s. These components form a quadruple of real numbers, so you have the specification of a linear transformation from a 4-dimensional vector space to itself. This has a well-defined rank.
Basically, you should be unfazed by the fact that the space of linear operators on an n-dimensional vector space is a vector space of dimension n^2, whose components are the components of the matrices representing the linear operators. Thus linear operators acting on some vector space are vectors belonging to some other vector space. In particular a 2x2 matrix can be regarded as a 4-dimensional vector.0 -
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