Algebra or PDEs for Physics Major?

[jdstokes]

Hi all,

I'm facing a bit of a dilemma regarding subject choices for the second semester of second year univeristy and I was wondering if anyone could lend some of their advice. First a bit of context. I'm enrolled in a BSc (Advanced) degree at The University of Sydney, Australia with a real passion for physics so this will definitely be my major (followed by honours year and PhD in some aspect of plasma physics). Although I've already started to specialise with the research I'm doing for the plasma physics group, I have always found the most interesting part of physics to be the deep underlying theories (eg special and general relativity) and in particular, their connection with sophisticated mathematics. I should stress, however, that I'm no maths wiz: the only reason I liked linear algebra was that I saw connections with almost every concept and special relativity. Although I finding mathematics by itself devoid of meaning, I gain great satisfaction applying it to some physical problem (the more advanced and abstract the maths the better :). At heart, maybe I want to be a theoretician, but I appreciate the need for applications which is why I'm changing tack next year to work in industry on first-wall (blanket) materials for use in the ITER project.

So, as for my subject choices, each semester we are required to do 4 subjects worth 6 credit points each, totalling 24 credits per semester. The prerequisite for third year physics is a 6 credit course which focuses on quantum physics and electromagnetic properties of materials. I also want to do 6 credit points of French for both interest and for my future ITER aspirations. That leaves 12. I think it's a good idea to do the ``other'' physics course which is a course in astrophysics, special relativity and experimental physics, although quite honestly relativity is the only part of this course which interests me and I'm not even sure how much I'd learn which I haven't already taught myself from my copy of Taylor and Wheeler. Now, filling up that 6 credit hole with maths is the tricky part. There are two courses of interest, one is on abstract algebra and the other on partial differential equations. Here are the outlines

Algebra (Advanced)

This unit provides an introduction to modern abstract algebra, via linear algebra
and group theory. It starts with a revision of linear algebra concepts from
junior mathematics and MATH2961, and proceeds with a detailed investigation
of inner product spaces over the real and complex fields. Applications
here include least squares lines and curves of best fit, and approximation of
continuous functions by finite Fourier series. Further topics in linear algebra
covered in this unit include dual space, quotient spaces and (if time permits)
possibly tensor products. The second part of the unit is concerned with introductory
group theory, motivated by examples of matrix groups and permutation
groups. Topics include actions of groups on sets, including linear
actions on vector spaces. Subgroups, homomorphisms and quotient groups
are investigated, and the First Isomorphism Theorem is proved.

Introduction to Partial Differential Equations (Advanced)

This unit of study is essentially an advanced version of MATH2065, the emphasis
being on solutions of differential equations in applied mathematics.
The theory of ordinary differential equations is developed for second order
linear equations, including series solutions, special functions and Laplace transforms.
Some use is made of computer programs such as Mathematica. Methods
for PDEs (partial differential equations) and boundary-value problems
include separation of variables, Fourier series and Fourier transforms.

From the applied physics perspective the choice is obvious, since I've already run into the problem of solving partial differential equations in Poisson's equation in my work on plasma physics. On the other hand, I suspect that Algebra could be more beneficial in terms of understanding abstract theories like general relativity and so on.

Ideally, I would do both, but this would mean sacrificing the relativity subject which is probably not a good idea since it is ``strongly recommended for those majoring in physics''. The question is then, which of these subjects would be the easiest to self-learn? If I can answer this question then I guess the hardest subject would be the one to study at uni.

What do you think?

Thanks

James

 

[SeReNiTy]

Very hard choice, I myself is studying second semester for second year at Monash Universtiy. I also am pursuing a path of physics with a intense interest in the mathematics.

Personally I think you should just overload and complete 5 units next semester...

 

[Kiley Dean]

I find algebra is simple and easy to learn, you can just make a simple plan that fits your schedule, timetable, I am sure you can get fruitful results from some of the books on the subject.
The hardest one is probably PDE. Because I once failed the general PDE course for the complex problems. The introductory part of PDE as you mention is truly 'narrow' but applying its use to solving problems in other fields may be a big pain in the butt.

 

[Kiley Dean]

Also, relativity is not only a necessity for physics but a pretty interesting stuff to learn as well. It too is really simple if you already grasp basic knowledge of vector space, simple calculus and derivative.

 

[J77]

In terms of relativity, you'll need to fully understand the tensor stuff.

Like you say though, in terms of general applicability, the PDE course would give you more 'hands-on' techniques. Plus you'll learn manipulation of matrices from stability calculations and discretisation techniques.

At uni stage, I'd find the PDEs more interesting - you can always brush up on the algebra when and if needed.

Up to you though

 

[jdstokes]

Hi Kiley,

At first glance, last year's tutorial exercises for PDEs do not ``look'' that hard, whereas the lecture notes for algebra are an abstract mess of symbols (not that there's anything wrong with that because I love abstract mathematics in a physical setting).

Wouldn't you say that although PDEs are difficult, that their solution is basically a methodical task which can be handled by a computer package like Mathematica? On the other hand, the proofs of algebra can only be done by a person, requiring more creative thought and thus making the subject inherently more difficult?

Thanks for everyone's comments.

James

 

[arildno]

Numerical methods are by no means easy!
What do you think applied mathematicians worry over?
Or, what do you think one of the main task of an engineer (who is to suit a general software package to his firms particular needs) is?

True, there exist several methods that is easily learnt, but to gauge the effectiveness, accuracy of such for simulation of complex, real-life problems is by no means an easy task. It is quite challenging.

 

[J77]

Depends what you mean by 'solutions of'.

In a physics context, pdes appear all over the place; particularly w.r.t. nonlinear optics.

These can be analysed analytically through, eg., an amplitude equation or numerically - where, in the latter case, initial spatial conditions play a vital role.

For me (someone who does a lot of numerics), algebra can just as easily be done through, eg., LAPACK (or indeed mathematica/maple).

You may want to look up equations like: Swift-Hohenburg, Nonlinear Schrodinger and Ginzburg-Landau to get a feel for the more complicated pde stuff.

edit: agree with arildno!

 

[jdstokes]

Depends what you mean by `Algebra'.

As someone who uses Mathematica all the time, I appreciate it's symbolic math capability. But I'm sure you'll agree that the best software package in the world is no substitute for the humain brain as far as understanding and proving vital theorems is concerned.

 

[J77]

OK - to me, algebra is mainly numerical algebra - which is basically matrix manipulation.

I've worked with theorists, and while I can follow their work, I don't always have the patience (concentration?) to - similarly, they don't always have what it takes to code the ideas/produce telling results.

Again, it's up to you but you say you're no maths whiz. In which case I'd say go for the more application based course.

If you want an intro into LA w.r.t. application I'd suggest the Trefethen book (who's title slips my mind).

 

[jdstokes]

I'm definitely not a maths genius, but I love theory, which is basically pure maths, does that make sense? I'm not saying I want to be a string theorist, but I like the idea of being able to at least read and critically evaluate the latest papers in the field. That means understanding things like groups, Lie algebras etc etc. Do you think I'm asking too much of myself?

 

[J77]

Do you think I'm asking too much of myself?

If you want to do it, go for it!

I think Matt Grime's the pure maths whiz on these boards - he may be able to give you some pure leanings...

 

[matt grime]

In terms of relativity, you'll need to fully understand the tensor stuff.

what an algebraist calls a tensor bears only passing relation to what a physicist means by tensor.

If U and V are two finite dimensional vector spaces, then $U\otimes V$ is a vector space through which any blinear map UxV-->W factors as a linear map.

It is not immediately obvious that this even exists, but it can be described as the set of all symbols $u\otimes v$ for u (and v) in u (and V) modulo the 'obvious relations like $(a+b)\otimes v = a\otimes v +b\otimes v$.

One can if necessary argue by bases.

The tensor product is unique (up to unique isomoprhism as ever in a category) and it is relatively clear that $U\otimes V$ is symmetric in the arguments.

If U and V are not finite dimensional then analysts have all kinds of funny things to say about (completed) tensor products that I've never bothered to understand.

 

[matt grime]

OK - to me, algebra is mainly numerical algebra - which is basically matrix manipulation.

this is a tricky one to address.

In some sense all (simple) algebra is about matrix multiplication, but I suspect not in the (derogatory?) sense that you mean here.

(Any finite dimensional algebra modulo its radical is a product of matrix algebras over a division ring - Wedderburns structure theorem.)

 

[matt grime]

I'm definitely not a maths genius, but I love theory, which is basically pure maths, does that make sense? I'm not saying I want to be a string theorist, but I like the idea of being able to at least read and critically evaluate the latest papers in the field. That means understanding things like groups, Lie algebras etc etc. Do you think I'm asking too much of myself?

If you truly want to understand the latest stuff in string theory then yes you need to do proper algebra and geometry. I egotisitically recommend my thread in maths on quivers and representations. And then I''d point you in the direction of McKay correspondence as something that the string theorists are interested in. (You won't be able to understand it all, but it would be a good idea of the utility of algebra in physics. Roughly speaking, to understand strings one should try to understand categories of sheaves on manifolds, which are equivalent in a suitable sense to representations of quivers, conjecturally.)

 

[jdstokes]

Hi Matt,

I don't intend to `truly' understand string theory any time soon. It would be nice, however, to have to done some preliminary work on groups, eg, before I try to attack harder concepts like Lie groups.

So far my mathematical knowledge is depressingly limited, consisting of first year differential and integral calculus, first year stats and second year vector calculus. I didn't do graph theory or real and complex analysis, are these terribly important in mathematical/theoretical physics?

The current path I'm considering is second year Alegra (outlined above), followed by third year Metric spaces and Differential Geometry.

http://www.maths.usyd.edu.au/u/UG/SM/

Do you think this will comprise a sufficient stepping stone to start studying the harder stuff relevant to mathematical/theoretical physics?

 

[tmc]

real analysis is definitely a very important subject, and is especially crucial if you want to go towards String theory or other such purely theoretical subjects.

 

[jdstokes]

Oh well, I guess that's yet another subject I'll have to go back and teach myself one day *sigh*.

 

[J77]

In some sense all (simple) algebra is about matrix multiplication, but I suspect not in the (derogatory?) sense that you mean here.

I certainly don't mean that to be derogatory!

I thrive on the manipulation of matrices. It's just that (naturally) most of my final work is numerical.