Courses Math course selection

Jenga · Dec 14, 2005

Regarding upper division math courses, I was hoping I could get some advice from you guys as to what I should take. This spring, I have room for one and only one of these classes:
Complex Analysis
A study of complex valued functions: Cauchy’s Theorem and residue theorem, Laurent series, and analytic continuation. Already have some experience in this area, but by no means extensive... and I'd like to know how useful it might be from a physicist's perspective.
Topics in Differential Equations
An introduction to the theory of ordinary differential equations. Existence and uniqueness theorems, global behavior of solutions, qualitative theory, numerical methods. This is being taught by an amazing guy, who told me today that much of the focus will be on different classes of problems such as PDEs in fluid dynamics and GR, classical formulations of the N-body problem, etc... Sounds cool, but one concern of mine is that I have already studied diffeqs in some depth, only not as rigorously as I'd like.
Linear Algebra
A brief introduction to field structures, followed by presentation of the algebraic theory of finite dimensional vector spaces. Geometry of inner product spaces is examined in the setting of real and complex fields. This is the usual course taken by people in my position here. That said, I am not sure it will be very illuminating beyond what I already know of linear (most of the practical stuff was addressed way back in sophomore year physics). But then again, I could use some practice, and I'm sure a rigorous treatment would be interesting as well.
What is your experience with courses of the nature of those listed above? I will likely cover everything else in grad school at some point, so I guess the prudent question is, "what will be most useful to get a head start on," and also, based on my interests, "what will help me the most in plasma physics research this summer?"
Thanks!

gravenewworld · Dec 15, 2005

Jenga said:

Regarding upper division math courses, I was hoping I could get some advice from you guys as to what I should take. This spring, I have room for one and only one of these classes:
Complex Analysis
A study of complex valued functions: Cauchy’s Theorem and residue theorem, Laurent series, and analytic continuation. Already have some experience in this area, but by no means extensive... and I'd like to know how useful it might be from a physicist's perspective.
Topics in Differential Equations
An introduction to the theory of ordinary differential equations. Existence and uniqueness theorems, global behavior of solutions, qualitative theory, numerical methods. This is being taught by an amazing guy, who told me today that much of the focus will be on different classes of problems such as PDEs in fluid dynamics and GR, classical formulations of the N-body problem, etc... Sounds cool, but one concern of mine is that I have already studied diffeqs in some depth, only not as rigorously as I'd like.
Linear Algebra
A brief introduction to field structures, followed by presentation of the algebraic theory of finite dimensional vector spaces. Geometry of inner product spaces is examined in the setting of real and complex fields. This is the usual course taken by people in my position here. That said, I am not sure it will be very illuminating beyond what I already know of linear (most of the practical stuff was addressed way back in sophomore year physics). But then again, I could use some practice, and I'm sure a rigorous treatment would be interesting as well.
What is your experience with courses of the nature of those listed above? I will likely cover everything else in grad school at some point, so I guess the prudent question is, "what will be most useful to get a head start on," and also, based on my interests, "what will help me the most in plasma physics research this summer?"

Thanks!

Linear algebra is one of the most important classes for a math/science undergrad, especially for the science major. I just did an independent study on Hilbert Spaces (which is the very core for quantum mechanics), and it uses hardcore infinite dimensional linear algebra along with analysis. Linear algebra is used a lot in many different fields of science, economics, and business.

neurocomp2003 · Dec 15, 2005

depends on rwhat programme your in...
LinAlg and Both DiffQs are really important i say if you can take both of them concurrently do so...I'm surprised your able to take complex analsysi or DiffQ without LinAlg.

dicerandom · Dec 15, 2005

In your plasma physics research you'll likely be dealing mostly with fluid equations and Maxwell's equations, unless the specifics of your research are centered around a non-MHD plasma. Of all the classes you listed I think DE might be the most applicable here, but honestly what you really need is a solid foundation in E&M and fluid mechanics.

As far as future studies go, I'll go with the crowd and reccomend that you take Linear. As gravenewworld mentioned it's extremely important in Quantum Mechanics.

JasonRox · Dec 15, 2005

neurocomp2003 said:

depends on rwhat programme your in...
LinAlg and Both DiffQs are really important i say if you can take both of them concurrently do so...I'm surprised your able to take complex analsysi or DiffQ without LinAlg.

Why would you need those for Complex Analysis?

You can do Complex Analysis without those backgrounds. It's not necessary.

It's only an introduction.

Jenga · Dec 19, 2005

My only concern with taking linear or diffeqs is that I already have a fair amount of experience with both of those subjects. Will a rigorous treatment of linear really be that revealing? Will a full-blown proof of the existence theorem give me insight into diffeqs? I've been told by the faculty that I already know as much on both of these subjects as I'll need to for just about anything physics-related... but I'm still inclined to think that doing the 'dirty work' will be rewarding in some way.

I may just audit complex analysis and take linear, or some similar combination.

devious_ · Dec 19, 2005

You could always take complex analysis and read books that treat linear algebra and ODEs rigorously in your own time. That way if you find the "dirty work" to be a bit on the dull side, you could simply stop and move on, and suffer relatively little consequences.

Edit:
In fact, http://www.math.uga.edu/~roy/rev.lin.alg.pdf" [Broken] is a little ebook on linear algebra written by our own mathwonk. Gloss through it - maybe you can find something of interest.

leon1127 · Dec 19, 2005

Jenga said:

Regarding upper division math courses, I was hoping I could get some advice from you guys as to what I should take. This spring, I have room for one and only one of these classes:
Complex Analysis
A study of complex valued functions: Cauchy’s Theorem and residue theorem, Laurent series, and analytic continuation. Already have some experience in this area, but by no means extensive... and I'd like to know how useful it might be from a physicist's perspective.
Topics in Differential Equations
An introduction to the theory of ordinary differential equations. Existence and uniqueness theorems, global behavior of solutions, qualitative theory, numerical methods. This is being taught by an amazing guy, who told me today that much of the focus will be on different classes of problems such as PDEs in fluid dynamics and GR, classical formulations of the N-body problem, etc... Sounds cool, but one concern of mine is that I have already studied diffeqs in some depth, only not as rigorously as I'd like.
Linear Algebra
A brief introduction to field structures, followed by presentation of the algebraic theory of finite dimensional vector spaces. Geometry of inner product spaces is examined in the setting of real and complex fields. This is the usual course taken by people in my position here. That said, I am not sure it will be very illuminating beyond what I already know of linear (most of the practical stuff was addressed way back in sophomore year physics). But then again, I could use some practice, and I'm sure a rigorous treatment would be interesting as well.
What is your experience with courses of the nature of those listed above? I will likely cover everything else in grad school at some point, so I guess the prudent question is, "what will be most useful to get a head start on," and also, based on my interests, "what will help me the most in plasma physics research this summer?"
Thanks!

linalg for sure... you want to know how to dual with n-dimensional space.
If your ODE course isnt advance. you might want to just self-study yourself. As far as you understand the concept of ODE, you should have no trouble to understand in class (plus, most of the ODE can be done on computer if you need to).