Advanced Math Course Options for Physics Majors

[Isaiah Gray]

Hi,

I am a sophomore physics major at the University of Minnesota. I have several options for technical electives and I will probably fill them up with math. Past the four-course math requirement, would it be better to take assorted courses such as PDEs or complex variables, or should I try for hardcore real analysis?

 

[Kevin_Axion]

Hi,

I am a sophomore physics major at the University of Minnesota. I have several options for technical electives and I will probably fill them up with math. Past the four-course math requirement, would it be better to take assorted courses such as PDEs or complex variables, or should I try for hardcore real analysis?

PDEs and complex variables are more important at this point. Real analysis is useful but I think PDEs and complex variables will be more relevant and useful.

 

[nonequilibrium]

Seeing complex analysis before real analysis seems odd though?

 

[Kevin_Axion]

Seeing complex analysis before real analysis seems odd though?

Complex variables is generally different than complex analysis. Physics majors usually take a complex variables course after their series of calculus courses. It generally covers the following:

Theory of functions of one complex variable, analytic and meromorphic functions. Cauchy's theorem, residue calculus, conformal mappings, introduction to analytic continuation and harmonic functions.

I don't think you need a firm grasp of real analysis or even a course in it to learn these things but I can be wrong.

 

[DivisionByZro]

I think any good course on complex analysis MUST be preceded by a course in real analysis. I suggest you take the "hard" road, and take real analysis.

 

[Kevin_Axion]

I think any good course on complex analysis MUST be preceded by a course in real analysis. I suggest you take the "hard" road, and take real analysis.

I would wait for the course description but I can assure you that it is not a complex analysis course but just an overview of basic complex analysis topics that are useful to a physics major.

 

[DivisionByZro]

Complex variables is generally different than complex analysis. Physics majors usually take a complex variables course after their series of calculus courses. It generally covers the following:

Theory of functions of one complex variable, analytic and meromorphic functions. Cauchy's theorem, residue calculus, conformal mappings, introduction to analytic continuation and harmonic functions.

I don't think you need a firm grasp of real analysis or even a course in it to learn these things but I can be wrong.

Can you explain to the OP and I what those terms are, I'm really not familiar with this stuff. It would be very helpful. Mainly, give us some examples of how they are useful too.

 

[Kevin_Axion]

Can you explain to the OP and I what those terms are, I'm really not familiar with this stuff. It would be very helpful. Mainly, give us some examples of how they are useful too.

I'm not familiar with them either, I got them from a 'Complex Variables' course description form a physics major curriculum. My point is Real Analysis isn't a pre-requisite for the course.

 

[DivisionByZro]

I'm not familiar with them either, I got them from a 'Complex Variables' course description form a physics major curriculum. My point is Real Analysis isn't a pre-requisite for the course.

Why not? I'd also like this question answered, as I don't know much math.

 

[Kevin_Axion]

Why not? I'd also like this question answered, as I don't know much math.

Because it says so in the course description.

 

[micromass]

I'm not familiar with them either, I got them from a 'Complex Variables' course description form a physics major curriculum. My point is Real Analysis isn't a pre-requisite for the course.

So you're not familiar with the terms and you still feel qualified to give other people advise?? You never studied complex or real analysis, so how would you know what the OP should do?? Your "advise" is very dangerous and people could fail classes because of it.

 

[Kevin_Axion]

So you're not familiar with the terms and you still feel qualified to give other people advise?? You never studied complex or real analysis, so how would you know what the OP should do?? Your "advise" is very dangerous and people could fail classes because of it.

I say we wait for the course description. I'm basing it off of a curriculum I was reading and telling him to do a Real Analysis course could be just as destructive.

 

[DivisionByZro]

I say we wait for the course description. I'm basing it off of a curriculum I was reading and telling him to do a Real Analysis course could be just as destructive.

How could taking a Real Analysis course be destructive? If he's at the level where on takes PDEs or Complez Analysis or Real Analysis, then he is certainly prepared for at least the Real Analysis course. Are you suggesting he take none of them?

 

[micromass]

I say we wait for the course description. I'm basing it off of a curriculum I was reading and telling him to do a Real Analysis course could be just as destructive.

And what would the course description tell you?? You admitted you didn't know anything about it. Please stop giving advise about things you don't know anything about.

 

[Kevin_Axion]

How could taking a Real Analysis course be destructive? If he's at the level where on takes PDEs or Complez Analysis or Real Analysis, then he is certainly prepared for at least the Real Analysis course. Are you suggesting he take none of them?

The point is your presuming it's a complex analysis course where in many cases I have seen two separate courses. A complex variables course and a complex analysis course. The former is for physics majors that doesn't require real analysis the latter does. The point I'm making is that we need the course description. My suggestion was a suggestion and as I stated I could be wrong.

 

[Kevin_Axion]

And what would the course description tell you?? You admitted you didn't know anything about it. Please stop giving advise about things you don't know anything about.

Very well. I digress. We still need the course description though for anyone to give a reasonable response and my response was qualified under the condition that real analysis isn't a pre-requisite for it. The fact that OP states "hard core" real analysis seems to suggest that the course is on a higher level than the others.

 

[nonequilibrium]

As a physics student I've had a one-semester course "Mathematical Methods in Physics" where one third of the course was about complex analysis. Basically, this is about integration in the complex plane and how it can help you to solve integrals on the real line which are too hard to solve without the tricks of complex analysis. It has other applications too, for example easily solving Laplace's equation in 2D systems (or 3D systems where the solution is symmetric in one direction).

You might not understand what I'm exactly talking about, but let me state that complex analysis should belong to the backpack of any mathematically/theoretically inclined physicist.

It is true that for this course real analysis was no prerequisite, but by far most of the students had taken the class (probably simply because it was offered sooner in the curriculum than this course, quite logically). Having followed the course, I can say that you can indeed get by without real analysis, the reason for this being explained later in this post.

That being said, I also took a complex variables course in the mathematics department (also a one-semester course, but this time of course not simply a third of it). For this you definitely needed a real analysis course.

Why the difference? Well, actually, both courses talked about the same things, the latter just in far more depth. The first didn't pay attention to any rigour, it simply wanted to get the mathematical theorems across and how you could use them. The second course was a course for math students, i.e. everything you used you had to prove. Personally, I preferred the latter, as the extra time and attention for rigour showed the interconnection between the different theorems and at the end gave a deep feeling of elegancy: complex analysis is probably the most elegant course I've taken so far. In the "mathematical methods in physics" course I often felt like a robot applying mysterious tricks, not knowing why they worked (well okay, I did as I had taken the mathematical course before this one, but all the rest felt like a robot then).

That being said, I can't decide based on the name of your course which of the two types it is, but now at least you know what the two possibilities are. I hope that helped.

PS: sometimes I've called it "complex analysis", sometimes "complex variables"; they're synonyms!

PPS: I haven't answered your question "which to take: PDE and complex variables, or something like real analysis" and I find it very hard to answer! PDE is without a doubt the most useful one, but a course like real analysis makes you more mature mathematically speaking, which is also very useful, but maybe in a less direct way. I've already described the importance of complex variables. If you have the time for all three, I'd do all three! (and put real analysis first, in that case) But maybe more precise advise can be given if you describe where your interests lie, where you want to go (maybe rigour is, say, repulsive to you).

 

[Isaiah Gray]

Wow; thanks for all the advice. I'm new here, and did not expect so many posts within a few hours. Here real analysis is a two-semester course and is supposed to be extremely difficult. That doesn't scare me, but I just wanted to know if it'd be worth the effort.

 

[nonequilibrium]

Oh, it was a one-semester course in my university. Do you have a summary of the course somewhere? (the topics treated)

And it's hard to say if "it'd be worth the effort": that totally depends on in what way you're interested in physics and math. Are you interested in math for its own right, or is it as a tool for physics? And what kind of physics are you leaning to? Experimental or theoretical? Maybe too soon to be able to answer?

 

[Isaiah Gray]

Introduction to Analysis I

Axiomatic treatment of real/complex number systems. Introduction to metric spaces: convergence, connectedness, compactness. Convergence of sequences/series of real/complex numbers, Cauchy criterion, root/ratio tests. Continuity in metric spaces. Rigorous treatment of differentiation of single-variable functions, Taylor's Theorem.

Introduction to Analysis II

Rigorous treatment of Riemann-Stieltjes integration. Sequences/series of functions, uniform convergence, equicontinuous families, Stone-Weierstrass Theorem, power series. Rigorous treatment of differentiation/integration of multivariable functions, Implicit Function Theorem, Stokes' Theorem. Additional topics as time permits.

 

[micromass]

Real analysis is quite a difficult choice, but it certainly pays of. First of all, you understand the math behind several concepts. Secondly, you will be more flexible. Do you want to study functional analysis because you want to use it for quantum mechanics?? Well, you will be able to do that if you followed a real analysis course. You'll be ready to study things like differential geometry, complex analysis, etc.

Of course, PDE's and complex analysis is more useful in the short term.

Not everybody will agree with me, but if you want to be a theoretical physicst then I find that you SHOULD take real analysis