Theorectical Physics and Studying Math Independently

[Qbit42]

Hello, I've come to seek opinions on what mathematics I should study to become a good theorectical physicist.

Some background on myself: I started out my university career (I'm a 2nd year student) interested in both physics and computer science. During my 1st semester I began to become interested in mathematics, and have since fostered a growing love for the subject. However taking 3 majors would be ridiculous, so I decieded on a math minor and to supplement that with independent study. To that end I've begun taking out mathematics books from my university library (with intentions of studying a single topic a semester, kinda like an extra course). The mathematics I'm planning to start with will be mathematics relavent to theoretcial physics, but time permitting I'd also like to study some pure mathematics as well. To that end I've devised a list of math subjects I would like to become familar with:

Math topics I wish to cover independently (Chronologically Ordered)

• Linear Algebra (currently studying)
• Non-Euclidian Geometry
• Differential Geometry
• Tenser Calculus
• Real Analysis
• Complex Analysis
• Group Theory
• Manifolds
• Algebraic Topology

At the rate of one topic a semester (including summer) this will take me up to the completion of my undergraduate degree. My hope is to have a comprehensive mathematical background for graduate school

Topics covered through mathematics minor

• 3 courses in Calculus (Done)
• Linear Algebra (Done, although it wasn't comprehensive enough for me, thus I'm studying it independently)
• Basic statistics (In progress)
• Vector Calculus
• Discrete Math
• Ordinary Differential Equations (Done)

Does anybody have any suggestions on what other topics I should cover, or what order I should study them in? Has anyone attempted anything like this before and if so how did it work out? Have I bitten off more than I can chew? I have a feeling I should devote some time to studying PDE's more comprehensively but am unsure what topic I should drop, if anything.

Edit: Oh yeah I guess I should mention that my university offers 2 courses in mathematical physics that I entend to take when I meet the prerequisites.

Mathematical Physics II: examines the functions of a complex variable; residue calculus. Introduction to Cartesian tensor analysis. Matrix eigenvalues and eigenvectors. Diagonalization of tensors. Matrix formulation of quantum mechanics. Quantum mechanical spin. Vector differential operators in curvilinear coordinate systems. Partial differential equations of Mathematical Physics and boundary value problems; derivation of the classical equations, separation of variables; Helmholtz equation in spherical polar coordinates.

Mathematical Physics III: covers further topics on partial differential equations of Mathematical Physics and boundary value problems; Sturm-Liouville theory, Fourier series, generalized Fourier series, introduction to the theory of distributions, Dirac delta function, Green's functions, Bessel functions, ' functions, Legendre functions, spherical harmonics.

...Oddly there isn't a Mathematical Physics I

 

[hellbike]

I'm pretty much the same case.

But i see very weak connection between math covered on physics courses and math that is actually math.

On my university all math courses(on physics) are about calculating. Are things different on your uni?

Because i think that pure math have almost nothing to do with calculating things.

 

[Qbit42]

Well at my university there are 5 math courses required, all of them Calculus based (Calc I, II, III, Vector Calc, and ODE's) so yeah I'd say that most of the math I do for my physics degree here is based around calculations. I can't really say I'd expect anything less though, they teach you applied math because, well, physics is about applying math lol. Your average undergraduate physics student doesn't really have a need for pure math, its only the few who are interested in theorectical physics (like myself) who could benifit from it. Even then you can always double major in math/physics, but since I'm already double majoring that isn't an option for myself (thus I study it independently).

 

[MathematicalPhysicist]

You can ofcourse change majors and leave CS as a minor.

Also in pure math you do calculations, after all it's not as if it's just proving theorems, there also calculational techniques, but they don't emphasize them in the exercises.

For example, I took a course in mathematical methods for theoretical physics with a course in pure maths complex analysis, thanks to the first course I have learned Cauchy principal integral and how to calculate, whereas in the second course we hardly covered it if we had.

 

[Klockan3]

You need tensors for differential geometry, also that subject is a quite high level maths so I would recommend you putting it more towards the end so that you got more mathematical maturity when you come to it. Real analysis, complex analysis and group theory are quite easy and really they cover a lot of basics that you will see in most advanced mathematical courses so you should do them the first thing you do after linear algebra in my opinion.

 

[Qbit42]

You can ofcourse change majors and leave CS as a minor.

I've bounced this idea around a lot actually, but fundamentally I still don't know wither I'd rather be a physicist or a computer scientist and I fear that if I make the major-minor switch I'll regret it later on, especially if it turns out that I don't have what it takes to make the cut as a theoretical physicist (up until now I've been doing well but I'm only a 2nd year student).

You need tensors for differential geometry, also that subject is a quite high level maths so I would recommend you putting it more towards the end so that you got more mathematical maturity when you come to it. Real analysis, complex analysis and group theory are quite easy and really they cover a lot of basics that you will see in most advanced mathematical courses so you should do them the first thing you do after linear algebra in my opinion.

thanks for the info, I'll be sure to switch those around in my plan

 

[some_dude]

Hello,

I'd suggest putting a basic analysis course first on your list of topics - it would be required to cover almost everything else you named. You could also merge manifolds and tensors into differential geometry. But basic real analysis should come before any of those. How rigorous was your calculus? Do you have much experience with proofs? That will determine where exactly you should begin with analysis.

 

[some_dude]

One more thing,

A math major is a lot more general than a CS major, and usually a CS major has fewer required courses. So, it's possible that majoring in math would be a better compromise between your various interests. Just something to consider.

 

[Klockan3]

Hello,

I'd suggest putting a basic analysis course first on your list of topics - it would be required to cover almost everything else you named.

Wouldn't that be real analysis?

 

[some_dude]

Wouldn't that be real analysis?

Yeah, that's probably what I meant. I notice some people tend to reserve "Real Analysis" for study on the level of Folland or Royden. But by "basic analysis" what I had in mind was things like metric spaces, compactness, proving the standard calculus theorems in a general topological settings, very basic functional analysis, etc.

 

[Qbit42]

Ok so Real/Complex analysis are quite obviously going to be my next areas of study from the look of it.

You could also merge manifolds and tensors into differential geometry.

thanks for the note, I'll keep it mind for after I finish up with Analysis and Group theory.

How rigorous was your calculus? Do you have much experience with proofs? That will determine where exactly you should begin with analysis.

My first 2 calculus courses required us to memorize proofs, my 3rd didn't. However I do tend to read the proofs in my textbooks whenever I come across them so I guess that puts me above basic proofs.

A math major is a lot more general than a CS major, and usually a CS major has fewer required courses. So, it's possible that majoring in math would be a better compromise between your various interests. Just something to consider.

I will consider this very thourghly, I think I'll look through my university calendar and figure out when preciely is the point of no return course wise (ie when it would become difficult to switch majors without a lot of make up time)

Also would operator theory be worth the time to learn from a physics standpoint?

 

[Landau]

Also would operator theory be worth the time to learn from a physics standpoint?

Sure, but first you will have to learn (real, introductory) analysis and basic functional analysis. By then, you will be able to decide for yourself whether you think it is important.

The problem is, there's just too much interesting mathematics. And if you look hard enough, everything may in some way or another be useful in (theoretical) physics, except perhaps areas like number theory. You could spend a whole lifetime doing it. If you learn most of the stuff you listed, you'll have about the knowledge of a bachelor's in mathematics (maybe except for algebra, logic, numerical math,...). By then you'll have a sufficient 'bird's eye view' to choose specialised topics, persuing your own particular interests. My point being: don't include too much advanced areas in your list yet, follow your schedule freely, just do what you like now. It's good to have goal in mind, but it will certainly change along your way, and it can be frustrating to have a schedule so tight you can't stick to it.

 

[Qbit42]

Sure, but first you will have to learn (real, introductory) analysis and basic functional analysis. By then, you will be able to decide for yourself whether you think it is important.

The problem is, there's just too much interesting mathematics. And if you look hard enough, everything may in some way or another be useful in (theoretical) physics, except perhaps areas like number theory. You could spend a whole lifetime doing it. If you learn most of the stuff you listed, you'll have about the knowledge of a bachelor's in mathematics (maybe except for algebra, logic, numerical math,...). By then you'll have a sufficient 'bird's eye view' to choose specialised topics, persuing your own particular interests. My point being: don't include too much advanced areas in your list yet, follow your schedule freely, just do what you like now. It's good to have goal in mind, but it will certainly change along your way, and it can be frustrating to have a schedule so tight you can't stick to it.

Thanks for the advice, you may just be right (I'm kinda OCD when it comes to plans). There are other areas that I'd like to explore but have excluded for the sake of relavence (Abstract Algebra, Set theory, Number Theory,etc)

 

[Landau]

There are other areas that I'd like to explore but have excluded for the sake of relavence (Abstract Algebra, Set theory, Number Theory,etc)

Sounds like you should major in mathematics instead of CS, you might want to reconsider MathematicalPhysicist's advice ;)

 

[MathematicalPhysicist]

Sure, but first you will have to learn (real, introductory) analysis and basic functional analysis. By then, you will be able to decide for yourself whether you think it is important.

The problem is, there's just too much interesting mathematics. And if you look hard enough, everything may in some way or another be useful in (theoretical) physics, except perhaps areas like number theory.

even number theory has applications in TP, mainly in QC.