# Courses Next undergrad math course

1. May 4, 2005

### The_Brain

I'm trying to decide what math course to take next year. I am a freshman at an ivy league school who as far as I can tell right now, wants to go into theoretical physics; either GR or String Theory (I'm sure you hear this everday ). I'm trying to decide on what math course to take next year, namely, whether I should take an advanced linear algebra course and some other math course (in the spring) next year and put off real & complex analysis until junior year or vice versa. Just talking to grad students I hear that analysis was generally their least favorite math course and also one of the least applicable to physics. Thus, I am heavily leaning towards the advanced linear algebra route. Right now I'm finishing up a year long course that combines both linear algebra and mutlivariable; we are integrating forms over manifolds and have just proved the generalized stokes' theorem which i think is pretty fun and will be applicable later in theoretical physics (i think). If this is the case, I want to try to take more courses like this one then, however I do not know... that's why I am asking you guys!

2. May 4, 2005

### Pyrrhus

I don't know... I liked Real Analysis... There were plenty proofs.

3. May 5, 2005

### semidevil

I imagine the algebra will help in understanding in physics and applications....real analysis helps in understanding how continuous functions, derivative, and integration works....

4. May 5, 2005

### gravenewworld

No matter what anyone tells you Analysis sux, period. If you like proofs take an abstract/modern algebra class it is MUCH more interesting and probably more useful to you as a physicist seeing how group and field theory and linear algebra are used a lot in science.

5. May 7, 2005

The grad students are leading you astray. Key concepts in real analysis such as Cauchy sequences, convergence tests, uniform convergence, and the Lebesgue integral will recur in physical applications, e.g. when you're using Fourier series and integrals. As a physicist, you have to come to terms with Banach space and Hilbert space theory sooner or later: they are used in ODEs and PDEs, and quantum theory formalism is in the language of Hilbert spaces. At an even higher level, operator algebras are used frequently in theoretical physics. Complex analysis is almost as essential. And of course, don't forget the linear algebra: the important structure theorems for Jordan and rational canonical forms, the spectral theorem, modules over a PID, tensopr algebras, and so on. And remember linear algebra and real analysis coalesce in Hilbert space theory.

6. May 8, 2005

### Kalimaa23

BS.
I've seen many American quantum mechanics courses twist needlesly around difficulties that could be easily resolved if students were required to take more analysis courses...

I'd take every class you've mentioned, they're all essential if you want to be a theoretical physicist.

7. May 8, 2005

### marlon

Don't the Americans call analysis Calculus ? :uhh:

marlon

8. May 9, 2005

### mathwonk

Agreed, you need all those courses. Theoretical physicists use lots of analysis, and linear and abstract algebra, as well as real and complex analysis.

I have been a lecturer at the internation center for theoretical physics in trieste, and i was lecturing on riemann surfaces for string theorists. i used algebraic topology and complex analysis (of several variables) in my lectures as well as algebraic geometry. other lecturers used partial diff eq, such as laplacians, hodge theory, higgs fields, etc...

you are just beginning. all that stuff you mentioned is basic.

9. May 17, 2005

### The_Brain

Thanks for the replies! I think I got the wrong point across. Indeed, I am going to take both an advanced linear algebra and both real and complex analysis courses. I just wanted to know which one to take first, however, I am sure it's arbitrary at this point. I guess then the question is whether I should take one before the other for any specific reason. For example: taking linear algebra before multivariable this year was required to do caluclus on manifolds. Just wondering if the same thing applied to the next level up.

10. May 17, 2005

### neurocomp2003

REAL analysis or LinAlg first. COmplex last

11. May 17, 2005

### mathwonk

linear algebra is a prerequisite for real analysis, if reals contains any banach spaces, i.e. real analysis is the science oif using linear algebra to approximate non linear analysis, but complex can be taken independently in most cases.

sorry to disagree neurocomp. what is your opinion based on? [mine is based having taught all these courses, not that that makes it more accurate, but it does make me wonder where you are comingfrom.]

oh i bet i know, i bet you are calling "real analysis" what i call advanced calculus, if so then i agree, as i am calling "real analysis" a grad course on infinite dimensional linear algebra and measure theory that comes after advanced calc.

by the way, although we are flattered, why are you asking us instead of the professors teaching those courses? trust me, they should know.

Last edited: May 17, 2005
12. May 17, 2005

### neurocomp2003

guess it differs from school to school

mathwonk: is this sarcasm? or your real def'n?
"h i bet i know, i bet you are calling "real analysis" what i call advanced calculus, if so then i agree, as i am calling "real analysis" a grad course on infinite dimensional linear algebra and measure theory that comes after advanced calc."
He is an undegrad so i don't know if that applies

at MAC-the first two real analysis are based on studying series,sequence,metric spaces compactness & completeness. The 3rd one studies banach and lesbegue
though the linalg is a prereq for the 2nd analysis...its not necessary...if the student can handle it. Complex Analysis required the first LinAlg and RealAnalysis. So guess it depends on whats taught int hte courses like you said.

13. May 18, 2005

### The_Brain

Although I have not asked so I cannot say for sure, I am almost positive the professors will say it's arbitrary; I wanted to ask students who may have gone through a similar decision. Mathwonk, it's interesting that you say "real analysis" should be a course on infinite dimensional linear algebra and measure theory because the course I just took, "Theoretical Linear Algebra and Multivariable Calculus", went over these although probably not the the depth you are suggesting. We also did Lesbegue integrals, Funademental Theorem of Algebra, etc... The decision is between taking an advanced, axiomatic linear algebra and some other math and then complex/real analysis the next year or vice versa. However it seems that this thread has come the same conclusion as everything else: it doesn't really matter.

14. May 20, 2005

### The Reverend BigBoa

Have you considered talking to some of the profs/instructors in the math and physics departments at your school to see what they suggest? I don't know how well you know them, but they generally seem to be pretty helpful if they're decent profs in the first place.

15. May 21, 2005

### mathwonk

i was not being sarcastic. I went to harvard in the early 1960's and the courses there were as follows [for students planning to go to math grad school]:

Freshman year math 11: a rigorous proof based course ala Spivak calculus, convergence of sequences and series including complex numbers for the first semester, and moving into coordinate free linear algebra and inner product spaces and hilbert space the second semester, and concepts of topology such as compactness, completeness, and proofs of existence of maxima and minima.

sophomore year: math 55: honors advanced calculus ala Dieudonne, infinite dimensional calculus in banach space (frechet derivative), including content theory and integration, plus intro to differentiable manifolds, spectral theory for compact hermitian operators and application to sturm - liouville theory.

junior year: real analysis math 212: i.e. abstract and lebesgue measure and integration, and general topology, a la Halmos.
complex analysis math 213: a la Ahlfors (who taught the course)

senior year: grad algebra math 250 ala Lang's algebra
grad alg top math 272: al la Spanier.

so to me "real analysis means the content of Royden, or Halmos, or Wheeden? and Zygmund. or lang's real analysis.

i was guessing he might mean something like "baby reals" al la Rudin's little book, which is commonly referred to as "baby rudin". I.e. elementary stuff like metric spaces and convergence of sequences and series, such as we covered actually in freshman calc.

by the way, the program i described was only for people with NO calculus coming in. those who already knew calculus started with math 55, and in fact math 55 was more than 1/2 freshmen.

one remark: if you are at an ivy league school you need to get some advice out of those guys and gals who work there on the faculty, and not let them get away with ignoring your needs, the way the faculty did back in my day. we were wimps. do not be like that. go meet them and ask some questions, and insist on some answers.

there is a lot of pressure at ivy schools and some junior faculty try to make time for research by shortchanging their undergrad teaching and advising duties. so be polite but persistent.

when i was an undergrad, my math advisor (a grad student) declined ever to make an appointment to see and advise me, and when i reached him by phone, he asked me to send my study card to him by mail to sign. he offered no advice at all on my program.

i hope things are not still that bad.

good luck.

Last edited: May 22, 2005
16. May 22, 2005

### mathwonk

I did not at first read that you were at an ivy school. that makes all comparisons with most schools out of line. I.e. most places they refer to basic theoretical ideas such as convergence of sequences, or metric spaces, as "real analysis", and reserve it for senior math majors. Whereas in math 11 we started with this stuff as freshmen.

The course you took sounds very sophisticated. It is hard to know what to advise from a distance and from a perspective of teaching much less ambitious courses for decades. It is also hard to know how approrpiate those high level courses are for you personally, without knowing you. For me for example, those super high powered courses at harvard were both heady and exciting, but simultaneously left me without a sound grounding in elementary calculus, which would have been useful.

I am still glad I took them because at harvard in the 60's the choice was: either take high powered courses and skip low level stuff, which you then fill in later while teaching, or take lower level courses and never get the high powered version, which is a loss you have trouble ever filling in.

Please make an effort to talk to your faculty, and maybe even more helpful, to older math and physics majors there at the school.