Courses What additional math course should I take?

[Shackleford]

Physics major with math minor. I need one 4000-level math course. Which should I pick?


MATH 4320: Introduction to Stochastic Processes
Cr. 3. (3-0). Prerequisite: MATH 3338. Generating functions, discrete and continuous versions of Poisson and Markov processes, branching and renewal processes, introduction to stochastic calculus and diffusion.

4331;4332: Introduction to Real Analysis
Cr. 3 per semester. (3-0). Prerequisite: MATH 3334 or consent of instructor. Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals.

MATH 4333: Advanced Abstract Algebra
Cr. 3. (3-0). Prerequisites: MATH 3330 and consent of instructor. Direct products, Sylow theory, ideals, extensions of rings, factorization of ring elements, modules, and Galois theory.

MATH 4335;4336: Partial Differential Equations
Cr. 3 per semester. (3-0). Prerequisite: MATH 3331. Existence and uniqueness for Cauchy and Dirichlet problems; classification of equations; potential-theoretic methods; other topics at the discretion of the instructor.

MATH 4337: Topology
Cr. 3. (3-0). Prerequisite: MATH 3333 or MATH 3334 or consent of instructor. Metric spaces, completeness, general topological spaces, continuity, compactness, connectedness.

MATH 4340: Nonlinear Dynamics and Chaos
Cr. 3. (3-0). Prerequisite: MATH 3331 or consent of instructor. Dynamical systems associated with one-dimensional maps of the interval and the circle; elementary bifurcation theory; modeling of real phenomena.

MATH 4350;4351: Differential Geometry
Cr. 3 per semester. (3-0). Prerequisites: MATH 2433 and MATH 2331 (formerly 2431) or equivalent. Frenet frames, metric tensors, Christoffel symbols, Gaussian curvature, differential forms, moving frames, Euler characteristics, the Gauss-Bonnet theorem and the Euler-Poincare index theorem.

MATH 4355: Mathematics of Signal Representation
Cr. 3. (3-0). Prerequisites: MATH 2433 and either MATH 2331 (formerly 2431) or MATH 3321. Fourier series of real-valued functions, the integral Fourier transform, time-invariant linear systems, band-limited and time-limited signals, filtering and its connection with Fourier inversion, Shannon's sampling theorem, discrete and fast Fourier transforms, relationship with signal processing.

MATH 4360: Integral Equations
Cr. 3. (3-0). Prerequisites: MATH 3331 and MATH 3334. Relation to differential equations; Fredholm, Hilbert-Schmidt, and Volterra type equations; special devices and approximation methods.

MATH 4362: Theory of Ordinary Differential Equations
Cr. 3. (3-0). Prerequisites: MATH 3331 and MATH 3334. Existence, uniqueness, and continuity of solutions of single equations and systems of equations; other topics at the discretion of the instructor.

MATH 4364;4365: Numerical Analysis
Cr. 3 per semester. (3-0). Prerequisites: MATH 2331 (formerly 2431), MATH 3331; COSC 1301 or COSC 2101 or equivalent; or consent of instructor. Topics selected from numerical linear algebra, approximation of functions, numerical integration and differentiation, interpolation, approximate solutions of ordinary and partial differential equations, Fourier methods, optimization.

MATH 4370: Mathematics of Financial Derivatives
Cr. 3. (3-0). Prerequisites: MATH 2433 and either MATH 3338 or MATH 3341. Stochastic processes for modeling the dynamics of returns of financial instruments and commodities. Use of Ito's calculus and Black-Scholes Model to value contingent claims and real options in capital budgeting.

MATH 4377;4378: Advanced Linear Algebra
Cr. 3 per semester. (3-0). Prerequisites: MATH 2331 (formerly 2431) and a minimum of three semester hours of 3000-level mathematics. Matrices, eigen-values, and canonical forms.

MATH 4380: A Mathematical Introduction to Options
Cr. 3. (3-0). Prerequisites: MATH 2433 and MATH 3338. Arbitrage-free pricing, stock price dynamics, call-put parity, Black-Scholes formula, hedging, pricing of European and American options.

MATH 4383: Number Theory
Cr. 3. (3-0). Prerequisite: MATH 3330 or consent of instructor. Perfect numbers, quadratic reciprocity, quadratic residues, algebraic numbers, and continued fractions.

 

[nicksauce]

I took a course on Non-linear dynamics and chaos, and it is possibly my favorite course at university so far. PDES is also very useful.

 

[tronter]

Take Introduction to Stochastic Processes. You will probably use the book by Ross. Its a very cool subject. Markov Processes are important in modeling real world situations (i.e probabilities of going into another state of some process).

 

[Shackleford]

The Non-linear Dynamics and Chaos course sounds interesting.

I am completely unfamiliar with Stochastic Processes.

It would make sense, though, to take PDE, since I am already taking Intro. to Partial Diff. Eq.

 

[bravernix]

I took a course on Non-linear dynamics and chaos, and it is possibly my favorite course at university so far. PDES is also very useful.

I concur with this sentiment. So much so that I am attempting to go to grad school in a field that uses a lot of this type of analysis.

 

[Shackleford]

Another important question, down the road, I think I might like to do some kind of financial work part-time while I go to school for a master's. Would this class be helpful?

MATH 4370: Mathematics of Financial Derivatives

 

[Shackleford]

Personally i would chose from (depending on how many one can chose)

Advanced Abstract Algebra
Partial Differential Equations
Topology
Differential Geometry
Integral Equations
Number Theory

Which classes would be the best for graduate school preparation?

 

[PowerIso]

Hehe you go to UH. Anyway, the professor who tends to teach Non-linear dynamics is rather cool and the class is fun. It's interesting, but it's mostly an introduction/

If you want to prepare for graduate school, I would suggest Stochastic Processes or Differential Geometry. Although the PDE class seems tempting, it's mostly a theory on PDE. It isn't very useful in regard to your major. If you do go to UH you should have some "free electives" around senior year, so if you want, you could take PDE and non-linear dynamics during those times.

Oh yea, I hear Numerical Analysis tends to be a useful skill.

 

[Shackleford]

Hehe you go to UH. Anyway, the professor who tends to teach Non-linear dynamics is rather cool and the class is fun. It's interesting, but it's mostly an introduction/

If you want to prepare for graduate school, I would suggest Stochastic Processes or Differential Geometry. Although the PDE class seems tempting, it's mostly a theory on PDE. It isn't very useful in regard to your major. If you do go to UH you should have some "free electives" around senior year, so if you want, you could take PDE and non-linear dynamics during those times.

Oh yea, I hear Numerical Analysis tends to be a useful skill.

Why the Hehe? Actually, I don't go there yet. lol. The PDE consists of two classes. I've already picked my other electives and everything, so I might do the non-linear dynamics instead of PDE altogether.

 

[zhentil]

I would say a solid course in PDE (not just basic separation of variables) and a solid course in differential geometry (that got into riemannian geometry) would make grad school a lot easier for you.

With PDE, I mean that if your intro course didn't go into the theory of Green's functions, convergence of Fourier series, etc., it might be more useful than a course on stochastic processes, which is quite elementary when taught at the undergrad level (at least at the level of Ross' book, which was mentioned).

 

[mgiddy911]

what math have you taken?
I'm sure most of those classes are hard, but Real Analysis is supposed to be one of the hardest classes to take. If you want the challenge i'd say take the analysis class, but it won't apply to very much for your physics major. I liked the class, took it because
I have a dual physics maths degree.

As far as what is most applicable, as others have said, choose from the PDE, Nonlinear Dynamics, the financial one since that's one of your interests, and the numerical analysis

 

[Shackleford]

what math have you taken?
I'm sure most of those classes are hard, but Real Analysis is supposed to be one of the hardest classes to take. If you want the challenge i'd say take the analysis class, but it won't apply to very much for your physics major. I liked the class, took it because
I have a dual physics maths degree.

As far as what is most applicable, as others have said, choose from the PDE, Nonlinear Dynamics, the financial one since that's one of your interests, and the numerical analysis

As of right now, I have Partial Differential Equations in there. That's all. I just mentioned finance because I want an "easy" job while I'm graduate school. I don't really know if a finance job would be "easy." I just assume it'd be a little easier than an engineering job or other technical job I would qualify for with a Physics BS.

 

[tim_lou]

if I were you I would take real analysis or topology. After finishing them this semester, all of mathematics come together... seriously, in physics, one deals with integrals, derivatives...etc all the time but do most of the people really know what exactly that they are doing? real analysis will tell you exactly what things are. Seriously, Real Analysis is what separates math people from normal people. It basically open up your eyes and after you finish the course, you will no longer fear really abstract mathematics. Also, ask around and see if the professors teaching the courses are any good. If not, you might be better off learning analysis yourself.

Honestly, after finishing more than half of rudin's analysis book, and some of munkre's topology book, difficult things like tensors in general relativity become manageable. Be warned, however, analysis and topology are no easy matters. PDE is an insignificant piece of dust compare to them.

 

[Shackleford]

if I were you I would take real analysis or topology. After finishing them this semester, all of mathematics come together... seriously, in physics, one deals with integrals, derivatives...etc all the time but do most of the people really know what exactly they are doing? real analysis will tell you exactly what they are. It basically open up your eyes and when you are reading really abstract mathematics book, you will no longer fear them.

Hmm. I really would like an in-depth mathematics understanding like that. This course is only an Introduction to Real Analysis. I'm still a ways off from this. Maybe next year or so.

So, you recommend Introduction to Real Analysis over the two PDE courses?

 

[tim_lou]

it depends on your personal motivation... I took PDE this semester and felt that I could've learned the material on my own; well, I had to take a PDE course as a physics major so no complaint there. Your PDE course might be better depending on the professor.

To be honest, yes, analysis would not be as important as some others in terms of calculations in real life. But as a personal thing--to understand calculus once and for all, real analysis is a must.

Anyway, as I can tell, hardcore physics majors take real analysis. Even though some of my professors are against it, the top kids take analysis. But do know that it will be a struggle. If you are worried about your grades dropping... think about it first.

Also, does your school require you to take PDE? if so, then take it. Otherwise, if you are motivated enough, learn it over the summer after analysis. Anyway, I cannot imagine learning PDE without knowing what exactly partial derivatives do... it's a shame, really...learning PDE before even knowing what volume integrals are (all the divergence, curl, stokes theorem).

I think I'll add an overview of the material covered in first semester analysis (yours might be a bit different):
basically, the course starts with general things like set theories, countability, and then it goes over the real number system--in a theoretical sense--as a completion of the rational number.

then, one learns about metric space, open sets, closed sets, limit points, compactness... topology stuff.

after the fundamentals, the real fun begin, personally, I found the fundamental stuffs the most difficult part of the course, partly because I haven't seen such notations and rigorous definitions before.

sequences, series, limits.

then continuity, limits of functions, derivatives, Riemann integrals, then sequences of functions, sets of functions (Ascoli theorem). If you work hard on the exercises, you will go over some theories about Fourier series, differential equations and other goodies.

and that's about the first semester.

 

[Shackleford]

it depends on your personal motivation... I took PDE this semester and felt that I could've learned the material on my own; well, I had to take a PDE course as a physics major so no complaint there. Your PDE course might be better depending on the professor.

To be honest, yes, analysis would not be as important as some others in terms of calculations in real life. But as a personal thing--to understand calculus once and for all, real analysis is a must.

Anyway, as I can tell, hardcore physics majors take real analysis. Even though some of my professors are against it, the top kids take analysis. But do know that it will be a struggle. If you are worried about your grades dropping... think about it first.

Also, does your school require you to take PDE? if so, then take it. Otherwise, if you are motivated enough, learn it over the summer after analysis. Anyway, I cannot imagine learning PDE without knowing what exactly partial derivatives do... it's a shame, really...learning PDE before even knowing what volume integrals are (all the divergence, curl, stokes theorem).

As a physics major, I am required to take Introduction to Partial Differential Equations. As an elective/math minor, I'm taking Introduction to Complex Analysis. Will that be just as hard as Introduction to Real Analysis?

 

[tim_lou]

Since you are required to take PDE, then take it.

As for Complex Analysis, I haven't taken that course (and I don't think I will take it) so no comment personally. But I doubt it will be as hard as real analysis.

Actually, it depends on what you mean by Complex Analysis. If you mean the kind where they go over Cauchy theorem, contour integrals.. then no, it would be as hard as analysis. Or if you are talking about Complex analysis as real analysis done using complex numbers, then heck yes... though I doubt it refers to the latter.

 

[Shackleford]

Since you are required to take PDE, then take it.

As for Complex Analysis, I haven't taken that course (and I don't think I will take it) so no comment personally. But I doubt it will be as hard as real analysis.

I am required to take Introduction to Partial Differential Equations. That's an entirely different class than the TWO PDE mentioned here.

 

[zhentil]

if I were you I would take real analysis or topology. After finishing them this semester, all of mathematics come together... seriously, in physics, one deals with integrals, derivatives...etc all the time but do most of the people really know what exactly that they are doing? real analysis will tell you exactly what things are. Seriously, Real Analysis is what separates math people from normal people. It basically open up your eyes and after you finish the course, you will no longer fear really abstract mathematics. Also, ask around and see if the professors teaching the courses are any good. If not, you might be better off learning analysis yourself.

Honestly, after finishing more than half of rudin's analysis book, and some of munkre's topology book, difficult things like tensors in general relativity become manageable. Be warned, however, analysis and topology are no easy matters. PDE is an insignificant piece of dust compare to them.

It depends on what your PDE course is. Baby Rudin is pedantic compared to Evans' PDE book.

 

[tim_lou]

I am required to take Introduction to Partial Differential Equations. That's an entirely different class than the TWO PDE mentioned here.

Do you mean the stochastic process and chaos theory class?

As I haven't taken those classes, I cannot comment on them. Anyway, I say that those class aren't as fundamentally important... meaning you could probably learn them on your own or they are not prerequisites of many higher level course. One thing about real analysis is that you learn the ability to read math, which I think beats any kind of math application classes.

 

[Shackleford]

Do you mean the stochastic process and chaos theory class?

As I haven't taken those classes, I cannot comment on them. Anyway, I say that those class aren't as fundamentally important... meaning you could probably learn them on your own or they are not prerequisites of many higher level course. One thing about real analysis is that you learn the ability to read math, which I think beats any kind of math application classes.

I am required to take this course:

MATH 3363: Introduction to Partial Differential Equations
Cr. 3. (3-0). Prerequisites: MATH 2433 and either MATH 3321 or MATH 3331. Partial differential equations and boundary value problems, Fourier series, the heat equation, vibrations of continuous systems, the potential equation, spectral methods.

If I throw out the two 4000-level PDE courses, I'll have to pick a different 4000-level MATH course, and I can pick another elective.

 

[FrogPad]

With the right book you can always self teach yourself nonlinear dynamics. If you are interested, check out https://www.amazon.com/dp/0738204536/?tag=pfamazon01-20 book -- it is extremely easy (and fun) to follow. The book is a delight to read!

Some advice I've tried to follow when scheduling my final classes is to choose a group of great professors, and then take one of the offered courses that needs a classroom to grasp the material.

 

[leon1127]

Advance linear algebra will get you so deep in physics... deep as the quantum world...

 

[Shackleford]

So, maybe Intro. to Real Analysis and Advanced Linear Algebra?

How hard will the Intro. Real Analysis course be?

 

[PowerIso]

Intro Analysis can be rather difficult if it's your first real exposer to a proof writing class. I wouldn't say it comes really easy, but some people seem to get it better than others. If you really only have the required math courses you need for physics, it might be a bit tough for you, especially if you have no idea what-so-ever how to write a proof. So if you do plan to take the more abstract math courses, I suggest taking one or two lower level math classes to learn how to write proofs.

My choices still stand as S process, PDE. Although advance linear algebra helps, I really doubt you'll use it to much in your undergrad studies, but if you plan to study things at the quantum level, it's a nice tool.

 

[egypte]

I find dynamical systems / Bifurcation theorie and chaos a very interesting course. It gives a total other view on differential equations. (I study physics in belgium)

http://en.wikipedia.org/wiki/Dynamical_system

 

[mgiddy911]

Real analysis can be very hard. I speak having just taken it last semester. It was my first proof oriented class, bad was very challenging, but also the most rewarded I have felt from any math class. Sure you will not use it much in physics, but it would sure look good to ave taken it, and even better to have done very well in it. It is a fairly abstract class, so it will show you can handle classes of that nature and caliber. I would be a good peparation for grad. level math classes simply speaking in terms of rigor. Its also nice to have a math class where you don't just plug and chug your way through DE's, Matrices, or integrals.

 

[sx70]

To confuse things further... If you think you'll do anything experimental take the Mathematics of Signal Representation. Speaking from ~25 years of work experience this stuff is incredibly useful all over physics and engineering in general. You might pick this up on your own, but I always find it useful to have a formal class so I don't skip over important points.

 

[Shackleford]

Ok, guys, I decided to resurrect this thread. Heh. I plan on going to graduate school for physics after my BS. I need one 4000-level math course to complete my math minor.

MATH 4320: Introduction to Stochastic Processes
Cr. 3. (3-0). Prerequisite: MATH 3338. Generating functions, discrete and continuous versions of Poisson and Markov processes, branching and renewal processes, introduction to stochastic calculus and diffusion.

4331;4332: Introduction to Real Analysis
Cr. 3 per semester. (3-0). Prerequisite: MATH 3334 or consent of instructor. Properties of continuous functions, partial differentiation, line integrals, improper integrals, infinite series, and Stieltjes integrals.

MATH 4333: Advanced Abstract Algebra
Cr. 3. (3-0). Prerequisites: MATH 3330 and consent of instructor. Direct products, Sylow theory, ideals, extensions of rings, factorization of ring elements, modules, and Galois theory.

MATH 4335;4336: Partial Differential Equations
Cr. 3 per semester. (3-0). Prerequisite: MATH 3331. Existence and uniqueness for Cauchy and Dirichlet problems; classification of equations; potential-theoretic methods; other topics at the discretion of the instructor.

MATH 4337: Topology
Cr. 3. (3-0). Prerequisite: MATH 3333 or MATH 3334 or consent of instructor. Metric spaces, completeness, general topological spaces, continuity, compactness, connectedness.

MATH 4340: Nonlinear Dynamics and Chaos
Cr. 3. (3-0). Prerequisite: MATH 3331 or consent of instructor. Dynamical systems associated with one-dimensional maps of the interval and the circle; elementary bifurcation theory; modeling of real phenomena.

MATH 4350;4351: Differential Geometry
Cr. 3 per semester. (3-0). Prerequisites: MATH 2433 and MATH 2331 (formerly 2431) or equivalent. Frenet frames, metric tensors, Christoffel symbols, Gaussian curvature, differential forms, moving frames, Euler characteristics, the Gauss-Bonnet theorem and the Euler-Poincare index theorem.

MATH 4355: Mathematics of Signal Representation
Cr. 3. (3-0). Prerequisites: MATH 2433 and either MATH 2331 (formerly 2431) or MATH 3321. Fourier series of real-valued functions, the integral Fourier transform, time-invariant linear systems, band-limited and time-limited signals, filtering and its connection with Fourier inversion, Shannon's sampling theorem, discrete and fast Fourier transforms, relationship with signal processing.

MATH 4360: Integral Equations
Cr. 3. (3-0). Prerequisites: MATH 3331 and MATH 3334. Relation to differential equations; Fredholm, Hilbert-Schmidt, and Volterra type equations; special devices and approximation methods.

MATH 4362: Theory of Ordinary Differential Equations
Cr. 3. (3-0). Prerequisites: MATH 3331 and MATH 3334. Existence, uniqueness, and continuity of solutions of single equations and systems of equations; other topics at the discretion of the instructor.

MATH 4364;4365: Numerical Analysis
Cr. 3 per semester. (3-0). Prerequisites: MATH 2331 (formerly 2431), MATH 3331; COSC 1301 or COSC 2101 or equivalent; or consent of instructor. Topics selected from numerical linear algebra, approximation of functions, numerical integration and differentiation, interpolation, approximate solutions of ordinary and partial differential equations, Fourier methods, optimization.

MATH 4370: Mathematics of Financial Derivatives
Cr. 3. (3-0). Prerequisites: MATH 2433 and either MATH 3338 or MATH 3341. Stochastic processes for modeling the dynamics of returns of financial instruments and commodities. Use of Ito's calculus and Black-Scholes Model to value contingent claims and real options in capital budgeting.

MATH 4377: Advanced Linear Algebra I
Cr. 3. (3-0). Prerequisites: MATH 2331 and a minimum of three semester hours of 3000-level mathematics. Linear systems of equations, matrices, determinants, vector spaces and linear transformations, eigenvalues and eigenvectors.

4378: Advanced Linear Algebra II
Cr. 3. (3-0). Prerequisite: MATH 4377. Similarity of matrices, diagonalization, hermitian and positive definite matrices, normal matrices, and canonical forms, with applications.

MATH 4380: A Mathematical Introduction to Options
Cr. 3. (3-0). Prerequisites: MATH 2433 and MATH 3338. Arbitrage-free pricing, stock price dynamics, call-put parity, Black-Scholes formula, hedging, pricing of European and American options.

MATH 4383: Number Theory
Cr. 3. (3-0). Prerequisite: MATH 3330 or consent of instructor. Perfect numbers, quadratic reciprocity, quadratic residues, algebraic numbers, and continued fractions.

MATH 4385;4386: Mathematical Statistics
Cr. 3 per semester. (3-0). Prerequisite: MATH 3339 or equivalent. Linear models-estimation, testing and application to designs of experiments, nonparametric statistical models.

MATH 4389: Survey of Undergraduate Mathematics
Cr. 3. (3-0). Prerequisites: MATH 3330, 3331, 3333, and three hours of 4000-level Mathematics. A review of some of the most important topics in the undergraduate mathematics curriculum.

 

[Hunterbender]

I would choose between partial differential equations or differential geometry. I would keep away from real anay. (just because there isn't much use in physics)

 

[Shackleford]

I would choose between partial differential equations or differential geometry. I would keep away from real anay. (just because there isn't much use in physics)

Would just one semester be all right even though they have two semesters of PDEs?

 

[Nolen Ryba]

My suggestion is MATH 4355: Mathematics of Signal Representation. Why? Because I think it will pack the most punch for the money. The idea of a signal is incredibly useful and unifying, and you don't really have go into full rigor to understand things conceptually. I was lucky to learn these ideas outside of class early on and I can hardly think of any more useful in understanding physics in general. Think of just a few examples of physical things that are signals:

position of particle(s) as function of time
string
membrane
sound field
electromagnetic field
voltage as function of time at a point
temperature field
any time series data

A few places where these ideas pop up:

Heisenburg uncertainty principle (a property of the Fourier transform which applies to any signal)
decomposition of string, membrane, sound field, quantum wavefunction, or EM field into modes
condensed matter (phonons)
electronics/communications
spectroscopy
imaging (CT scans, holography, etc)
system identification and control
general interpretation and storage of data

As far as I can tell, every branch of physics uses these ideas in a nontrivial way. In math, this ties together linear algebra, functional analysis, and ODEs/PDEs.

I don't see how the other courses could be this useful. Things like real analysis and (point-set) topology are good to know, but you'll get very little working knowledge out of the entire semester- mainly you'll learn how to make what you already know rigorous. If you'll immediately be doing research in GR you should probably take differential geometry but otherwise it'll give you little milage. The linear algebra and differential equations courses are probably useful, but I think the signals course would be better- for example, it will make those courses easier to learn on your own later. The nonlinear dynamics course would also be useful, but I still think the signals would be better- for NLD you could pick up a copy of Strogatz and learn it on your own. The abstract algebra and number theory courses might be fun (if you like that sort of thing) but are more or less useless for physics.

I don't know much about the financial stuff or how you might use it, so I can't comment there. Maybe one of those courses would be worth more to you than my suggestion- I don't know.

 

[Shackleford]

Hmm. Each suggestion has validity.

 

[Nolen Ryba]

Hmm. Each suggestion has validity.

Nope; just mine.