Is real analysis helpful in physics?

[cytochrome]

I have tons of room for electives and I'm filling them up with math classes. I can see how complex analysis and even abstract algebra would be helpful, but would real analysis be helpful for research in the future? Other than getting a deeper understanding and proof based re-introduction to calculus, I don't see how it would help a physicist (especially when one can learn these proofs in their own time).

I'm interested in pursuing condensed matter physics or physical chemistry in grad school, and maybe even on the theoretical side. I was thinking maybe real analysis would be useful for getting more practice at mathematical writing for publishing papers, but is that about it?

 

[Simon Bridge]

By "Real analysis" you mean: the branch of mathematical analysis dealing with the real numbers and real valued functions of a real variable.

It will certainly help - though most physics courses include the specific math you'll need.
The way mathematicians and physicists approach math is a bit different. If you enjoy the formalism of pure math, then this will be good for you pretty much anywhere is physics.

 

[mathwonk]

i only log in because it is late at night and i have nothing else to do, and i hope someone more knowledgeable will chime in. but i think it depends which aspects of real analysis you refer to. i find it hard to believe that careful studies of how to complete the rational numbers to the real numbers by taking cauchy sequences has much value in physics, nor other foundational topics, but I do think it likely that integration theory and Fourier analysis are very useful. in particular i think groups and symmetry are fundamental in physics and Fourier analysis is about analysis on groups.

 

[Robert1986]

Where I did my undergrad, there was a one year analysis sequence. The first half we did lots of stuff that made absolutely no sense to me at the time: a bunch of open sets, closed sets, closure operators, neighbourhoods (essentially very basic topology in R^n.) Then we started talking about continuous functions, series and sequences. None of this part was brand-new to me, but it was a deeper look at series and sequences than I did in calc. Essentially, all of the proofs that my calc book said were too advanced to calculus were proven in this class (at least the ones dealing with series and sequences.) The last three weeks or so was all about Fourier stuff.

The second semester we dealt with differentiation and integration. Both done in general R^p spaces (though we only integrated over R^p, not anything more general than that.) This was very useful to me for several reasons. For example, I never really understood the whole change of variables thing from calc 3, and the theory helped me understand it more. Also, you will really start to see the derivative as a linear approximation to the difference of a function evaluated at two points and this makes some numerical approximation theory a little easier to understand. Also, in the book we used, there were lots of "projects". The projects were harder than homework, and broken down into several parts and you proved "real math things". For example, one was proving that every ODE has a unique solution. (Well, it wasn't every ODE, it was a particular class of ODEs, I forget which, but the proof was non-trivial.) During the course, we re-visted some Fourier stuff and the last three weeks was Lebesgueish stuff.So, if this course is like the one you can take, then it seems like the second one might be helpful. The first one probably isn't as helpful, but you need it to do the second one.

 

[bcrowell]

I have tons of room for electives and I'm filling them up with math classes. I can see how complex analysis and even abstract algebra would be helpful, but would real analysis be helpful for research in the future?

It might be useful for some theorists, but the chances are you would never use it. If you have tons of room for electives, why don't you take some liberal arts classes?

 

[ahsanxr]

Depends on your interests, but at worst it'll improve your mathematical maturity.

The biggest advantage of classes like Real Analysis, Abstract Algebra and Topology for physicists is not that the material in those classes is directly useful, but instead is necessary for the advanced stuff which is. For example, if you ever want to learn about the mathematics behind quantum mechanics rigorously i.e functional analysis, you will certainly need a background in real analysis. If you want to learn about Algebraic Topology and Differential Geometry which have applications in string theory (I hear at least) and general relativity a knowledge of general topology is very helpful. And obviously lie groups and representation theory is an important topic which requires knowledge of abstract algebra.

So if you're interested in mathematical physics it's a must. If you're interested in theory, you should still take them if you have the time, but don't give them preference over your physics courses. If you're going into experimental though, it's going to be almost completely useless.