# What mathematics do undergraduate physics majors need

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• shinobi20

#### shinobi20

I have already studied Single and Multi-Variable Calculus, Linear Algebra and Differential Equations. I am self studying real analysis now and I find it hard, do physics majors really need to study real analysis? I am studying this because I just can't seem to remember a lot of what I've studied if I don't see the rigorous proof of those maths... Any suggestions?

You need complex analysis as well.

• Real/Complex Analysis
• Differential Equations
• Linear Algebra
• Calculus
• Differential Geometry
• Topology

• Real/Complex Analysis
• Differential Equations
• Linear Algebra
• Calculus
• Differential Geometry
• Topology

What do you need Topology and Differential geometry for?

I would say you definitely need calculus, ODE and PDE, linear algebra, and complex analysis. Most undergraduate curricula have a course or two in mathematical methods in physics, which covers all the math you'll need as an undergrad. I recommend "Mathematical Methods in the Physical Sciences" by Mary Boas. The book basically covers all math in undergrad physics, and also a lot of graduate-level physics (tensor analysis for relativity, etc...)

I don't think you need topology and differential geometry either. Look at requirements of various universities (and frankly, that should have been the 1st thing checked) and I would be surprised if any required this. I'd be shocked if more than a handful did.

MIT's requirement when I was there was the calculus sequence + differential equations + two courses.

But how should a physics major study real analysis? Should it be the same as mathematicians, what I mean is that, knowing how to prove every theorems and exercises in a real analysis book?

Doing things in a mathematician's style with proofs is more or less a defining quality of real analysis. If it didn't have that stuff, it would be calculus, not real analysis. You don't have to know how to prove EVERY theorem. Some have fairly nasty proofs. But at least the way I did it was to know how to prove most of the theorems and try to get some feel for the ones whose proofs were too complicated to just bang out on demand. Not every exercise either. Something like 4-5 exercises per problem set is typical.

Also, you shouldn't try to memorize the proofs as they are written. Just memorize the ideas that lead to the proofs.

• Linear Algebra

In which areas (except Quantum Mechanics) does extensive knowledge of it come handy?
I would prefer to take a Linear Algebra course in a later semester (before a Quantum Mechanics course), but I could take it earlier if it's helpful.

To the OP: If you want to know what mathematics you might need for an undergraduate physics major, you may start by looking at the topics covered in Mary Boas's "Mathematical Methods in the Physical Sciences" text.

Zz.

The math you need depends on the area you are going to study.

For example: Fluid mechanics is going to really work your calculus, up to at least multi-dimension partial differential equations. Probably you will want to know about curvilinear coordinates. And possibly you will want matrix algebra. And the computational parts will get you into a variety of algebra, discrete math, and finite difference or finite element mathematical methods.

Electronic circuits may cause you to want some topology. Maybe quite simple topology such as introductory graph theory. It will also likely cause you to want some formal logic.

Crystallography will likely get you into some interesting geometry, group theory, and again some matrix methods. And again probably some computational methods.

If you want to study general relativity then you want differential geometry.

And if you are doing quantum mechanical anything, you are going to be getting into matrix algebra, group theory, topology, computational methods, and just loads of other things.

Math is a tool. You should try to get the best tool set you can manage. You probably need to focus on the areas you are specifically working on rather than studying all the math you can find. Unless you are mutant smart. So look at the course catalog for your school and find the courses you are interested in for later years. Look at the math they use. Plan ahead and get the tools you will need. But don't close doors you can keep open.

Example: I mentioned group theory for crystals. But it is often useful in any situation in which you find symmetry. And symmetry exists in a large variety of problems in physics. So knowing group theory, at least a term's worth, will give you one more quite powerful tool in your tool box.

Obviously you'll need calculus up through ordinary differential equations, linear algebra, and numerical methods. I've found, though, that those seem to be the only typical hard and fast requirements, the rest being just a certain number of credits in junior, senior, and advanced-level math courses.

The physics electives you choose will dictate what exactly you should take.

The only strictly required math courses for my physics degree are calculus 1-3, ordinary differential equations, and linear algebra. So far I've taken calc 1, calc 2, and differential equations (at the 200 level). I'm currently in calc 3. Linear algebra will be next semester. I want to continue in math though, so I'm planning to declare a math minor. I'm in the same situation...trying to decide what to take.

This is what I've got in mind currently: Applied linear algebra, ordinary differential equations (400 level), partial differential equations, and differential geometry.

Aside from those, I'll have space for 1-2 more math classes. I'm considering taking both real analysis and complex analysis. I've also considered a probability theory class, but I'm not sure how necessary that really is for physics. I know it's important in some areas. A lot of you are mentioning topology as well, so now I'm considering that.

Sometimes I think I should just declare a double major in physics and math.

And that is why Mary Boas, in the Preface of her book, said that she thinks that often times, a physics major needs more math than a math major. That was one of the impetus for her to write her classic text.

Zz.

And that is why Mary Boas, in the Preface of her book, said that she thinks that often times, a physics major needs more math than a math major. That was one of the impetus for her to write her classic text.

Zz.

Heh. It's sort of like, who needs more saws? A carpenter or a metal smith?