Math Education for the Physics Major

[PhysicsWiz4]

Dear PF, I've been a lurker for about a year now and decided I needed a bit more advice:

I'm a freshman physics major getting ready to pick classes for this spring quarter.
So far I've taken/taking:

Fall:
Multivariable Calculus: A
Honors Intro Mechanics: A

Winter:
Vector Calculus
Linear Algebra
Honors Intro SR and Thermo

Next quarter I'm going to take basic Differential Equations, the last mathematics course required by my major. However, having relatively few math courses under my belt makes me uneasy. I hope to be a competitive graduate school applicant, but I also want to be prepared for graduate school. My impression is that the two are mutually exclusive to a degree.

Background:
My math grades were poor in high school, ranging from a C- to a B at best. I was in the honors track all four years, advised to leave it because of my grades, but stuck with it out of stubbornness.
I didn't know what "physics" was until my senior year, when I chose to take AP Physics C as my first physics course (skipping the prerequisite) because I wanted to be with my friends. I also heard that since calculus was involved, it might help Calc BC topics stick.
I learned that "physics" referred to all that had interested me my whole life (QM, SR, E&M, etc), and the calculus basis led to my first A in math.

The Dilemma:
I'd like to take as much mathematics as I can to prepare myself for graduate school (and lifelong physics learning in general), but my GPA would probably suffer as a result because this would mean taking an extra load. However, I feel as though without the math class my understanding will never be what it could potentially be with the class.

My question is: which math do I need to learn in math classes, which math can I pick up from my physics classes, and which math can I learn on my own?

The reason I'm asking this question now is because ultimately this boils down to: Should I take Advanced Calculus spring quarter or not? (In addition to basic DEs)
Description of Advanced Calculus: Introduction to the rigorous treatment of abstract mathematical analysis. Proofs in mathematics, induction, sets, cardinality; real number system, theory of convergence of sequences

My options for sophomore year:
PDEs: Prerequisite is only basic DEs, so I already plan to take this course.
This is where it gets messy though -
Real Analysis: Prereq is advanced calc
Complex Analysis: Prereq is real analysis
Numerical Analysis: Prereq is advanced calc
Fourier Analysis: Prereq is advanced calc
Probability: Prereq is real analysis
Stochastic Processes: prereq is probability
Topology: prereq is real analysis
Differential Geometry: prereq is real analysis
Algebra: it starts in fall, and the prereq is abstract linear algebra, which isn't offered in spring, so I can't take it in sophomore year.

Advanced calc/real analysis doesn't sound that pleasant to me, but it's a necessary evil if I want to take any of the classes listed above, except PDEs. I've also heard real analysis isn't all that beneficial to physics. However, I also hear probability and complex analysis are useful. I can't see myself getting an A in advanced calc/real analysis, since I hear it's quite hard. It's unfortunate to have so many math classes with that as a prereq, but I'll feel disadvantaged if I don't take those classes. Please don't think that I don't want to risk my GPA in favor of taking more advanced classes. I'm already planning on taking five other classes spring quarter at reasonable risk to my GPA.

Other plans:
Self study the necessary mathematics on my own time/ audit the classes. The only problem is that I can only motivate myself enough to read the text and to do only some of the problems. However, my work ethic has been improving every year, and maybe I'll be able to find the inclination to do more problems by next summer. I've been looking at books recently, specifically
Visual Complex Analysis by Needham
Complex Variables and Applications by Brown and Churchill. Does anyone have experience learning from these books? I've also read that probability and the like is good for grad schools. Would they want me to have taken a specific class in probability? Or would teaching myself suffice? If someone could recommend a good inexpensive book I'd be grateful. I'm also looking at Schaum's outline of tensor calculus. I'm aware that there are also online classes I could also watch like the OCW but I like having a physical book to read.

The physics department offers a junior level class called mathematical methods and uses the book by Mary L. Boas. I already own the book, as recommended by ZapperZ (thank you) and read it from time to time. While I am certain that the book will give me the tools I need, can I be confident that it's a complete substitute for more in depth math classes? I guess this comes down to breadth vs depth, and we all have a limited time in college (as well as in life).


Take math classes in physics graduate school. I actually know very little about this. How does picking classes in grad school work? I know that there is the core physics classes to take, but how about electives? Is it possible to take math classes (whether upper division or grad level) while in physics grad school? If so, I'll be much more at ease.

I apologize for an extremely drawn out post but I thought it'd be helpful to provide the extra information I did. I'm also quite nervous because the accessibility to all those math courses depends on whether or not I take advanced calculus this quarter, so it's a "now or never" kind of situation =/ Thank you all in advance for your help!

 

[nlsherrill]

I think you are over thinking here. Really, for the undergraduate physics major one should know how to deal with:

Calculus
Linear Algebra
Differential equations
Probability/Statistics

I suppose it really depends on what you want to study in graduate school. Some areas of physics are somewhat light on advanced math, while other areas such as field theory are very heavy on advanced mathematics. What do you want to study?

In any event, for grad school physics(I hear) you need to be very proficient at multivariable calculus, ODEs, and PDEs. By proficient I mean able to use concepts in these topics very well(not prove them)

As a physics major in real analysis right now, I will tell you it has absolutely zero relevance to any physics I have taken or read about so far. Apparently it has use in graduate level theoretical physics courses, though. I have also talked with seniors who have taken analysis and algebra, and they say it hasnt popped up anywhere in quantum or e&m

So in all, it kind of depends what you want to study. Also, frequently in physics courses the professor(or book) will teach you necessary mathematics along the way.

 

[PhysicsWiz4]

Thank you for your reply! It was just what I was looking for. I think I'll be satisfied with just the PDEs class and find time to learn real analysis/algebra at my leisure. After all, if they aren't much use in undergrad physics, I could use the time to do other things. Since I'm only in my first year I can't say what I'm most interested in, maybe particle physics. But even if it uses advanced math, I'll learn enough of it along the way huh? Thanks so much ^^

An edit: can someone enlighten me on what the math situation is like in grad school? Is it all physics classes? Or is there an option to take math electives and the like?

 

[PhysicsWiz4]

One Year Later - Update!

Hi, everyone! I'm back with an update a year for those people who wonder "What ever happened to that person?" I hope such an update doesn't break any forum rules.

• I ended up not taking Advanced Calculus that spring ('12)
• I ended up not taking PDEs this sophomore year ('13)
• I ended up taking math methods from the physics department in fall ('12)
• I am taking Advanced Calculus this spring ('13)

I am going to be using Elementary Real Analysis by Thomson, Bruckner, and Bruckner. Does anybody have any experience using this book? The reviews on amazon seem favorable, but does anyone want to add anything?

I was originally taking real analysis in order to take complex analysis afterward, but I realized the physics department offers a complex variables course. Would complex analysis add anything that I would be missing from a complex variables course?

I am still taking real analysis because I find it interesting in its own right and hope that it will become relevant in later physics. I'm leaning toward the theoretical side so I'm also planning on taking algebra at some point in my undergraduate career.

Another question, how much would real analysis help me in understanding general relativity/differential geometry? I plan on having a year's worth of RA experience before taking a GR course in '14.

 

[vela]

I don't think you'll find taking real analysis beforehand will have helped you a lot when you take GR next year. That said, it never hurts to take math courses even if they're not directly relevant to your immediate physics education. It's useful to be exposed to techniques and ideas you will not typically see just taking physics courses.

 

[espen180]

Another question, how much would real analysis help me in understanding general relativity/differential geometry? I plan on having a year's worth of RA experience before taking a GR course in '14.

Real analysis will not help you very much in and of itself. However, it gives you the required prerequisites to take two VERY important courses: topology and differential geometry. The latter is prerequisite for understanding current theoretical physics. The former prepares you for manifold theory, geometric topology and algebraic topology. These have applications mainly in statistical physics and quantum field theory. Since you are leaning toward theory, I expect you will deal a lot with at least one of these.