PhD paths for BS Physics vs. Applied Physics

Most physics undergrad degrees require the calculus sequence, differential equations, and usually but not always linear algebra. That’s more or less what grad schools expect and they usually have a class called “Math Methods” or similar which is “all the advanced calculus and analysis you are likely to run across”. I would not recommend taking a lot of additional math just for the sake of getting into grad school - it’s unlikely to help, and a low-ish grade could even potentially hurt you. I would recommend a class in probability and statistics, as this is useful and has seemed to have fallen into a crack.

I’m not saying don’t study math if you’re interested in math for math’s sake. I’m saying that it’s not nearly as helpful in getting into grad school as other alternatives, like research or more advanced physics classes.

Indeed, I’d say that the grad Math Methods class might be a better option than anything in the math department. It’s a subset known to be useful, it shows the student can handle a grad-paced class, and while it probably won’t eliminate the need to take it in grad school, it will make it easier, and an easy class in the first year of grad school is a big advantage in adjusting.

Furthermore, it makes little difference in selecting an undergrad physics program. If the school has a math major as well as a physics major (which is likely 99.99% of the schools with a physics major) the math courses intended for physics majors will be good enough. Finally, at that level math department courses are different than the needs of physicists. “Physics math” is about what you can calculate. “Math math” is more about what you can prove.

Math is necessary for advancement in physics, but is far from sufficient. A black hole is, after all, a mathematical (and physical) singularity where math breaks down. What’s most important in physics is physical intuition (and sometimes counter-intuition) that a student acquires by understanding how nature works and why it works certain way. The math needed in physics is all over the place (not just breadth but also depth) that no student is likely to have acquired them all in her/his undergraduate studies. That’s why most physics departments would offer a course called “Mathematical methods for physicists” or something similar. It’s only an introduction and students can obviously pursue more depth in each of the topics covered (and more) on their own or in other more in-depth courses.

Students in physics need to comprehend much more than the introductory physics taught at the freshmen and sophomore levels before moving on to advanced topics in modern physics. To get into general relativity and high energy astrophysics, physics students need to take courses (all year-long courses) in advanced mechanics at the level of Goldstein (Classical Mechanics (Goldstein) - Wikipedia), advanced electromagnetism at the level of Jackson (Classical Electrodynamics (book) - Wikipedia), not to mention advanced quantum mechanics. These courses are easily among the most difficult courses on any college campus.