[relaytheurgency]

I can't speak to a doctoral level of knowledge, it's probably not really formalized and will differ on your specialization. But just for my bachelor's degree I took the following courses: Calculus 1, Calculus 2, Vector Calculus, Elementary Linear Algebra, Differential Equations, Linear Algebra.

Those are just the courses specifically in the mathematics department. You also cover mathematics within the physics courses themselves of course. Especially in quantum mechanics and E&M. You also might be required to take more math depending on the structure of your degree program, I did a focus in chemistry as well so took more chemistry and less math than other students.

I also can't say how many hours I spent on this. But the overwhelming amount of my homework time, every night, was spent writing proofs and solving mathematical equations. To a lay person physics work probably looks no different than mathematics. It was all math all the time :) Sometimes I would have a homework assignment that was only a few "problems" that would take me a dozen hours to solve. As for aptitude, I was probably in the middle amongst other physics students, but that group overall was above average already.

[wly_cdgr]

Thank you!