Maths to fill the gaps

[LightPhoton]

TL;DR

Asking for sources and general mathematical topics useful for grad school in physics.

I have around one free year before my grad school starts. I have completed classical mechanics and 1 semester worth of QM and EM, currently on stat mech. There are some things that i would call as "mathematical gaps", while others aren't. For example

1) In classical mechanics saying $$\partial_q \dot q=0$$ is highly unsatisfying until and unless one invokes differential geometry. Same is true for being able to switch from $$(q,\dot q) \rightarrow (q,p)$$ and demanding $$\partial_{\dot q} p=\partial_q p=0$$
2) In quantum mechanics saying one needs to use sums in discrete cases (such as that of spin wavefunctions) and integrals in continuous cases, without any sound mathematical basis (and not even calling it a postulate of quantum mechanics; since it isn't and can be "derived" from spectral theorem).

What is not a mathematical gap:
1) Essentially everything done in electrodynamics. I won't consider being pedantic about a particular integral or convergence as a "gap", since there is nothing that is stopping me to be able to understand the logic, given the prerequisites of the course. That is, with all the mathematical prerequisites assumed to take an electrodynamics course, there is nothing that cannot be derived from scratch.


This gives me a way to summarize what i mean by mathematical gaps: Any mathematical result invoked in the course, that does not comes directly from the prerequisite of the said course, and used in crucial moments to derive the key results of the course.

Now of-course this all depends on what the course is, what it prerequisites are, and what is actually taught in the syllabus of the course. To that end, i'll assume the standard books, so for classical let it be Goldstein, for electrodynamics Jackson or Zangwill and for quantum Sakurai.


Given that i have a year, what all mathematical topics should i cover so that these mathematical gaps can be filled, not only limited to above 3-4 subjects, but relevant to whole grad school, including QFT and GR? To be clear, i am not expecting a mastery, nor the rigour of a mathematician. Simply enough knowledge to understand why this works and what is the key understanding pinning the result.

 

[renormalize]

1) In classical mechanics saying $$\partial_q \dot q=0$$ is highly unsatisfying until and unless one invokes differential geometry.

Huh? How can it be otherwise? Given the continuity conditions that physicists commonly assume to hold, ordinary/partial derivatives must commute, so:$$\partial_{q}\dot{q}\equiv\frac{\partial}{\partial q\left(t\right)}\frac{d}{dt}q\left(t\right)=\frac{d}{dt}\frac{\partial}{\partial q\left(t\right)}q\left(t\right)=\frac{d}{dt}1=0$$Why does this require differential geometry to see?

 

[Matterwave]

It is not clear to me what the gaps actually are since I don't really know what you know and don't know.

Have you tried standard mathematical methods for physics books? E.g. Arfken and Weber.

 

[Matterwave]

If you want to learn differential geometry because it's interesting, I would recommend starting with Schutz Geometrical Methods for Mathematical Physics. It's a good gentle introduction. Unfortunately, it does have some errors... but there are errata pages online you could look at.

Differential geometry is often used heavily mostly in GR. It is also heavily used in QFT (a lot of Lie theory), but often more at the foundations level. For "everyday" calculations in QFT, you're looking more at stuff like complex integration, differential equations (a lot of Greens functions), hilbert spaces, and the like.