How to start reading math books, namely bishop and goldberg's tensor analysis on manifolds

• Studying

Main Question or Discussion Point

I am a graduate student in physics. One of my biggest frustrations in my education is that I often find that my mathematical background is lacking for the work I do. Sure I can make calculations adequately, well enough to even do well in my courses, but I don't feel like I really understand what's going on. To combat this problem I have decided to learn mathematics a bit more rigorously.

At this point I would like to learn a bit of differential geometry, some abstract algebra, and some functional analysis. The problem is, I don't really know where to begin. As an undergraduate I took linear algebra, ODEs, PDEs, and vector analysis. Those were more or less the only courses I took beyond Calculus I-III. As a graduate student I have taken your typical math methods course out of Arfken and Weber. These courses have not prepared me to read a book written for mathematicians.

I recently picked up Bishop and Goldberg's Tensor Analysis on Manifolds, however the book looks quite daunting to me. It is in a language I am not entirely familiar with. My question is what are the mathematical prerequisites to begin reading a book such as that. What should I read before I ever pick up these math books? Is there a quick intro into the language of mathematicians for physicists?

I'd start with real analysis and maybe differential geometry of curves and surfaces. I don't know that there's a quick into.

Another thing you can do is try to find someone like me who knows a bit about both worlds and can translate between them to some degree.

I'm not sure what it is you're doing, but you might question whether more math is really the answer to finding the understanding you seek. There are definitely places where it will help, but it might not solve all your problems. Sometimes, I was able to do everything rigorously and solve all the textbook problems, but I still didn't feel like I really understood it. So, in math, you'll probably be confronted by some of the same problems as in physics.

I googled Bishop and Goldberg and came across this post and thought, wow, this guy has the exact same issues I did. It's remarkable, he's expressing this in almost the same way I would. I then looked at who wrote this post and realized it was me.

Anyway, if anyone else does have the same issue, I can't recommend this lecture series highly enough: