Help me use my library of textbooks to form a study plan

[brpetrucci]

Hi everyone! Hope your week is going well. I'm an ex-physics and math student, now getting my PhD in mathematical biology, and I've recently come back to the subjects because I miss them and feel like it'd be fun to get proficient in some of this again. I've been mostly working on building my way up to GR in physics since that's what I'm most interested in, but in the case of math I kinda wanted to focus on making the best use of the textbooks I already have (some of which I never touched :d) to brush up on analysis, linear algebra, and (eventually, hopefully) differential geometry. I wanted to check with y'all what you thought of my reading order here, and if you'd recommend any intermediaries if a jump between two books is too much. I'm ok buying some new books eventually, just trying to structure my study around what I already have to start.

The plan would be to start with Spivak's Calculus, then Apostol's Calculus Vol. 2. I don't know why I only have 2, but I assume past me just figured that having gone through Spivak, Apostol's volume on Linear algebra and multivariate stuff would be a more useful follow up. Then, in no particular order I wanted to go through Rudin's Principles of Mathematical Analysis, and Hoffman and Kunze's Linear Algebra. I also have Apostol's Mathematical Analysis, which I might read after baby Rudin as well, not sure if that's supposed to be a level above or the same. Finally, and I believe there's a jump here--though I'd love to be wrong--, I'd go through O'Neill's Elementary Differential Geometry. I assume I'd need to first read some topology book or something of the sort, would love to hear what you guys think. This basically exhausts my math textbooks, outside of Piskunov's Differential and Integral Calculus which I've mostly used for references up to this point, maybe I'll read it.

Again, I have no particular objective here besides just learning and having fun! I guess eventually it would be nice to have a good enough understanding of DG that I could understand the more mathematical-view texts on GR, maybe. In any case, I have no time limit and would love any recommendations. Thanks in advance!

 

[MidgetDwarf]

A nice intro to Topology is the topology book written by Simmons. It was recommended to me by Mathwonk, and I covered most of the book. A really nice, gentle, and interesting read. I liked that Simmons took his time to explain the why we care, instead of trying to figure it out on my own. Looking at you Rudin!

You can have a look at Munkres after Simmons, or just skip Simmons.

There is a nice book by Bartle : Elements of Real Analysis. This is a multi-variable analysis book, which is a bit gentler than Apostol's corresponding sections. Maybe reference Bartle when you get stuck with Apostol.

I found Rudin unreadable when I was learning Analysis for the time. It's a bit clearer, but I hate that book lol. But I am sure working through Spivak and Apostol, will prepare you for Rudin, something I was not at that point.

Apostol is easier to work through than Rudin. Reads clearer too!

 

[brpetrucci]

A nice intro to Topology is the topology book written by Simmons. It was recommended to me by Mathwonk, and I covered most of the book. A really nice, gentle, and interesting read. I liked that Simmons took his time to explain the why we care, instead of trying to figure it out on my own. Looking at you Rudin!

You can have a look at Munkres after Simmons, or just skip Simmons.

Nice! Thank you. So I'm assuming you'd recommend reading that between the analysis books and the DG book? Or is topology reasonably independent from analysis? I haven't looked into it too much before.

There is a nice book by Bartle : Elements of Real Analysis. This is a multi-variable analysis book, which is a bit gentler than Apostol's corresponding sections. Maybe reference Bartle when you get stuck with Apostol.

I found Rudin unreadable when I was learning Analysis for the time. It's a bit clearer, but I hate that book lol. But I am sure working through Spivak and Apostol, will prepare you for Rudin, something I was not at that point.

Apostol is easier to work through than Rudin. Reads clearer too!

Oh I get you 100%. I did the mistake of testing out of calculus in undergrad and going directly to analysis, and I was definitely unprepared. I had worked through Spivak but definitely not as carefully as I should. I'll definitely take a look at both Apostol and Rudin! If Rudin keeps giving me trouble I'll drop it lol.

Thanks for all the help!

 

[MidgetDwarf]

Nice! Thank you. So I'm assuming you'd recommend reading that between the analysis books and the DG book? Or is topology reasonably independent from analysis? I haven't looked into it too much before.Oh I get you 100%. I did the mistake of testing out of calculus in undergrad and going directly to analysis, and I was definitely unprepared. I had worked through Spivak but definitely not as carefully as I should. I'll definitely take a look at both Apostol and Rudin! If Rudin keeps giving me trouble I'll drop it lol.

Thanks for all the help!

Topology uses ideas of analysis. So they benefit each other. The topology book will introduce you to ideas of open/closed sets, compact and connected sets, continuity. Just to name a few. Which are things covered in Rudin/Apostol. Although, in topology you move away from R^n and focus on more general sets.

It is hard to say if you should start reading studying Topology. All depends if you are familiar and comfortable with the above terms, and are able to do proofs.

I took Topology at the same time I took Multivariable Analysis. The two classes complemented each other nicely. It's kinda cool to see what holds in R and R^n and not in general spaces.

If you want to skip Spivak Calculus. Then you can always read something like Abbot: Understanding Analysis.

It does analysis on R only. Very clear. I actually found this book easier to read than Spivak. You can always read both simultaneously. Although, Spivak has some outstanding exercises!

But in general. Learning is personal experience. Some people learn faster than others, or they are able to skip beginner/intermediate text. Which is fine. The important thing is that you start practicing now. Sometimes, you have to stop reading a book, and find a simpler one. Work through the simple one, then go back to the book you left.

The converse will become true someday too. You start with a book, but it is too simple, so you find a more rigorous book.