[thraxil]

Does math count as "technical?"

I was originally a Physics major and lately I've been on a kick of filling in the mathematics that was used in my Physics classes but that I feel like was never really gone into in much depth.

My current reading list is:

- "Analysis I" (and II) by Terence Tao (I finished the first volume and am now on the second, but I consider them really one book)

- "Understanding Analysis" by Stephen Abbott

- "Topology Through Inquiry" by Starbird and Su

- "Introduction To Topology And Modern Analysis" by George F. Simmons

The Terence Tao books are amazing so far. Extremely readable introduction to Real Analysis. Abbott also came highly recommended and from reading the first couple chapters I can see why as it also seems to be very readable. I don't know if it would be a better introduction than Tao, but it covers mostly the same material and I think having two different perspectives will really help solidify things for me.

Once I finish those, I'll see whether I want to go deeper into Topology or move to Complex Analysis or Differential Geometry.

I also have a copy of Emily Riehl's Category Theory in Context. I've read some Category Theory before and have a basic grasp, but after reading a few pages of her book, I put it aside until I feel like I'm much more well versed in Topology (the content looks amazing and I really want to read it, but it relies on Algegraic Topology more heavily than other Category Theory material I've seen). So I'll see where I'm at after getting through those Topology books.

[benrbray]

Riehl's "Category Theory in Context" is great. It's one of those books where every page really takes a couple hours to fully absorb. It pairs well with Milewski's "Category Theory for Programmers" [1] series, which is comparatively lighter and gives some concrete examples.

[1] https://bartoszmilewski.com/2014/10/28/category-theory-for-p...

[thraxil]

Yeah, I watched his video series on Youtube a couple years ago and really enjoyed them. Definitely a good introduction to the topic.

[musgravepeter]

Snap. I am also a physicist trying to learn math. In my case it's:

- Introduction to Analysis (Mattuck)

- Elementary Differential Geometry (Pressley)

both chosen because they have solutions in the book, which I find important for self-study.

Also just got (hot off the presses)

Visual Differential Geometry and Forms (Needham)

Looks fantastic!

Topology is next on the list...

[thraxil]

"A Geometric Approach to Differential Forms" by Bachman is the one that's in my Amazon shopping cart, but I'll take a look at those others as well.

[erikpl]

What would you say are the prerequisites to the Analysis books? I did single-variable calculus and linear algebra in university a few years ago, but I have to admit that I'm a bit rusty.

[thraxil]

I'd definitely recommend having taken a college level Calculus course before and you'll get more out of it if you are basically comfortable with Abstract Algebra (know what sets, groups, rings, and fields are, and be able to think fairly abstractly about operations on those kinds of objects). That said, I think you could get through the first one with just high school level pre-calc although I think it would be hard to motivate yourself if you don't know enough calculus to see where it's heading. It does an absolutely brilliant job of starting with Peano's axioms to define the natural numbers, using those to define integers, using integers to define the rationals, introducing Cauchy sequences and using them to define the Reals, then introducing limits, continuity, etc. and Riemann integrals. That whole part is pretty much self contained with each concept rigorously (but clearly) built out of the previous ones. Some basic algebra and the ability to follow a mathematical argument are pretty much all you need. As it gets into the second volume, it expects some familiarity with logarithms and starts building into more advanced Calculus. It's still fairly self-contained but you might struggle if you don't remember, eg, what integration by parts looks like.

[wheelinsupial]

You can check the analysis courses offered where you took calc and linear algebra to see if you have sufficient prereqs.

Depending on the level of comfort or exposure you've previously had to proofs, it could be good to have a book covering introduction to proof handy. A book on counterexamples in analysis is also something I've seen recommended.

For proofs, something that covers direct proof, proof by contrapositive, proof by contradiction, and mathematical induction are good to be familiar with.

Delta-epsilon proofs are also good to have alternative or more accessible explanations to draw on.

If you're covering topics in multivariable calculus, then it might also be handy to cover some of the calculations from a calc 3/4 book to see implementations of it.

[benrbray]

A good intro real analysis book will have no prerequisites, really! I learned from Ross, "Elementary Analysis: The Theory of Calculus", which starts with sequences of natural numbers.

(I'm of the opinion that the usual Calc1+2+3 sequence should be scrapped in favor of everyone taking "Advanced Calculus" first. It should probably even be taught in 10th/11th grade in place of the dreadful "pre-calc" courses many schools have. Calculus didn't click for me until my first proof-based course. That's also when I learned that a "proof" is just a detailed explanation of exactly why something is true.)