Self teaching higher math

>>8063128
Read Baby Rudin -> Munkres' Analysis on Manifolds -> Stein's 4 Princeton Lectures in Analysis (Fourier Analysis; Complex Analysis; Real Analysis: Measure Theory, Integration, and Hilbert Spaces; and Functional Analysis)

>>8063140
Will do. I have heard that Rudin's lower level book may be a good place to start. How about for PDEs? Or should I focus exclusively on analysis first? I really appreciate the advice, man.

>>8063128
OP I want to get into applied math at cal poly. how do I hone my skillz outside of self-study?

>>8063228
How old are you? High School? When I was in high school I just used self study over a long period of time to knock out multivariate calculus and intro to ODEs (still had to take them). You could also try to take advanced classes that will be able to be used for college credit (AP calculus, stats, etc.). There isn't really a lot besides that.

>>8063128
This book is hard if you do not have a solid understanding of multivariate calculus

>>8063273
I have (will have in less than a week) a degree in math, do you think it would likely be an appropriate level for me? I've taken multivar, vector calculus, and a couple of classes in ODEs, but virtually nothing in PDEs (I've seen how you can use separation of variables with the heat equation given some boundary conditions, but I might as well be a blank slate). Do you think I'd be able to use it or would it be over my head?

>>8063300
In my uni the make us do stats after ode and some computationally biology(used to strengthen our stats/probs skills) and then we do pde.

conclusion: My uni would say you are ready for it.

>>8063174
>How about for PDEs? Or should I focus exclusively on analysis first?

If you are talking about PDEs at the level Evans, then you 100% need a solid foundation in analysis first.

You need at least some functional analysis and differential geometry if you want to seriously learn PDE, it's pretty complicated.
A good book to learn real analysis is probably Rudin. For complex analysis, Needham's Visual Complex Analysis might be a good place to start in order to get a good idea of what is going on (complex analysis is very geometric). After that, you can read any good book on the subject (maybe Ahlfors, I heard it's good).
For functional analysis, I really like Brézis' book. The only problem is that it does not mention distributions. Yosida is good for that but it's pretty hard.

>>8064164
Ok, so I took a class in analysis (kind of built the foundations more than anything else but it was focused on rigorously proving stuff from calculus like continuity and the IVT), but I never took the 4000 level real analysis class.

>>8064176
What preparations are needed for functional analysis? Just real analysis or complex analysis too? A couple people have recommended Rudin, so I'll be buying that on Amazon within the next few days. I have a basic understanding in analysis but I'm pretty removed from it (that was like 2 years ago) and it was never the full version of real analysis.

>>8064490
If you plan on jumping directly into Evans, you should be familiar with Functional Analysis and the fundamentals of measure-theoretic real analysis.

If you really just want to jump into PDEs, this course here (despite being a course in QM) quickly covers all the higher level analysis you should need.
https://www.youtube.com/playlist?list=PLPH7f_7ZlzxQVx5jRjbfRGEzWY_upS5K6

>>8064490
>>8064176
>>8064164
Oh, I forgot to mention, the class in analysis I took used Understanding Analysis by Stephen Abbott and taught everything in the book.

Evans is a good book, but its a graduate-level proof treatement of PDEs. You need to start with a numerical/applied treatment.

I highly suggest "Partial Differential Equations for Scientists and Engineers" by Farlow. It is a Dover book and costs $10 on Amazon.

>>8064490
For functional analysis, you should be familiar with real analysis, have a strong foundation in linear algebra (especially duality) and general topology and it would help if you were familiar with normed vector spaces.
Basically, it is a study of infinite dimensional linear algebra. The problem is that there are a number of things that are not true (for example, not every one-to-one linear map is onto and vice versa, the powers of an endomorphism are not necessarily linearly dependent so all the study of diagonalization/Jordan goes out the window).
To remedy the problem, you have to require some sort of finiteness to have similar results, which generally means compactness. New problem: There are now many non-equivalent interesting topologies you can put on a vector space (that happened to coincide in the finite-dimensional case) and you have to juggle between them to gain info. Besides, the ones that are metric are not automatically complete anymore, which leads to continuity problems (not all endomorphisms are continuous etc.)
Hence the importance of Banach spaces, Hilbert spaces and all that.
So, pretty quickly, you are led to use a lot of stuff and you should have a solid background.

>>8064642
I feel pretty strong in linear algebra, my senior year I took a continuation of linear algebra where we talked about normed linear spaces, inner product spaces, SVDs, least squares, direct sums, etc. Hard class, but I learned a lot. I don't have a lot of experience with infinite dimensional spaces. We generally just focused on arbitrary finite dimensional spaces.

>>8064554
Same here and I am self studying out of Rudin now after having graduated.
The exposition is not that rough since I have seen Analysis before, the problems are still hard though, many of which I find just ineffective. I would supplement Rudin with Tao's UCLA classes and just do his problems.
http://www.math.ucla.edu/~tao/resource/general/131ah.1.03w/
and
http://www.math.ucla.edu/~tao/resource/general/131bh.1.03s/