Which order to read math books?

The cool thing about higher mathematics is that everything's related. To that end the first goal would be to get a basis in everything before getting really abstract. To that end, in no particular order

Munkres Topology

Artin Algebra

Gamelin Complex Analysis

Riesz Functional Analysis

Rudin Real and Complex Analysis

If you haven't seen it, I'd also recommend looking into Vector calculus. It might have been included in Multivariable, but that's doubtful. Vector will be required to understand Calculus of Variations. Also a good idea of geometry will be useful for Differential Geometry, but it's not required as it's based more on Linear Algebra.

After that you're probably ready for the other three. Analysis on Manifolds is a mixture of Topology and Analysis, Differential Geometry is a mixture of Calculus and Geometry (you've probably seen a lot of this in your multivariable calculus course, but this is more general), and Calculus of Variations is basically abstract calculus.

Like the other answerer said, if you get stuck in one try to find the material in another. You're getting to the point where everything overlaps and so going linearly isn't really an option anymore.

Also, you're missing some branches of mathematics, in particular Combinatorics, Probability Theory, Statistics, and Number Theory. Probability Theory and Statistics aren't very abstract, but they have huge impacts on our daily lives. Combinatorics and Number Theory are quite abstract, but are very interesting and in the case of Combinatorics has quite large implications on other branches of mathematics.

I didn't read these books, but their names are good hints of what's inside. My advice is:

1. Pick one at random, and start reading.

2. If you're stuck because of lack of required knowledge, search the other books for help, or use Google / Wikipedia to find other sources (there are many e-books out there, mostly in torrents - search for "MYRIAD Mathematical Library").

3. If you find repeated content from one book to another, feel free to skip it, to save time, although you can enjoy comparing styles of proving and explaining.

4. Back to step 1.

One visible gap in your books is abstract algebra (group theory, ring theory); other, less apparent, is geometry; see the source for more.

Good luck in your studies!

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