My major was physics, but i'm changing my major to mathematics this year.
I took 2 years off to study mathematics myself and now i'm going back in this year.
Here's the list of books i have studied for last 2 years:
Elementary Set Theory - Jech (I have studied one more set theory text, but i don't remember its name)
Linear Algebra - Friedberg / Linear Algebra - Hoffman & Kunze
Principle of Mathematical Analysis - Rudin (Only First 8 Chapters: I have not studied multivariable calculus)
Topology - Munkres (Only Point-set-topology, NOT algebraic topology)
Real & Complex Analysis - Rudin (First 3 chapters)
I'm sure that i have a concrete understanding of books listed above.
I really like abstract approach, so i didn't study multivariable calculus since i thought it could be done in differential manifold. (BUT i'm not sure if this is possible to study differential manifold WITHOUT studying multivariable calculus). For example, I really feel comfortable with Measure theory than basic calculus.
Here's the course descriptions in my school:
Differential Geometry - "In this course, we understand the theory of curves and surfaces, then we investigate differential and integral calculus on surfaces"
Differential Manifold - "Linear Algebra, calculus and 1 year analysis are prerequisites to this course"
My questions: is it possible to study differential manifold without studying differential geometry, complex analysis and multivariable calculus?
This is a very important question to me.. When i studied RCA-rudin, it's written in the index that "only first 8 chapters of PMA are enough to start this book", but frankly it was not. It required concrete understanding of linear algebra and topology.