Relearning advanced undergraduate/beginning graduate level mathematics

Question

I'm currently a first year graduate student in mathematics, and I'm trying to go back and relearn some advanced undergraduate level mathematics. The motivation comes form reading Terrence Tao's blog post, which states:

The “post-rigorous” stage, in which one has grown comfortable with all the rigorous foundations of one’s chosen field, and is now ready to revisit and refine one’s pre-rigorous intuition on the subject, but this time with the intuition solidly buttressed by rigorous theory. (For instance, in this stage one would be able to quickly and accurately perform computations in vector calculus by using analogies with scalar calculus, or informal and semi-rigorous use of infinitesimals, big-O notation, and so forth, and be able to convert all such calculations into a rigorous argument whenever required.) The emphasis is now on applications, intuition, and the “big picture”. This stage usually occupies the late graduate years and beyond.

For example, I'm currently learning a bit functional analysis, and I feel I have to go back and revise/re-learn linear algebra so I can differentiate between all details between finite/infinite dimensional setting etc. Now, I suppose if I were to pick up a linear algebra textbook, I could work through most of the details quite easily, which puts me in the "post-rigorous" stage, where I'm not only interested in learning about the details (proofs), but rather also learning about the the big picture: the intuition behind the proofs, the motivation behind the introduction of the definitions, the applications etc.

So, I'm looking for concrete book suggestions on such subject areas as calculus, linear algebra, abstract algebra, geometry (differential and otherwise), complex analysis, ODE's, probability, etc. that will help me:

1. Revise the basic foundations,
2. Understand the big picture of the subject,
3. Expose myself to various applications of the subject in mathematics or outside of mathematics.

I know one can't acquire all these skills from one book, and in most cases the big picture can only be seen if one studies non-math books: for example, physics book that put all these ideas to practice. However, my aim behind generating this thread is to get as good a list of textbooks that will help me achieve the aforementioned goals, and perhaps make me understand the subject more holistically.

Edit: Edit: I subscribe to Arnold's philosophy on teaching of mathematics which in one sense or the other argues that the study of mathematics shouldn't be separated from its real world applications. I'm interested in the inter-disciplinary application of mathematics, and, after some time when I have forgotten the working details of a subject, I tend to feel very uneasy and feel the need to go back and revise and better understand the big picture.

Answer 1

Below are some possible books, depending on your background and interests. Probably [3] would be the best overview that doesn't dig very deep into any specific topic and for which the goal is similar to what you seem to want. The first 4 are generally well known, but I'd recommend also looking at [5] and [6] despite the fact that they don't seem to be mentioned much on the internet.

[1] Infinite Dimensional Analysis: A Hitchhiker's Guide by Charalambos D. Aliprantis and Kim C. Border (1994)

[2] Encounter With Mathematics by Lars Garding (1977)

[3] All the Mathematics You Missed: But Need to Know for Graduate School by Thomas A. Garrity and Lori Pedersen (2001)

[4] The Road to Reality by Roger Penrose (2004)

[5] Some Modern Mathematics for Physicists and Other Outsiders, Volume 1 (contents) and Volume 2, by Paul Roman (1975)

Note: I notice that amazon.com's "Look inside" for Volume 1 also shows the inside of Volume 1. See pp. x-xi for the contents of Volume 2.

[6] 100 Years of Mathematics by George Temple (1981)