Modern education not only depletes one's purse but also stifles one's thoughts. Modern buildings and air conditioners stifle one's brain. A natural breeze may refresh an exhausted brain and has an magic power to help the brain solve mathematical problems. I have found many times that if I stay indoors all day, the problem remains unsolved; once I step outdoors, the solution immediately becomes clear to me. [Lan1, p. xvii] contains a photograph of Landau sitting under a tree and reading a book. Perhaps the book is telling one that studying outdoors is ten times as effective as studying indoors. If one cannot find a tree, find a couch used by shrinks then. It may also help one's thoughts flow.

Reading one excellent book is better than reading ten average books. One will save tremendous time and effort if one can study only good books. However, one often finds a better book only after one has spent years studying one book. Allowing students to circle around the maze by reading books of poor quality is not only a personal loss but also a huge loss for the entire nation. Consequently, it is important to choose good books. One's understanding will be more objective if one can read two books on the same subject that take different approaches. In the following, I list some good textbooks for each subject.

- Classic mechanics: [Lan1] and [Fomi]. [Cou, vol.1] and [Sag, chapters 1-3 & 7] can be used as reference books: study them when needed. Gelfand is a knowledgable mathematician. His writings are "concise and structured" with emphasis on key ideas and the big picture. Sagan uses a great length to describe the competing functions, admissible variations and other details. Consequently, his statement and proof of a theorem are often long and rambling. However, in order to understand a deep theorem, an elaborate analysis like Sagan's is often indispensable [1].
- Topology: [Per], [Dug, chapters I-XVII] & [Mun00, part II]. [Per] is an easy read. [Dug, chapters XV-XVII] can be delayed until one studies [Spi].
- Advanced calculus: [Ru1] & [Spi1]. [Wid] & [Wat1] can be used as reference books.
- Real and complex analysis: [Ru2]. [Con] & [Gon] can be used as reference
books.

The Chebyshev polynomials of the second kind:

http://www.mathematik.tu-dortmund.de/lsviii/new/media/veranstaltungen/sose2010/orthpoly10/OPW7.pdf - Functional analysis: [Ru3].
- Analytic geometry: [Fin] (2-dim) and [Bel63] (3-dim).
- Differential geometry: [Wea1, chapters I-VII & X], [O'N], [Kre], and [Spi, vol. 1, chapters 1-8].
- Ordinary differential equations: By considering accessibility; viewpoints;
breadth, depth, and organization of contents; the effectiveness of methods; the
interactions with physics; and the potential for development, I divide the
reference books
on differential equations into the following three levels:

Level 1: [Edw] ® [Col] ® [Pon]; Level 2: [Arn1] ® [Bir] ® [Har] ® [Cod]; Level 3: [Inc1].

One may start with [Inc1], but one must update one's knowledge by reading the following supplements:- [Sne, p.20, Theorem 3] may help one read [Inc1, §1.22].
- For [Inc1, §3.23], see [Pon, chap. 2, §16].
- Analytic continuation: [Bir, chap. 9, §1 & §4].
- Asymptotic expansions: [Was].

- Partial differential equations: [Cou, vol.2]. One does need the background of [Cou, vol.1] in order to study [Cou, vol.2].
- Algebra: [Jaco, 3 vols].
- Electromagnetism: [Wangs], [Jack] and [Lan2].
- Optics: [Hec] & [Born].
- The method of steepest descent

For general details, read

http://www.math.osu.edu/~gerlach/math/BVtypset/node128.html

The following webpage explain clearly why the chosen path is the path of steepest descent through a critical point:

http://galileo.phys.virginia.edu/classes/751.mf1i.fall02/ComplexVariable.htm

- The method of steepest descent
- Thermal physics: [Reif], [Kit], [Zem], and [Hua]. [Pat] can be used as a reference book.
- Quantum mechanics: [Coh], [Mer2], and [Lan3]. [Schi] can be used as a reference book.
- Solid state physics: [Kit2] & [Ashc].
- Links {1}