First, the contents of the course "Advanced Mathematics" usually include:
Elementary calculus (different from advanced mathematical analysis such as complex variable function, real variable function and functional analysis) and simple differential equations, linear algebra, analytic geometry in space or plane, elementary probability theory and mathematical statistics. The above list is not exhaustive, it may be divided into more than one course. Sometimes "Advanced Mathematics" is only the content of elementary calculus and differential equations.
This kind of content mainly depends on the choice of teaching materials. The common ones are as follows. The content is arranged from shallow to deep, and can be selected according to the introduction. Personally, I think this is the simplest book, and understanding the way of thinking is the most important:
1, Advanced Mathematics for Liberal Arts published by Peking University or National People's Congress. If you want to quickly understand some ideas, principles and calculation methods of high numbers, these two books are good choices. There is basically no difficulty, and there will be no obstacles for high school students to read. Another great advantage is that the content is quite complicated, such as calculus, algebra, geometry and statistics.
2. Higher Education Press, the old version is People's Education Press, "Advanced Mathematics Lecture Notes" written by Fan Yingchuan. This book was widely used as a teaching material from 1950s to 1980s, especially in normal universities. The explanation is quite detailed, the examples are selected accurately, and there are no exercises. Another advantage of this book is that it has a large space to talk about spatial analytic geometry first, and then calculus.
3. Tongji Edition (the new version is the fifth edition) Advanced Mathematics. It is a textbook widely used in domestic engineering universities, and it is also a textbook for the tenth five-year plan of the country. It is better in similar textbooks, and the calculation examples are more detailed. But I think it may be a bit too long as an "extracurricular book". (to correct santiagomunez's point of view, there are a large number of advanced mathematics textbooks in China, which are not based on it. Tongji uses more, but it is not authoritative. )
4.Xi Jiaotong University Edition, or National University of Defense Technology Edition "Engineering Mathematical Analysis". The content is quite deep, which is difficult to chew down as an extracurricular book. The characteristic of engineering mathematical analysis is that all problems can basically make you "know why", leaving no logical loopholes, but paying attention to image thinking, not as formal as math books.
5. Advanced Mathematics (Physics) edited by Jong Li, Peking University Edition. The difficulty of science books is similar to that of 4, emphasizing theoretical derivation. It also includes the content of spatial analytic geometry.
6. Peking University Edition, New Lecture Notes on Mathematical Analysis written by Zhang Zhusheng (three volumes). This is the book used by the math department. This version is characterized by paying more attention to visualization and putting some difficult things behind. But you must have good training after learning.
7. The new edition of Science Press, Fichkingolz, Calculus Course (three volumes). Classic teaching materials. Soviet books are very detailed and there is nothing to say. All theorems are discussed in detail. The disadvantage is that there is too much space, sometimes too much.
8. American R. Courand wrote Introduction to Calculus and Mathematical Analysis, Science Press. This is a famous math book, which talks about many applied principles rarely mentioned in other books, and its style of writing is more distinct than that of Fichkin Goelz's book. It was after reading this book that I understood the strict definition of "area". Although it is difficult, there are many interesting contents worth reading.
9.W.Rudin Principles of Mathematical Analysis, Mechanical Industry Press, English photocopy and translation. This is a famous math book. It's hard. All of them are analyzed mathematically from the perspective of abstraction and generalization. The style is simple. Reading is not recommended for beginners.
If you want to teach yourself, you may still need problem sets or problem sets with answers, also in order of difficulty:
1, Tongji Edition, high math problem solving or synchronous tutoring. I guess you're not interested.
2. People's Education Press, Jimmy Dovich, Mathematical Analysis Problem Set; Shandong Education Press, Answers to Mathematical Analysis Problem Sets. There are more than 4000 questions in this book, which are too many in any case, and there are many repetitions of the same type. I have seen the condensed version of this book, with more than 1000 copies.
3. Fang, Peking University, Guide to Solving Problems in Mathematical Analysis. The content is similar to the last one, and the difficulty is similar. The quantity is small but fine.
4. Higher education, Pei, typical problems and methods in mathematical analysis. It's hard. I talked a lot about skills. Novices are not recommended.
5.Paulia and Xie Gui, Problems and Theorems in Mathematical Analysis. This is a famous math book. It's super difficult, but it's not a book for exams. If you are interested in entering the field of mathematics, it is worth seeing. However, it is not recommended for beginners.
Second, the content is broad.
Everything after middle school is basically a big category of advanced mathematics, and it is difficult to list them one by one. But as an introductory and extracurricular reading, the better ones are:
1, Peking University, the beauty and reason of mathematics, the source and flow of mathematics. They all talk about the ideas, methods and applications of mathematics, and the former is particularly easy to understand.
2. Science Press, Alexander Love, Mathematics-Its Content, Method and Significance (three volumes). This is a famous math book. This paper introduces the main contents, thinking methods and applications of various branches of mathematics. Because it is popular science, high school knowledge can be understood. The content of this book is very extensive and comprehensive, and the fields involved are by no means shallow. This is a rare good book.
3. Shanghai Science and Technology Education Press, Klein, Ancient and Modern Mathematical Thoughts (four volumes). A very classic book on the history of mathematics, mainly the achievements of western mathematics, couldn't be better. Like the above book, they are all encyclopedias, but the difference is that the previous book emphasizes "theory" and this one emphasizes "history".