First of all, I want to say that the level of mathematics undergraduates in English-speaking countries may not be high ... and many well-ranked schools are simple as far as I know. I don't know why I want to reach the level of others ... just learn math by myself. Mathematics, language is just a carrier, in fact, it doesn't matter what you read, as long as it is a good textbook. Many textbooks in China are really good, definitely exceeding most textbooks abroad. Why do you have to read foreign textbooks? Note that you can't practice English just by reading math textbooks. . .

Closer to home, zero foundation began to contact high numbers. Are there any good textbooks to read? Many people recommend more than a dozen textbooks for a course, all of which are good and have their own characteristics. But which one should I read? How many books are you reading? To what extent do you read each book? Is 24 hours a day enough? Still stupid, so I try to recommend only one textbook per course, recommend several books as reference books and explain how to refer to them.

In what order should I teach myself English undergraduate and graduate math books? -Mathematics book recommendation

Basic courses:

First, mathematical analysis.

Textbook: Mathematical Analysis, 2nd Edition by Chen Jixiu, Yu Chonghua and Jin Lu.

There are too many textbooks for mathematical analysis, and there are many classics, but this is the best one in my opinion, not because it is more exciting than other classic textbooks, but because it is suitable, comfortable and just right in all aspects. The first and second volumes are very detailed, but they are not as scary as Fichkin, Goelz and Zoroach (in fact, Zoroach is ok). This is definitely not a simple list of definition theorems. What's the significance of the historical process of concepts and theorems? What are the special or interesting examples that illustrate the problems? It's all very clear and concise. The difficulty and quantity of the exercises are just right. The detailed answers to all the exercises in Volume 1 and Volume 2 are specially published, which is very important for self-learners. Using this textbook, in the case of consistent efforts and talents, will never be worse than using other textbooks.

Reference book: Fekhingingolz's calculus course (8th edition).

To tell the truth, no one dares to call his textbook the first if he says his textbook is the second. However, this book is too perfect, too detailed and too thick. . . Therefore, it is more suitable as a reference book and can be used as a step-by-step reading machine. If you don't know where to order and feel that you still don't understand the textbook after reading it, you can open Fichkingolz, open the directory and find the corresponding content, which will look good. There are many examples in Fichkingolz's book, which can always be understood one by one. In addition, this book is a textbook of the Soviet Union, and the original must be in Russian. Here, I acquiesce that you don't know Russian, so it depends on the translated version. Do you choose Chinese or English? Send sub-questions, class!

Reference book: typical problems and methods in Pei's mathematical analysis

A set of crying exercises for a group of mathematics undergraduates, the content of which is Gao Daquan, who has spare capacity to study and is interested in Shu Fenxin, can be used for self-abuse. Not much to say, just study, the first two books are enough.

Second, advanced algebra

Compared with analysis, the content of algebra is not so easy to master. Personally, I think it is better to start with linear algebra.

Textbook: Linear Algebra and Its Applications, Third Edition, Washington, D.C.

The perfect textbook for self-study introduction. It's definitely not just a few words, and then we need to use the matrix, and then we start to define the theorem, the angle he cuts in, various examples, application examples, and of course, rigorous knowledge exposition, which makes people have an impulse to clap their thighs. You will really feel that you have learned. I know the ins and outs of linear algebra, not a bunch of definition theorems in my head.

In addition, post a website, Math 1 15A, which is the website of the linear algebra course taught by the great god Terunsutao in UCLA. A course has all kinds of materials, but it is very concise and can be used as a summary and exercise after class.

Of course, algebra requires higher mathematics majors than this. However, the textbook I choose here mainly introduces the knowledge of linear algebra framework, and the part related to abstract algebra is answered at the extraction end.

Reference book: Yao Musheng's Advanced Algebra, the third edition and the corresponding explanation books.

I chose this book because of its characteristics. This book has a higher point of view than the textbook, so when you read it, just ignore the calculation part and read the other parts directly. This book attaches great importance to the view of geometry, and the accompanying explanation book (commonly known as "white paper") is very unique in the quality of exercises and the way of solving problems. It can be said that it will be completely upgraded to a higher level after reading it. When the two books add up, the study of advanced algebra is enough.

Thirdly, analytic geometry.

I feel that most analytic geometry in China is almost equivalent to the application of linear algebra ... there are almost no pure geometric ideas and opinions. I don't know much about this. I'm free for the time being. Welcome the Great God to add.

Fourth, ordinary differential equations

The course of ordinary differential equations is a connecting link to some extent. Chang Wei is the first advanced course based on Shu Dai knowledge and closely combined with Shu Dai after Shu Dai. This course is easy to learn, and it can also test the performance level of Shu Dai.

Textbook: Jin Fulin and Li, waiting for ordinary differential equations, Shanghai Science and Technology Press.

An old book, probably from the 1960s. The content is detailed and rich, and the exercise configuration can also consolidate your knowledge. Unlike many reference books and textbooks on so-called ordinary differential equations, almost the whole book is a fancy equation solution. This book introduces Chang Wei's theory in detail. This book has a shortcoming. Solving linear ordinary differential equations with constant coefficients is tedious and tedious. In this part, it is recommended to refer to other textbooks (just make one, anyway, everyone is solving equations in a fancy way).

Reference book: Arnold of ordinary differential equations

Needless to say, Arnold's books are all worth reading, all worth reading! The classic among the classics, and it is perfectly connected with the last one. The preparatory chapter of qualitative theory can also read Arnold's book directly. Don't look at the manifold part. Not much to say, some books are really wonderful after reading.

Fifth, abstract algebra.

As one of the subjects I hate most, I also refuse to recommend his textbook. ...

Textbook: Aden algebra

It is a classic textbook. Anyone who has studied mathematics should have heard of it. This book is rich in content and introduces a lot of linear algebra. Of course, you don't have to look at it. The so-called substitution, simply speaking of group ring field+Galois theory, is easy to say, but it is terrible to learn ... (Personally, it seems that all the exercises in this book have answers. Algebra is not a brush problem, but an understanding of thinking methods.

Reference book: what reference book do you need ... after all, it is artin's excellent boy/girl, and algebra is enough. ...

Six, real analysis/real variable function

Textbook: H.L. Roden P.M. fitzpatrick's Real Analysis, 4th Edition.

Friends! Friends! Good book! Personally, I suggest you understand the first part, chapter 1-8, so as to introduce the real variable function, and then do the problem well. There are some answers on the website of the University of Maryland, which seems to be the old answers circulating on the Internet. If you feel that you need to be advanced, please continue reading the third part, General Measurement Theory.

Reference books: Xia Daoxing, etc. Theory of real variable functions and functional analysis Volume 1.

This book has a history of several decades, and it is definitely one of the best mathematics textbooks in China. Moreover, its content is much richer than Roy's first film, and the ideas in many places are different. Worth reference/broadening one's horizons. After class, the questions are rich, the difficulty span is large, and the answers to the exercises are few. It is suggested that we should look at the key theorems and definitions, and we will certainly gain something. Of course, if you don't need English, just use this book as a textbook.

Seven, probability theory

Textbook: Probability: Theory and Examples.

The classical textbook of probability theory is rich in content. You don't need any theoretical basis for measurement. People tell you clearly and easily. If we look at it according to one of his definition theorems, it doesn't feel steep and smooth at all. After reading it, I found that I understood so many things. That's the feeling. The exercises are also rich and worth doing seriously. This book is always new and worth reading several times. You'll get something every time.

Reference books: Basis of Measurement Theory and Probability Theory by Cheng/Basis of Modern Probability Theory by Wang Jiagang.

You can read either of these two books. Wang's book is a little more difficult, with rich exercises and answers. Deleutre's books are relatively easy to understand in terms of measure theory and probability theory, but they are still a little simple, and the introduction of measure theory is more rigorous, and you will find that they are still different in the introduction and development of measure and Deleutre's narrative, with different ideas, which can broaden your horizons.

Eight, functional analysis

Textbook: Theory of Real Variable Functions and Functional Analysis by Xia Daoxing et al. Volume II.

Ah, there is no way to be functional. I don't know much about basic foreign textbooks. I think this book is a good introduction. In addition, Liu Peide's functional analysis foundation is also very good. Both can be used, and they are also widely used books in China.

Reference book: functional analysis W. Rudin

There is nothing to say. The functional analysis of the great god Rudin is necessary for learning functional, and its content is rich and in-depth. It can be seen as an extension of the textbook, and there are many exercises. It's up to you to do it or not. I suggest discussing it with people who also study functional.

Nine, partial differential equations

Textbook: Mathematical and Physical Equations, 3rd Edition, Gu Chaohao, Li Daqian, Chen Shuxing, Zheng Songmu, Tan Yongji.

Friends, just look at the author's dream all-star lineup. I don't need to spend too much time on Amway to get the quality of this book. Three academicians and two masters, dare to dream more? Don't play and review multivariate calculus, Green's formula and so on in mathematical analysis before taking this course.