Fudan's own teaching materials should be counted from Shanghai Science and Technology Publishing in 1960s. It is republished in Hong Kong and other places, and it is said that the response is very good. Mr Qiu Chengtong seems to have benefited from it when he was a student.

Among the textbooks that can still be seen in the market in the 1990s, there is a set compiled by Mr. Chen and others. It may be a new edition of the book above. The experimental class of Jiaotong University has used this book as a teaching material for several years. In addition, the Shanghai Science and Technology Edition also has textbooks edited by Ouyang Guangzhong (Mr. Gu's brother-in-law), Qin Zengfu and Zhu Xueyan. It seems that the department of mathematics was not used later, but the department of computer is still in use. It is said that the statement of the second mean value theorem of integral in that book is a bit wrong.

Generally speaking, one of these books can be seen, that is, Principles of Mathematical Analysis written by Fickin Goldz. According to Teacher Qin, the reason is that when the textbook was first built, the "model essay" chosen by Peking University was "A Brief Course of Mathematical Analysis", while Fudan chose "Principles of Mathematical Analysis".

Later, naturally, there was the mathematical analysis of Mr Ouyang and Mr Yao Yunlong. I don't deny that this is an attempt, but I always feel a little embarrassed. From a relatively new point of view, the classic content of mathematical analysis is indeed the world trend, but in this sense, this book is not very good, on the whole.

In the curriculum system, it is debatable whether it is necessary to introduce Lebesgue integral when there is a real variable function course.

Let's start with some textbooks or reference books:

Fekhkin Goldz

Course of Calculus and Principles of Mathematical Analysis.

The previous book, three volumes in Russian and eight volumes in Chinese;

The latter book consists of two volumes in Russian and four volumes in Chinese.

This book is a classic.

In fact, even the author (a professor at Moscow or Leningrad University with many disciples, including the famous mathematician Kantrovich who later won the Nobel Prize in Economics) admitted that it was not suitable as a teaching material, so he gave it.

The latter set of books, which can be used as teaching materials, can be said to be a simplified version (finally supplemented by the introduction of subsequent courses).

I believe that until today, many teachers will still look for "calculus course" when they start classes, because there are too many examples in it. If you want to lay a solid foundation, you can consider making the example an exercise with an answer. Of course, not every problem can be done. Don't blame me if you finish all the questions there and then meet what you have done during the exam.

There is no doubt that this set of books represents the highest level of dealing with mathematical analysis in a classical way (meaning that the concepts of real variables and functionals are not introduced). Considering that there are hundreds of thousands of prints in China, only Gulsat's books in the world can compare with them.

Both sets of books are in the map.

Apostol

"mathematical analysis"

In the west (western Europe and the United States), this should be regarded as a relatively complete textbook, which is available in general libraries.

3.w. Rudin

"Principles of Mathematical Analysis"

(There is a Chinese translation: Principles of Mathematical Analysis by Lu Ding, which is shown in the picture)

This is also a very good book. As we can see later, this gentleman has written a series of teaching materials. The teaching method of this book (referring to the use of some symbols and terms) is also very good.

By the way, I read Advanced Mathematics compiled by Teacher Qin and Yu Yuenian from Ruding Publishing House. Although I have always felt that the book is very poor, I would like to quote a sentence from Mr. Qin here, hoping to help ddmm of non-mathematics majors: after learning Advanced Mathematics, I can find a book with advanced calculus level in the west, which can basically meet the requirements of the general mathematics department.

Speaking of AdvancedCalculus, there is a book under this heading that can also be read, that is, advanced Calculus by L.Loomis and S.Sternberg. The first edition is quite abundant in the General Library, and the second edition is in the foreign teaching material center of Litu. It is not clear whether it is in the department reference room. The viewpoint of this book is still very high. After all, it is a Harvard textbook.

4. Mathematical Analysis (Peking University Edition) Fang, Shen Xiechang, etc

Mathematical analysis exercises, textbook of mathematical analysis exercises.

This set of textbooks compiled by Peking University is OK, but the best one is about two exercises. As we all know, Jimidovich is not suitable for students majoring in mathematics. After all, most of them are calculation problems (one interesting thing is that the solution to the exercise problem scolded by the teacher actually has no answer to the second sub-question of one problem, because it seems that the person who compiled the book has not worked it out either. It seems to be a topic about series convergence). By contrast, this set of exercises of Peking University is much better, and it is really worth doing. That exercise textbook is also a very interesting book, including solutions to some rather difficult exercises. 1996 once at the press conference, I don't know how it is now.

5. Klebauer's "Mathematical Analysis"

I remember that it is an analysis book in the form of exercises, and the topic is also very good.

It's on the map

6. Zhang Zhusheng's "New Lecture on Mathematical Analysis" (* * * three volumes)

Personally, I think this is the latest mathematical analysis textbook written by China people. Teacher Zhang has really worked hard to write this book, and has written it back and forth for almost five times. Disabled people like him have paid much more than ordinary people for doing such a thing, so that he himself quoted "Dou Yun's author is crazy, who can understand the taste" in his postscript. In this set of books, many materials are treated differently from traditional methods. It's well worth reading.

Some of the following books may be novel.

7a。 Nicolschi's Mathematical Analysis (Tutorial? )"

There is one in the picture, which was translated by someone in Tsinghua. It seems that it has not been completely translated. It belongs to the representative of the new trend of the Soviet Union after the 1980s. Anyway, he is an academician of the Soviet Academy of Sciences.

7b。 "mathematical analysis"

I forget who wrote it, and it is also a textbook of Moscow University in the Soviet Union. There is a Chinese translation of the first volume in the picture, which is divided into two volumes. From the point of view of limit (for topological basis), people can obviously feel that the view is "high".

8. Fundamentals of Modern Analysis (Volume I)

It is the first volume of a set of textbooks written by people in the 20th century, and the terminology used is quite abstruse. Maybe it's better to look back at the functional after learning the real changes.

9. Say a few words about the high number of non-mathematics majors.

Some math books written by French people in Ricoh are highly recommended here. Because in the French higher education system, for the best students, studying for two years after graduating from high school is not a department, so their advanced mathematics (for example, the first volume of Advanced Mathematics by Academician J.Dixmier is in Ricoh) or General Mathematics (Ricoh has a set of books on this topic) is basically between domestic mathematics departments.

10. Add a technical question. For the convergence of function series, uniform convergence is sufficient, but not necessary. There is a necessary and sufficient condition called "sub-uniform convergence", which is mentioned in Calculus Course. Its detailed discussion seems to be only available in.

In Lucien's Theory of Functions of Real Variables, there are some in general libraries.

1 1. Mr. Hua's Introduction to Advanced Mathematics, Volume I

This set of books (in fact, the original plan has not yet been completed) is a handout given by Mr. Hua to college students in the early 1960 s with the help of Mr. Hua. At that time, they did an experiment, that is, a professor was responsible for the teaching of the first students, so Mr. Hua's book actually involved many aspects (incidentally, the other two were Mr. Guan and Mr. Wu Wenjun who were responsible for the first students). This is also an attempt. Some things in Mr. Hua's book do not belong to the traditional teaching content.

12. He Chen, Shi Jihuai, Xu Senlin

"mathematical analysis"

This should be HKUST's textbook. Although it doesn't seem to have much impact, I like it very much. It's the first time for senior one to use this set of books. I feel very organized and have good practice. The printing quality is also quite good. Unfortunately, it wasn't in school, so I put it last.

Spatial analytic geometry is really a classic or classic course. From the teaching content, it can be considered that it mainly describes some basic common sense in three-dimensional Euclidean space, including the most basic linear transformation (that is a special case of linear algebra).

And the invariant theory of second-order surfaces. In the current Fudan textbook Spatial Analytic Geometry edited by Su and Hu, there is a final chapter on projective geometry.

This book is very thin, but it is rich in content, especially some exercises are not easy. The projection of the last chapter is not very easy to read.

Of course, there is also a book that was used as a textbook about ten years ago:

Xiang Wuyi, Pan, etc

Classical geometry.

The content of this book is not quite the same as the textbook, but the handling method is still very good. Xiang teacher should be the kind of person who can talk very well.

Reference books that can be considered include:

Chen (bird)

"Spatial Analytic Geometry"

The content is basically the same as the textbook, but it is much thicker and naturally reads better. Mr. Chen is the wife of Mr. Wu Daren (Mr. Wu's cousin, president of Nankai University for many years), and a female scholar who studied abroad in China in the early days.

Zhu dingxun

"analytic geometry"

This book basically only discusses the problems of Euclidean space. The advantage is that it is very easy to understand, and even the two-dimensional invariant theory is clearly stated in the appendix. The exercises there are also reasonable and not difficult (if I remember correctly). Mr. Zhu was quite talented, but unfortunately he died young.

There is also a book about mathematical analysis exercises, which is

G. Paulia (Bulgaria), Seg (Xie Gui)

"Problems and Theorems in Mathematical Analysis"

In the stage of learning mathematical analysis, you can consider the first half of the first book, and the back is full of complicated things. The content of this book is still very rich. In history, this is a set of classic works that have benefited several algebras. One advantage of this book is that the title is difficult to answer, and there are answers or hints behind it.

One of the first volume of Calculus Course seems to be rare in the picture.

Go to the main library and have a look!

Loomis-Sternberg's book number is O 172 L863.

If you want to know more about "new" trends, you can consider

Postnikov

"Analytic geometry and linear algebra (? This is the new textbook of Moscow University. From the form of the course, we can see that analytic geometry will be eaten by linear algebra sooner or later if it is not used as a guide for students who have just entered the university and some specific objects are given. There is an English book in the overseas textbook center.

Personally, I think that the measures taken by the Education Commission to reduce the burden on students will be punished sooner or later. The level of secondary education in China is better than the worst secondary schools in the United States, and worse than that in Europe as a whole. I believe that the so-called three-dimensional "analytic" geometry will be decentralized to high school one day.

If the above books don't satisfy you and you don't want to take other courses, you can consider the following two classics. The advantage is that you can have a deep understanding of many geometric objects (of course, the quadric surface in three-dimensional space) after reading it.

Tommy Tam Nie

"(analytic) geometry"

This three-volume book contains many very interesting discussions. I remember five years.

I find it interesting when I look forward. This academician of the Soviet Academy of Sciences is really capable.

It's written. It is in the main library.

Mosharishvili

Course of Analytic Geometry

This set of books has been quoted many times in the aforementioned book by Mr. Chen. Specifically, it is worth referring to some viewpoints and explanations about projection (for example, an ellipse also has an asymptote, but it is only "imaginary").

Higher algebra can be regarded as a theory dealing with finite dimensional linear space. If strictly speaking, the theory of linear space should be called linear algebra, plus a little polynomial theory (which can be calculated as the content of algebra), it is called higher algebra. The western counterpart of this course is generally called linear algebra, which is a favorite word used by the Soviets. You can find a copy of Kurosh's Advanced Algebra in the foreign textbook center.

The textbook used now seems to be Advanced Algebra of Peking University (second edition? It is not common to use textbooks from other schools for basic courses. This book can be said to be slow and steady, and basically everything has been said. But if you want to say that it is particularly good, I am afraid I can't say it.

It is worth noting that in the 95-96 academic year, Mr. Wang Jie (a disciple of Mr. Duan Xuefu), who is currently the organization minister of the Party Committee of Peking University, gave a lecture to Class 95 1 of the School of Mathematical Sciences of Peking University.

I wrote a supplementary material at the beginning of the class, which made the theory of space clear. If anyone can get it, it's also a good thing to reprint it (my teacher Shu Wuchang gave it to him at the beginning of school in 1996, so I guess I can't find it).

There seems to be a problem, that is, Peking University's books are still in the first edition. The second edition seems to have been seen in a bookstore.

Judging from the content of this course, there are many kinds of statements. The emphasis of linear space is naturally linear transformation, so if a set of bases is set in the definition space and image space, there will be matrix representation. Therefore, this course can be based on matrix theory, and it must be done if you want to bind with numerical values. Fudan used to have two textbooks to do this.

Jiang Erxiong, Wu Jingkun, etc

"linear algebra"

This was the teaching material of computational mathematics specialty at that time, and it was said that the teaching requirements were higher than the corresponding courses of mathematics specialty. Because it is biased towards calculation, we can find some commonly used algorithms. Personally, I think it's quite interesting. There are some on the map.

Tu Boyu and others.

Advanced algebra

This is the one about advanced algebra in a set of Fudan Mathematics Department textbooks published by Shanghai Science and Technology. I don't remember it in the library, but it may be reprinted in the department.

This book devotes 80% of its space to matrix theory. There are many exercises, especially the "Selected Exercises" at the end of each chapter. It is very beneficial to understand the various properties of the matrix to be able to complete the exercise independently.

Of course, this is not easy:

It is said that when Teacher Tu retired, he left this sentence: "Whoever opens advanced algebra in the future will use this book as a teaching material, and you can come to me if you have difficulties in exercises." This can be seen. If the above book is too bad from the perspective of exercises, then the following book should be said to be more appropriate.

Tu Boyu and others.

"Linear Algebra-A Guide to Methods"

This book may be easier to find and more "practical" than the one above. It's worth doing.

In addition, when it comes to matrix theory, it is necessary to mention Gantemacher's "matrix theory"

I think this is probably the most authoritative book in this field. The translator is Mr. Ke Zhao. In this two-volume book, there are many things that are not in the textbooks at ordinary times. For example, we all know that matrices have Jordan standard, but how to find the transformation matrix from a matrix to its Jordan standard? Please look at Matrix Theory. There are some interesting discussions about matrix equations in this book. There are some in the general library.

There is also a book in the library whose name is related to matrix theory.

Xu yichao

"Linear Algebra and Matrix Theory"

Although xu teacher is not very friendly to Fudan (he told me that he was going to Peking University or studying domestic mathematics in his senior year-he graduated from Peking University and now works in the Institute of Mathematics-I didn't listen to him), he must admit that he wrote well and did well in the exercises. It must be pointed out that the concept of space actually attaches great importance to it. Anyway, he is still a disciple of Mr. Hua.

Hua

Introduction to Advanced Mathematics

The characteristic of Mr. Hua's mathematical research is that his elementary intuitive method is unique.

He also has a good job in matrix theory. You can only find the names of two China people in Guntmacher's book, one is Mr. Ky Fan and the other is Mr. Hua. It may be the first time that he introduced the following idea into China's mathematics textbook (I don't remember whether it was in this book): the determinant of order n is the only antisymmetric linear function that maps a set of standard bases to 1 on the Cartesian product of n N-dimensional linear spaces.