Calculus is the general name of differential calculus and integral calculus, which is abbreviated as calculation in English, because early calculus was mainly used for calculation problems in astronomy, mechanics and geometry. Later, people also called calculus analysis, or infinitesimal analysis, especially the knowledge of using extreme processes such as infinitesimal or infinity to analyze and deal with calculation problems.
The early calculus was not developed for a long time because it could not convincingly explain the concept of infinitesimal. Cauchy and later Wilstrass perfected the limit theory as the theoretical basis, which made calculus gradually evolve into a basic mathematical discipline with strict logic, called "mathematical analysis" and translated into Chinese as "mathematical analysis".
The basis of mathematical analysis is real number theory. The most important feature of real number system is continuity. Only with the continuity of real numbers can limit, continuity, differential and integral be discussed. It is in the process of discussing the legitimacy of various limit operations of functions that people gradually establish a strict theoretical system of mathematical analysis.
Mathematical analysis is a basic course for mathematics majors. Learning mathematical analysis (and advanced algebra) well is the necessary basis for learning other subsequent mathematics courses such as differential geometry, differential equations, complex variable functions, real variable functions and functional analysis, calculation methods, probability theory and mathematical statistics.
As one of the most important basic courses in the department of mathematics, the logic and historical inheritance of mathematical science determine the decisive position of mathematical analysis in mathematical science, and many new ideas and applications of mathematics come from this solid foundation. Mathematical analysis is based on the rigor and accuracy of calculus in the theoretical system, thus establishing its basic position in the whole natural science and applying it to various fields of natural science. At the same time, the subject of mathematical research is the object after abstraction, and the mathematical thinking mode has distinct characteristics, including abstraction, logical reasoning, optimal analysis, symbolic operation and so on. The cultivation of these knowledge and abilities needs to be realized through systematic, solid and strict basic education, and the course of mathematical analysis is the most important link.
We are based on cultivating outstanding talents with solid mathematical foundation, wide knowledge, innovative consciousness, pioneering spirit and application ability to meet the requirements of the new century. From the perspective of personnel training, whether a student can learn mathematics well depends largely on whether he can really master the course of mathematical analysis after entering the university.
The goal of this course is to master the basic theoretical knowledge of mathematical analysis through systematic study and strict training; Cultivate strict logical thinking ability and reasoning ability; Skilled computing ability and skills; Improve the ability of establishing mathematical model and applying calculus to solve practical application problems.
Calculus theory is inseparable from the development of physics, astronomy, geometry and other disciplines. Calculus theory has shown great application vitality since its birth. Therefore, in the teaching of mathematical analysis, we should strengthen the connection between calculus and adjacent disciplines, emphasize the application background and enrich the application content of theory. The teaching of mathematical analysis should not only reflect the strict logical system of this course, but also reflect the development trend of modern mathematics, absorb and adopt modern mathematical ideas and advanced processing methods, and improve students' mathematical literacy. Many people say it's hard to tell the difference, which is true. However, it is quite simple compared with the last question of mathematics in the college entrance examination. I mean, compared to complexity. It is very important to learn a subject well through thinking and understanding, especially a subject with strong mathematical logic thinking. Of course, there is a lot of hard work. I think if a person only holds one book in class every day but rarely turns over the books, he will be at a loss. After all, it is not difficult to learn, but as long as he studies hard, it is actually a very basic course, laying the foundation for many mathematics majors in the future. I recommend some books that you can read. I recommend Fudan Chen's book and Chen Jixiu's book, but the topic after class is better than the last one. It is best not to use Tongji version of calculus. I don't think even novices look at it. Reference books, this is the most important.
The first book is Jimidovich. Although there are many topics in this book, there are not many valuable topics, at least it can be compressed to the original 1/3. There is a book "Selected Examples of Mathematical Analysis" (3 books), which is to compress this set of books, and the level is quite high. In addition, Jimidovich's method is not very good. It is better to believe in books than to have no books. You'd better think of a good way yourself. This book is specially designed for students with moderate learning, and of course experts can also refer to it.
Besides, Guidance for Postgraduate Entrance Examination (Mathematical Analysis), Shandong Science and Technology Publishing House, is hard to find, but it is much better than Jimidovich, and there are almost no non-classic problems. There are more than 300 questions in the book. I suggest reading every question. The same problem will be simpler (or even simpler) than Jimidovich. There are several difficult questions in chapter 6, which are impossible to test. This book is for students above average.
Finally, look at the proof method and solution of difficult problems in mathematical analysis. The coverage of the topic is not very comprehensive, but the solution is classic and much more concise than the above. If you can't finish reading this book, it means that your level is too high Go and compile a textbook!
Because my level is not very high, that's all I can do.