Basic introduction Chinese name: mathematical analysis mbth: the subject of mathematical analysis: mathematical research content: theoretical basis of function, limit, calculus and series: the subject characteristics of limit theory: abstract, rigorous, extensive introduction, development history, early development, early establishment, research object, basic methods and correlation. The main content of mathematical analysis is calculus, and the theoretical basis of calculus is limit theory. Calculus is a general term for 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. Early calculus has been used by mathematicians and astronomers to solve a large number of practical problems, but it has not been developed for a long time because it cannot give a convincing explanation for the concept of infinitesimal. Many mathematicians are skeptical about this theory. Cauchy and later Wilstrass perfected the limit theory as the theoretical basis. Get rid of the vague description of limit, such as "the smaller the better" and "infinite tendency", and use accurate mathematical language to describe the definition of limit, so that calculus gradually evolves into a basic mathematical discipline with strict logic, which is called "mathematical analysis" and translated into Chinese as "mathematical analysis". 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. From the early stage of development history to the early stage of ancient Greek mathematics, the results of mathematical analysis are implicitly given. For example, Zhi Nuo's dichotomy paradox implies the sum of geometric series. Later, ancient Greek mathematicians such as eudoxus and Archimedes made the mathematical analysis more explicit, but it was not very formal. When they calculate the area and volume of regions and solids by exhaustive method, they use the concepts of limit and convergence. In the early days of ancient Indian mathematics, Pashgaro, a mathematician in the12nd century, gave a second example of derivative. Archimedes' early establishment of mathematical analysis began with the pioneering work represented by Newton (I) and Leibniz (G.W) in the17th century, and was completed with the pioneering work represented by Cauchy and Weisstras in the19th century. Calculus and its related contents have been called analysis since Newton. Since then, the field of calculus has been expanding, but many mathematicians still use this name. Today, although many contents have been separated from calculus and become independent disciplines, people still call it analysis. Mathematical analysis is also called analysis for short. The research object of mathematical analysis is function, and the basic behavior of function is studied from both local and whole aspects, thus forming the basic content of differential calculus and integral calculus. Differential calculus studies the local characteristics of functions, such as the rate of change. Derivative and differential are its main concepts, and the process of finding derivative is differential method. The main content of differential calculus is formed around the properties, calculation and direct application of derivative and differential. Integral studies the total effect of small changes (especially non-uniform changes) as a whole. Its basic concepts are original function (anti-derivative) and definite integral, and the process of finding integral is integral method. The nature, calculation, popularization and direct application of integral constitute the whole content of integral. Newton and Leibniz's outstanding contribution to mathematics is that around 1670, they summed up a series of basic laws of finding derivatives and integrals, and found that finding derivatives and integrals are two reciprocal operations, and reflected this reciprocal relationship through the famous formula named after them later, thus combining differential calculus and integral calculus, which were originally independently developed, into a new discipline. Because of their contributions and those of some later scholars (especially Euler), only a few mathematicians knew that the differential integral method, which could only deal with some specific problems quite difficultly, became a nearly mechanical method that ordinary people could master with a little training, which opened the door for its wide application in the field of science and technology, and its influence was immeasurable. Therefore, the appearance and development of calculus is regarded as one of the epoch-making events in the history of human civilization. Compared with integral, infinite series is also the superposition and accumulation of tiny quantities, but it takes discrete form (integral is continuous form). Therefore, in mathematical analysis, infinite series and calculus have always been inseparable and complementary. In history, infinite series has been used for a long time, but it was not until it became a part of mathematical analysis that it was really developed and widely used. Newton's basic method The basic method of mathematical analysis is the limit method, or infinitesimal analysis. 1696 The word infinitesimal analysis appears in the titles of L'Hospital's first calculus textbook published in Paris and Euler's Communication between Calculus and Elementary Analysis published in 1748. In the early stage of the development of calculus, this new method showed great power and obtained a lot of important results. Many new branches of mathematics related to calculus, such as variational methods, differential equations, differential geometry and the theory of complex variable functions, were developed in the early18-19th century. However, the initial analysis was relatively rough, and mathematicians inspired by the power of the new method often ignored the logical basis of deduction and used intuitive inference and contradictory reasoning, so that people generally doubted the rationality of this method throughout the18th century. These doubts are largely caused by the meaning and usage of infinitesimal that was often used at that time. Random use and explanation of infinitesimal leads to confusion and mystery. Many people participated in the debate about the essence of infinitesimal, and some of them, such as Lagrange, tried to rule out infinitesimal sum limit and algebraic calculus. Debate makes the concepts of function and limit gradually clear. More and more mathematicians realize that the concept of mathematical analysis must be distinguished from its prototype in the objective world and human intuition. Therefore, from the19th century, Euler began a critical transition period of new mathematical analysis characterized by arithmetic analysis (making analysis a deductive system like arithmetic). Cauchy's analysis course published in 182 1 is a sign of rigorous analysis. In this book, Cauchy established the limit close to the modern form, and defined infinitesimal as a variable tending to zero, thus ending the century-long struggle. Cauchy defined the continuity, derivative, integral and convergence of series of continuous functions on the basis of limit (it was later known that Porzano had also done similar work). Furthermore, Dirichlet put forward the strict definition of function in 1837, and Weierstrass introduced the ε-δ definition of limit. Basically, the arithmetic of analysis is realized, and the analysis is liberated from the limitation of geometric intuition, thus dispelling the mysterious cloud hanging over calculus in the17-18th century. On this basis, Cauchy established the strict integration theory of bounded functions in 1854 and Dabu in 1875. /kloc-in the second half of the 9th century, Dai Dejin and others completed the strict real number theory. At this point, the theories and methods of mathematical analysis are completely based on a solid foundation, basically forming a complete system, paving the way for the development of modern analysis in the 20th century. The generation of related calculus theory is inseparable from the development of physics, astronomy, economics, geometry and other disciplines. Calculus theory has shown great application vitality since it came into being. 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.