Also called advanced calculus, the oldest and most basic branch of analysis. Generally speaking, it refers to a relatively complete mathematical subject with the general theory of calculus and infinite series as the main content, including their theoretical basis (basic theory of real number, function and limit). It is also a basic course for college mathematics majors. The branch of analysis in mathematics is a branch of mathematics that specializes in studying real numbers and complex numbers and their functions. Its development began with calculus and extended to the continuity, differentiability and integrability of functions. These characteristics help us to study the material world and discover the laws of nature.
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, called "mathematical analysis", which is translated into Chinese.
Basic method:
The basic method of mathematical analysis is 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 gradually clarified the concepts of function and limit. 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, a critical transition period of new mathematical analysis characterized by arithmeticization of analysis (making analysis a deductive system like arithmetic) began. 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, riemann sum Dabu established the strict integral theory of bounded functions in 1854 and 1875, and Dai Dejin and others completed the strict real number theory in the second half of19th century. 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.