The Signalizational Progress Of Mathematical Analysis

Mathematical analysis is an important basic course in China's higher school mathematics, an important course that trains student's thinking, a necessary course for student's subsequent learning. As a huge system today, Mathematical analysis is an achievement by numerous mathematicians to experience successively all sorts of hardship after more than 2000 years of history stage. Particularly ever since Newton and Leibniz invented calculus before more than 300 year, western mathematics including mathematical analysis obtained a huge progress, whereas Chinese mathematics in the interval was placed in opposite fall behind of status, this makes people have to investigate reasons in it.By researching and tidying up the theory and standpoint of numerous experts and scholars, by describing the evolution process of some symbols in the mathematical analysis, and a brief analysis for Chinese mathematicians to usher in western mathematics in the late Ching dynasty, this paper reveals the huge role of the mathematical sign in the progress of mathematics.The full text is totally divided into ten chapters.Chapter 1 introduces four development stages of mathematics sign from the total top synopsis; Chapter 2 comments the early calculus thinking budding, particularly introduces the development process of the method of exhaustion and the application of Archimedes, supplementary to say the extreme limit of thought from The Warring States to the late Ching dynasty; Chapter 3 introduces briefly the foundation background of calculus in 17th century, Particularly the previous development of the integral calculus and differential calculus; Chapter 4 introduces the calculus work of Newton and Leibniz, comments three stages that Newton studied calculus, and analyzes delicately Leibniz's thought method and many calculus achievements, and carries on their work comparison; Chapter 5 elaborates the development process of function and introduces five times expansion of this concept:The analytic expression definition, picture type, science function form, junior high school form, senior high school and university definition etc.; Chapter 6 elaborates relevant sign in detail when the calculus establishes front and back, including the history of differential calculus and derivative, integral calculus, partial differential and partial derivative, indeterminate form and limit etc., and introduces the circumstance that the calculus streamed into China in the late Ching dynasty; Chapter 7 describes the mathematicians work in 18th century, including trying to make up the calculus loophole and the development of each analytic branch; Chapter 8 describes the severizational progress of mathematical analysis in 19th century, including the work of Bolzano and others, Cauchy's limit theory, Weierstrass's language system, Dedekind's partition theory, Peano's natural number axiom etc.; Chapter 9 introduces the application for mathematical logic sign in mathematical analysis; Chapter 10 sums up the whole process that mathematical analysis was signified.By studying the whole process in signifying of mathematical analysis to clear up sequence of events of each important sign in mathematical analysis, it can help to study the development regulation of mathematical sign in the whole maths, can provide good material for learner and fancier, and some teaching reference for the mathematical analysis teacher in college.