background
The concept of "infinity" sprouted in the ancient West and China. "Circlectomy" is the withdrawal of this idea. Archimedes got the value of pi from 223/7 1 to 22/7 by using the regular 96-sided shape in the circle. The famous mathematicians in Wei and Jin Dynasties even used the amazing regular 3072-sided shape in the circle to make the value of pi accurate to 3. 14 16. These methods all embody the calculus mathematical thought of "infinite division and infinite summation". However, due to the low level of production practice, it is difficult to further develop and improve these ideas.
Time soon reached the16th century, and the level of social production practice reached a new level. The rapid development of astronomy and physics has brought many mathematical problems, such as how to find instantaneous velocity and acceleration and how to calculate the area of curved triangle. After entering the 1 7th century, scientists gradually focused on four kinds of problems:1. Knowing the displacement-time relationship function of an object, find its velocity and acceleration at any time; Conversely, the acceleration-time function of the object is known, and the velocity and displacement are obtained. 2. Find the tangent of the known curve. 3. Find the maximum and minimum of the known function. 4. Find the length of the curve, the area enclosed by the curve, the volume enclosed by the curved surface, the position of the center of gravity of the object, and the attraction of an object (such as a planet) to another object. In the exploration of these problems, Descartes, Barrow (Newton's teacher at Cambridge University, one of the early pioneers of calculus), Kepler, cavalieri (Italian mathematician, discoverer of the western "origin principle") and other scientists have made groundbreaking contributions. However, there is still no complete theory. With the accumulation of a large number of knowledge and methods, a brand-new subject stands out.
Giants and Masters: Newton and Leibniz
Newton (1642- 1727) was born in a pure peasant family. After his father died young, his mother was forced to remarry to a priest, and Newton lived with his grandmother. The cruel family situation caused Newton's taciturn and stubborn character. Newton's grades in middle school were not outstanding, but he was quite curious and curious. The discerning middle school principal and Newton's uncle encouraged Newton to go to college, so Newton entered Trinity College of Cambridge University as a student with reduced tuition fees and began his road of scientific giants.
According to records, Newton's research on calculus began in 1664. At this time, he carefully studied Descartes' masterpiece Geometry and was fascinated by the method of finding the tangent of the curve in the book. Newton, who was eager for knowledge, urgently sought a more effective and universal method to solve this problem.
After two years of thinking, 1666+00, in June, Newton wrote the first calculus paper in the history of mathematics, Short Theory of Flow Number, and put forward the concept of "Flow Number" historically. Newton mapped the "flow number" to speed, that is, the WeChat service of displacement function to time, and then took the WeChat service of speed versus time as acceleration. After three years of careful consideration, Newton finished his second paper, Analysis with Infinite Polynomial Equation. In this paper, the general method of finding the instantaneous change rate of dependent variable to independent variable is given, and it is also proved that the inverse process of finding the change rate can get the area, which is actually very close to the basic theorem of calculus (Newton-Leibniz formula). 167 1 year, Newton perfected the first article in his third paper, Flow and Infinite Series, which made the discussion and method clearer. Five years later, Newton wrote his most mature calculus paper, Theory of Quadrature of Curves, which further improved the understanding of convection number, clearly described the basic theorem of calculus, and gave a series of marks invented by himself.
At this point, a generation of giants completed the great feat of creating calculus. However, due to his conservative and introverted personality, Newton did not publish his paper publicly for a long time, and only a few of his friends knew it. It was not until 1687 that Newton published his masterpiece Mathematical Principles of Natural Philosophy with the encouragement and request of his good friend Harley. Only then did Newton's calculus work become public. It was Newton's hesitation that triggered a century-long debate between Newton and Leibniz, the "father of calculus", and also caused a long-term gap between the British scientific community and the European scientific community.
Leibniz (1646- 17 16) was born in Leipzig, Germany. His research fields cover mathematics, physics, philosophy, history, biology, machinery and theology, and he is a rare genius and all-rounder in human history. At the same time, Leibniz is also a fanatic of China culture. In Leibniz era, Germany lagged behind Britain in science education and development.
1672, Leibniz came to Paris and began to study mathematics with Huygens' encouragement. A year later, Leibniz visited London, got a lecture on Barrow's geometry and heard about Newton's work from some mathematicians. After returning to Paris, Leibniz studied the works of Pascal, Descartes and cavalieri. Three years earlier than Newton, he published the first calculus paper in history, as if to prove the epoch-making significance of the paper, Leibniz took a long name: "A new method for finding minimax and tangents is also applicable to fractions and irrational numbers, and the wonderful type of calculation of this new method". In this paper, Leibniz gives the differential symbols and laws close to modern times. In the manuscript of 1677, Leibniz also gave a rough description of the basic theorem of calculus. Nine years later, Leibniz published the article "Analyzing abstruse geometry, indispensability and infinity", and discussed the relationship between integral and differential again.
At the same time, Leibniz is very keen to find simple symbols to simplify the calculation. Today, most calculus symbols come from Leibniz.
Newton studied calculus earlier, but Leibniz published the results earlier, but an argument was inevitable. Because of this, Britain, which was isolated overseas, cut off its contact with the European continent for almost a long time, which led to the backwardness of mathematics and even science in Britain.
However, Newton and Leibniz's description and use of the concept of infinitesimal are ambiguous, sometimes they regard it as an uncertain quantity, and sometimes they regard it as a qualitative "0", so the calculus theory has been criticized and questioned for a long time.
Stiffness of analysis
The appearance of calculus quickly gave birth to a series of new branches of mathematics, such as differential equation, differential geometry, function theory, variational analysis and so on. Mathematics belongs to the era of analysis, but the rigor of calculus theory is still a big problem for countless mathematicians.
The first mathematician who made a bold attempt in this field was Porzano (178 1- 1848). He gave a modern expression of the definition of continuous function, and pointed out that dy/dx is just a symbol and should not be understood as a ratio.
Cauchy made the greatest contribution (1789- 1857). 182 1 year, Cauchy published three important works: Analysis Course, Calculus Lecture Notes and Calculus Application in Geometry, and gave a series of strict definitions of calculus. First of all, he regards infinitesimal as a variable with a limit of 0, thus solving the long-term ambiguous situation of infinitesimal "like 0 is not 0" in one fell swoop. On this basis, he gave strict definitions of a series of concepts such as continuity, differentiation, integration and derivative. But his description of the definition of limit still uses a lot of literal things, which is not in line with the pursuit of mathematicians.
The famous "ε-δ" language about limit was put forward by the German mathematician Wilstras (18 15- 1897) half a century later. 19 century later, real number theory and set theory have been developed unprecedentedly. Dai Dejin (1831-1965438, a Gaussian student) and Cantor (1845-1965448). After decades of efforts, the historical task of analytical science rigor has finally come to a successful conclusion, ending the 300-year-long "melee" among all parties, making analytical science as rigorous as European geometry. The era of analysis has reached an unprecedented climax, and the development of various branches is becoming more and more prosperous.