First, the history of the development of calculus Calculus really became a mathematical discipline in the seventeenth century, but before that, calculus had developed slowly step by step with the pace of human history. Focusing on the whole development history of calculus, it is divided into four periods: 1. Early embryo. 2. Establish the molding cycle. 3. Mature and perfect period. 4. Modern development period.
Early germination stage:
1, the embryonic period of the ancient west: in the seventh century BC, Thales' research on the area, volume and length of graphics included early calculus ideas, although it was not obvious. In the 3rd century BC, Archimedes, a great all-round scientist, used the exhaustive method to calculate the area formulas of parabolic bows, spirals and circles, as well as the surface area and volume formulas of ellipsoid, paraboloid and other complex geometric bodies. His exhaustive method is similar to the limit in calculus now. In addition, he also calculated the approximate value of π, and Archimedes played a certain guiding role in the development of calculus.
2. China ancient budding period: Liu Hui invented the famous "secant" in the later period of the Three Kingdoms, that is, the method of finding the circumference and area of a circle by using inscribed or circumscribed regular polygons to exhaust the circumference. "If you cut it carefully, you will lose very little. If you cut it again, you won't be able to cut it. You will be in tune with the circumference without losing anything. " It is a great pioneering work in the history of mathematics in China to continuously increase the number of sides of a regular polygon and make the polygon closer to the area of a circle.
In addition, the outstanding grandfathers and sons of the Southern Dynasties calculated pi to seven digits after the decimal point, and their spirit is worth learning. In addition, Zu Xuan put forward Zu Qiu principle: "If the potentials are the same, the products cannot be different", that is, two geometric bodies bounded between two parallel planes are cut by any plane parallel to these two planes. If the areas of two sections are equal, the volumes of two geometric bodies are equal, which is ten centuries earlier than cavalieri's principle in Europe. Zu Xuan used Mohe square cover (the volume ratio of Mohe square cover to inscribed sphere is 4: π) to calculate the volume of sphere, and corrected the wrong formula of sphere volume in Liu Hui's Notes on Nine Chapters of Arithmetic.
Establish molding period:
1. The first half of the seventeenth century: During this period, almost all the masters of science devoted themselves to solving problems such as speed, extremum, tangent and area, especially the infinitesimal algorithm for describing motion and change, and made great progress in a short time. Astronomer Kepler discovered the three laws of planetary motion, and used the idea of summation of infinitesimal to find the area of curved polygon and the volume of rotator. Cavalieri, an Italian mathematician, also discovered cavalieri's principle and defined the integral formula of power function by the undivided method. In addition, cavalieri also proved Gilding's theorem (the volume of a three-dimensional figure obtained by rotating a plane figure around an axis is equal to the product of the circumference of a circle formed by the center of gravity of the plane figure and the area of the plane figure). ) has a profound influence on the formation of the rudiment of calculus.
In addition, the algebraic method of Descartes, the founder of analytic geometry, also greatly promoted the development of calculus. Fermat, a great French mathematician, has made great contributions to finding the tangent of curve and the extreme value of function. Among them, there is Fermat's theorem about mathematical analysis: let the function f(x) be defined on a certain interval χ, and take the maximum (minimum) value at the inner point c of this interval. If there is a finite derivative f'(c) at this point, there must be f'(c)=0.
2./kloc-the second half of the 7th century: British scientist Newton began to study calculus. Inspired by Wallis' arithmetica infinitorum, he extended algebra to the field of analysis for the first time. 1665, Newton invented downstream counting (difference), and the following year he invented countercurrent counting. After that, we summarized the flow number technology and wrote a brief introduction to flow number, which marked the birth of calculus.
Then, Newton studied the variable flow generation method, and thought that variables were generated by the continuous movement of points, lines or surfaces. Therefore, he called variable flow and variable rate flow. In the later period of Newton's creation of calculus, he denied that variables were a set of static infinitesimal elements, and no longer emphasized that mathematical quantities were composed of inseparable smallest units, but were generated by the constant movement of geometric elements. He no longer thinks that the flow number is the ratio of two real infinitesimals, but the initial ratio of the original quantity or the final ratio of the vanishing quantity, which leads to the infinite division process of the quantity from the original real infinitesimal point of view, that is, the potential infinity point of view.
At the same time, German mathematician Leibniz also independently founded calculus. He published his first differential paper in 1684, defined the concept of differential, and adopted differential symbols dx and dy. 1686 published an integral paper, which discussed differential and integral, and used the integral symbol ∫. The invention of symbols makes the expression of calculus easier. In addition, he also discovered the Leibniz formula for finding higher derivatives and Newton Leibniz formula for connecting differential and integral operations, which contributed as much to calculus as Newton.
Newton and Leibniz played an important role in the creation of calculus. We don't need to argue about who is the real founder of calculus. In the field of mathematics, this is really a boring thing, because every mathematical discovery is the wealth of all mankind, and real mathematicians will never have the mind to talk about such a list of problems!
Maturity period:
1. The beginning of the second mathematical crisis: Calculus gradually took shape in Newton and Leibniz's era, but the establishment of any new mathematical theory will arouse some people's strong doubts at first, and so will calculus. Due to the imprecision in the initial stage of the establishment of calculus, many restless molecules found loopholes in attacking calculus, the most famous of which was the attack on calculus by British Bishop Becquerel in the process of derivative (Δ x is neither 0 nor 0), thus starting the second mathematical crisis.
2. The solution of the second mathematical crisis: After the crisis, many mathematicians realized the theoretical rigor of calculus, and a large number of outstanding scientists appeared one after another. In the pre-crisis period, Bulcha Chanoh, a Czech mathematician, made a detailed study on the properties of functions, gave proper definitions of continuity and derivatives for the first time, put forward the correct concepts of convergence of sequence and series, and put forward the famous Bulcha Noh-Cauchy convergence principle (the necessary and sufficient condition for the finite limit of all sequence variables χ n is that there is always a sequence number n every ε > 0, so that when n > n and n' > n, Later, Cauchy, a great mathematician, established the limit close to the modern form, and defined infinitesimal as a variable close to 0, thus ending the century-old debate and defining the continuity, derivative, integral and convergence of series of continuous functions (at the same time as Bulcha). Cauchy has made great contributions to calculus (mathematical analysis): Cauchy mean value theorem, Cauchy inequality, Cauchy convergence criterion, Cauchy formula, Cauchy integral discrimination method.
In addition, Abel (whose greatest contribution is that he first thought of reverse thinking and opened up a vast world of elliptic integrals) pointed out that the abuse of series expansion and summation should be strictly limited, and Dirichlet gave a modern definition of function. In the later period of the crisis, the mathematician Wilstrass put forward a morbid function (a function that is continuous everywhere but differentiable everywhere), and then someone discovered a function that is discontinuous everywhere but integrable everywhere, which made people realize the relationship between continuity and differentiability again. He put forward the first and second theorems on continuous closed interval, introduced the definition of limit ε ~ δ, basically realized the arithmetic of analysis, and liberated from geometric intuitive limit.
Then, on this basis, Riemann, 1854 and Dabu established a strict theory of bounded function integration in 1875, and Dai Jinde and others established a strict theory of real numbers in the second half of19th century.
At this point, the theory and method of mathematical analysis (including the whole calculus) are completely based on a solid foundation.
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