If the whole mathematics is compared to a big tree, then elementary mathematics is the root of the tree, each branch of mathematics is the branch, and the main part of the trunk is calculus. Calculus is one of the greatest achievements of human wisdom. Since the17th century, with the progress of society and the development of productivity, as well as many problems to be solved in navigation, astronomy and mine construction, mathematics has also begun to study the quantity of change, and mathematics has entered the era of "variable mathematics", that is, calculus has been continuously improved and become a discipline. In the whole17th century, dozens of scientists did pioneering research for the creation of calculus, but it was Newton and Leibniz who made calculus an important branch of mathematics.
Calculus became a subject in17th century, but the idea of differential and integral existed in ancient times. In the 3rd century BC, Archimedes (287-2 12 BC), an ancient Greek mathematician and mechanic, had already contained the seeds of calculus in his works such as Measurement of Circle and On Spheres and Cylinders. When he studied and solved the problems of arcuate area under parabola, area under spiral and volume of hyperbola of revolution, he implied the idea of modern integration. As the basic limit theory of calculus, it was discussed in great detail as early as ancient China. For example, in the book "Zhuangzi" written by Zhuang Zhou, "One foot of space is inexhaustible." During the Three Kingdoms period, Liu Hui proposed in "Cutting Circle" that "if you cut it carefully, the damage will be small. If you can't cut it, you will be encircled and have nothing to lose". In his book 16 15, The New Science of Measuring the Volume of Barrels, he regarded the curve as a straight line with infinitely increasing sides. The area of a circle is the sum of the areas of an infinite number of triangles, which is a masterpiece of typical limit thought. Cavalieri, an Italian mathematician, published The Geometry of Continuity in 1635, and regarded the curve as an infinite number of line segments (indispensable). All these made ideological preparations for the birth of calculus later.
/kloc-the development of productivity in the 0/7th century promoted the development of natural science and technology. Not only the existing mathematical achievements have been further consolidated, enriched and expanded, but also because of the need of practice, we began to study the moving objects and the changing quantity, thus obtaining the concept of variables and studying the generality of the changing quantity and its dependence. /kloc-in the second half of the 7th century, isaac newton (1642- 1727), a great British mathematician and physicist, studied calculus from the perspective of physics on the basis of creative research by predecessors. In order to solve the problem of motion, he created a mathematical theory directly related to physical concepts, which Newton called "flow counting", but it was actually. Newton's main works on "flow counting" include "Finding the area of a curved polygon", "Calculation method using infinite polynomial equation" and "Flow counting and infinite pole number". These concepts are mathematical reflections of mechanical concepts. Newton thought that any movement exists in space and depends on time, so he took time as an independent variable and time-related solid variables as a flow. Not only that, he also regarded geometric figures-lines, angles and shapes-as the result of mechanical displacement. Therefore, all variables are flow.
Newton pointed out that "flow counting" basically includes three types of problems.
(l) "Knowing the relationship between streams and finding the relationship between them" is equivalent to differential calculus.
(2) Know the equation representing the relationship between streams and find the relationship between corresponding streams. This is equivalent to integral calculus. Newton integral method includes not only solving the original function, but also solving the differential equation.
(3) The application scope of "flow number technology" includes calculating the maximum and minimum values of curves, finding the tangent and curvature of curves, finding the length of curves and calculating the area of curved edges.
Newton has fully realized that the operations in the above two kinds of problems (1) and (2) are reciprocal operations, so he established the connection between differential calculus and integral calculus.
Newton mentioned "flow counting" in a manuscript on May 20, 1665, so some people took this day as the symbol of the birth of calculus.
Leibniz made calculus more concise and accurate.
German mathematician Leibniz (G.W. Leibniz1646-1716) discovered calculus independently from geometry. Before Newton and Leibniz, who made pioneering contributions to the birth of calculus, at least dozens of mathematicians had studied calculus. However, our work is fragmented, incoherent and lacks unity. Leibniz's way and method of establishing calculus are different from Newton's. Leibniz introduced the concept of calculus by studying the tangent of the curve and the area surrounded by the curve, and got the algorithm. Newton combined kinematics more in the application of calculus, and his attainments were better than Leibniz's, but Leibniz's expression was far better than Newton's, which not only revealed the essence of calculus concisely and accurately, but also effectively promoted the development of higher mathematics.
The symbols of calculus created by Leibniz promoted the development of calculus, just as Indo-Arabic numerals promoted the development of arithmetic and algebra. Leibniz is one of the most outstanding symbol creators in the history of mathematics.
Newton's differential and integral symbols are not used now, while Leibniz's symbols are still used today. Leibniz realized earlier and more clearly than others that good symbols can greatly save thinking labor, and the skill of using symbols is one of the keys to the success of mathematics.