What is calculus? It is a mathematical idea, in which' infinite subdivision' is differential and' infinite summation' is integral. Infinity is the limit, and the idea of limit is the basis of calculus. It looks at the problem with a concept of movement. For example, the instantaneous speed of a bullet flying out of a gun bore is a differential concept, and the sum of the distances traveled by a bullet at each moment is an integral concept.
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 productive forces, 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 "variable mathematics". That is, calculus has been constantly improved and become a discipline. In the whole17th century, dozens of scientists made pioneering research for the creation of calculus, but it was Newton and Leibniz who made calculus an important branch of mathematics.
Calculus became a discipline in the17th century, but the idea of differential and integral appeared in ancient times. In the 3rd century BC, the ancient Greek mathematician and mechanic Archimedes (287-2 BC12 BC) had already included the seeds of calculus in his works "Measurement of Circle" and "Measurement of Ball and Column". When he studied and solved the problems of arc area under parabola, area under spiral and volume of hyperbola, he implied the idea of modern integration. As the basic limit theory of calculus, it was discussed in detail as early as ancient China. For example, Zhuang Zhou's book "The World" wrote that "a foot of pestle, half a day, is inexhaustible". If you lose less, you can't cut it, but if you fit in with the circle, you won't lose. "In his book" The New Science of Measuring the Volume of Barrels "16 15, 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. Italian mathematician cavalieri was born in 15.
/kloc-the development of productivity in the 0/7th century has promoted the development of natural science and technology. Not only have the existing mathematical achievements been further consolidated, enriched and expanded, but also due to the needs of practice, the concept of variables has been obtained, and the generality of variables and their dependencies has been studied. In the second half of the17th century, isaac newton (1642- 1727), a great British mathematician and physicist, studied calculus from the perspective of physics on the basis of previous creative research. In order to solve the problem of motion, he founded a mathematical theory directly related to physical concepts, which Newton called "flow number". This is actually the theory of calculus. Newton's main works on "flow counting" include "Finding the area of a curved polygon", "Calculation method by using infinite polynomial equation" and "Flow counting and infinite number of points". 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 put the geometry-
Newton pointed out that "flow counting" basically includes three problems.
(l) "Knowing the relationship between streams and finding the relationship between them" is equivalent to differential calculus.
(2) Knowing the equation representing the relationship between streams, we can find the relationship between corresponding streams. This is equivalent to integral calculus. Newton's 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 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 of geometry. Before Newton and Leibniz, at least dozens of mathematicians studied it, and they made pioneering contributions to the birth of calculus. However, our work is fragmented and incoherent. Lack of 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, but Leibniz's symbols are still used today. Leibniz realized earlier and more clearly than others that good symbols can greatly save thinking.