history
Calculus became a subject in the17th century, but the idea of integral appeared in ancient times.
The early history of integral calculus
In the 7th century BC, Thales, an ancient Greek scientist and philosopher, studied the area, volume and length of a ball, which included the idea of calculus. In the 3rd century BC, Archimedes (287 ~ 2 BC12 BC), an ancient Greek mathematician and mechanic, wrote Measurement of a Circle and Measurement on a Sphere and a Cylinder, which contained the seeds of integration. When he studied and solved the problems such as the arcuate area under parabola, the area under spiral and the volume obtained by rotating hyperbola, he implied the idea of modern integration. China's ancient mathematicians also had the seeds of integral calculus, such as Liu Hui in the Three Kingdoms period. His thoughts on integral calculus mainly include two points: secant and volume.
Calculus generation
In the seventeenth century, there were many scientific problems to be solved, and these problems became the factors that prompted calculus. To sum up, there are mainly four kinds of problems: the first kind is the problem that appears directly when learning physical education, that is, the problem of finding the instantaneous speed. The second kind of problem is to find the tangent of the curve. The third kind of problem is to find the maximum and minimum of a function. The fourth problem is to find the length of the curve, the area enclosed by the curve, the volume enclosed by the surface, the center of gravity of the object, and the gravity of an object with a considerable volume acting on another object. Mathematics first introduces a basic concept (such as astronomy, navigation, etc.) from the study of sports. ), and in the next two hundred years, this concept occupies a central position in almost all works, which is the concept of function-or the relationship between variables. With the adoption of the concept of function, calculus came into being, which is the greatest creation in all mathematics after Euclidean geometry. Focusing on solving the above four core scientific problems, calculus problems were explored by at least a dozen largest mathematicians and dozens of smaller mathematicians in the17th century. Its founders are generally considered to be Newton and Leibniz. The works of these two masters are mainly introduced here.
In fact, before Newton and Leibniz sprinted, they had accumulated a lot of knowledge of calculus. /kloc-many famous mathematicians, astronomers and physicists in the 0/7th century did a lot of research work to solve the above problems, such as Fermat, Descartes, Roberts and Gilad Girard Desargues. Barrow and Varis in Britain; Kepler in Germany; Italian cavalieri and others put forward many fruitful theories. Contributed to the creation of calculus.
For example, Fermat, Barrow and Descartes all studied the tangent of the curve and the area surrounded by the curve in depth and got some results, but they didn't realize its importance. In the first two thirds of the seventeenth century, the work of calculus was lost in details, and they were exhausted by trivial reasoning. Only a few great mathematicians are aware of this problem. For example, James Gregory said, "The real division of mathematics is not divided into geometry and arithmetic, but into universality and particularity". This universal thing was provided by two all-encompassing thinkers Newton and Leibniz. /kloc-In the second half of the 7th century, Newton, a great British scientist, and Leibniz, a German mathematician, independently studied and completed the creation of calculus in their respective countries on the basis of their predecessors' work, although this was only a very preliminary work. Their greatest achievement is to connect two seemingly unrelated problems, one is the tangent problem (the central problem of differential calculus) and the other is the quadrature problem (the central problem of integral calculus).
Newton and Leibniz established calculus from the perspective of intuition and infinitesimal, so this subject was also called infinitesimal analysis in the early days, which is the origin of the name of the big branch of analysis in mathematics. Newton's research on calculus focused on kinematics, while Leibniz focused on geometry.
newton
Newton wrote Flow Number and Infinite Series at 167 1, and it was not published until 1736. In this book, Newton pointed out that variables are produced by the continuous motion of points, lines and surfaces, which negated his previous view that variables are static sets of infinitesimal elements. He called continuous variables flow, and the derivatives of these flows were called flow numbers. Newton's central problems in flow number technology are: knowing the path of continuous motion and finding the speed at a given moment (differential method); Given the speed of motion, find the distance traveled in a given time (integral method).
Leibniz
Leibniz of Germany (also translated as "Leibniz") is a knowledgeable scholar. 1684, he published what is considered to be the earliest calculus literature in the world. This article has a very long and strange name, "The new method of finding minimax and tangents is also applicable to fractions and irrational numbers, and the wonderful type of calculation of this new method". It is such a vague article, but it has epoch-making significance. It already contains modern differential symbols and basic differential laws. 1686, Leibniz published the first document on integral calculus. He is one of the greatest semiotics scholars in history, and his symbols are far superior to Newton's, which has a great influence on the development of calculus. The universal symbol of calculus that we use today was carefully chosen by Leibniz at that time.
basic content
mathematical analysis
It is the basic method of calculus to study the function and motion changes of things from the quantitative aspect. This method is called mathematical analysis.
Generalized mathematical analysis includes calculus, function theory and many other branches, but now it is generally customary to equate mathematical analysis with calculus, and mathematical analysis has become synonymous with calculus. When it comes to mathematical analysis, you know that it refers to calculus.
calculus
The basic concepts and contents of calculus include differential calculus and integral calculus.
The main contents of differential calculus include: limit theory, derivative, differential and so on.
The main contents of integral include definite integral, indefinite integral and so on.