Calculus usually refers to calculus, which is an important branch of mathematics.

First, what is calculus?

Calculus (Latin small stones for counting) is a branch of mathematics that studies limit, differential calculus, integral calculus and infinite series, and has become an important part of modern university education. Historically, calculus used to refer to infinitesimal calculations. More essentially, calculus is a science that studies changes, just as geometry is a science that studies shapes, and algebra is a science that studies algebraic operations and solutions.

Calculus is widely used in science, economy, engineering and other fields to solve problems that cannot be effectively solved by algebra alone. Calculus is based on algebra, trigonometry and analytic geometry, including differential calculus and integral calculus. Differential calculus, including the calculation of derivatives, is a set of theories about the rate of change. It enables functions, velocities, accelerations and slope of curves to be derived from a set of universal symbols. Integral calculus, including the calculation of integral, provides a set of general methods for defining and calculating area and volume. The basic theorem of calculus points out that differential and integral are inverse operations, which is why the two theories are unified into calculus. We can discuss calculus with either of them as the starting point, but we usually introduce differential calculus into teaching first. In the deeper field of mathematics, calculus is usually called analysis and is defined as the science of studying functions.

Second, the basic concepts

Calculus has three main branches: limit, differential calculus and integral calculus. The basic theory of calculus shows that differential and integral are reciprocal operations. Newton and Leibniz discovered this theorem, which caused other scholars to study calculus enthusiastically. This discovery also enabled us to convert between differential and integral. This basic theory also provides a method to calculate many integral problems by algebra, that is, to replace the limit operation method with indefinite integral method. This theory can also solve some problems of differential equations and solve the integration of unknowns. Differential problems are everywhere in science.

The basic concepts of calculus include function, infinite sequence, infinite series and continuity, and the operation methods mainly include symbolic operation skills, which are closely related to elementary algebra and mathematical induction.

Calculus has been extended to differential equations, vector analysis, variational methods, complex analysis, time domain differentiation and differential topology. Modern version of calculus is real analysis.

Third, the history of calculus

(1) Ancient times

The thought of ancient mathematics tends to be integral, but it is not rigorous and systematic. One of the tasks of integration, that is, calculating volume and area, can be found in Egyptian Moske papyrus (BC 1820). Its formula is also very simple, the method is not said, and the main components are not complete. The origin of integral is very early. In ancient Greece, eudoxus (408-355 BC) found a special graphic area by exhaustive method. Archimedes (287-2 BC12) exhausted the circumference of a circle with the circumference of an inscribed regular polygon, and obtained the approximate value of pi. A series of triangles are also used to fill the figure of parabola to find its area. These are classic examples of exhaustive method. After the 3rd century AD, Liu Hui of China also used the exhaustive method to find the area of a circle. Five centuries later, Zu Chongzhi came up with an algorithm to calculate the volume of a sphere, also called cavalieri formula.

(2) Modern

One motive force for developing modern calculus theory is to solve the tangent problem, and the other motive force is the area problem.

After the Renaissance, based on practical needs and theoretical discussion, integration skills have been further developed. For example, for the convenience of navigation, Gerardus Mercator invented the so-called Mercator projection method, which made the straight line on the map become a diagonal line to keep the orientation when sailing. In Europe, the basic argument comes from Bonaventura cavalieri, who thinks that volume and area should be calculated by finding the total number of infinitesimal cross sections. His idea was similar to Archimedes' methodology, but cavalieri's manuscript was lost and was not found until the early 20th century. Cavalieri's efforts were not recognized, because his method had great errors and infinitesimal was not taken seriously at that time.

/kloc-The first half of 0/7th century is the gestation period of calculus. In the process of exploring ideas, calculation is individual and application is individual. Subsequently, Gottfried Wilhelm Leibniz and isaac newton matured the concept of calculus almost at the same time, clarified the relationship between differential and integral, systematized calculation, and applied calculus to the study of geometry and physics on a large scale.

Before they founded calculus, people regarded differential and integral as independent subjects, and then they really divided this subject into "calculus".

In the formal study of calculus, pierre de fermat claimed that he borrowed the results of Diophantine and introduced the concept of "sufficiency", which is equivalent to infinitesimal error. Unfortunately, he failed to understand the close relationship between the two. John wallis (mathematician), Isaac Barrow and James Gregory completed the combinatorial argument. Isaac barrow, Newton's teacher, knew that there was a reciprocal relationship between them, but he could not understand the meaning of this relationship. One of the reasons is that there is no systematic derivative calculation method. The success of plane geometry in ancient Greece has a far-reaching influence on western mathematics: it is generally believed that only geometric demonstration methods are rigorous and real mathematics, and algebra is only an auxiliary tool. It was not until Descartes and Fermat advocated the study of geometry by algebraic method that this attitude gradually changed. However, on the one hand, the geometric thinking mode is deeply rooted in people's hearts, on the other hand, the algebraic method is not mature, and the real number system has not been established for a long time, so many mathematicians still stick to the geometric camp and cannot develop effective calculation methods, and Barrow is one of them. Although Newton gave up his teacher's view of pure geometry and developed an effective differential method, he dared not publish it. Newton used the skills of calculus to explain his cosmic system from the law of universal gravitation and the law of motion, and solved the problems of celestial motion, the surface of fluid rotation, the oblateness of the earth, the motion of heavy objects on the cycloid and so on. Newton used unique symbols to calculate when solving mathematical and physical problems. In fact, these are the product law, chain law, higher derivative, Taylor series and analytic equation. However, for fear of people's criticism at that time, he erased the trace of calculus in his masterpiece Mathematical Principles of Natural Philosophy (1687) and discussed it in a classical geometric way. In other works, Newton used the power of fractions and irrational numbers. Obviously, Newton knew Taylor series law. But he did not publish these findings, because infinitesimal was still controversial at that time.

The above viewpoints were integrated by Gottfried Wilhelm Leibniz into a truly infinitesimal version of calculus, while Newton accused the former of plagiarism. Leibniz is regarded as another person who invented calculus independently today. His contribution lies in his rigorous style, which makes it easy to calculate the second or higher order derivatives, and gives the product rule and chain rule in the form of differential and integral. Unlike Newton, Leibniz paid great attention to form and often studied appropriate symbols day after day.

Leibniz and Newton are both considered as independent inventors of calculus. Newton first applied calculus to general physics, while Leibniz created most of today's symbols. Newton and Leibniz both gave the basic methods of differential and integral, the second or higher derivative, the symbol of approximate value of sequence and so on. In Newton's time, the basic formula of calculus has been known to the world.

When Newton and Leibniz published their respective achievements for the first time, a long-lasting debate broke out in the field of mathematics about the ownership and priority of inventing calculus. Newton was the first to draw a conclusion, and Leibniz was the first to publish it. Newton claimed that Leibniz plagiarized his unpublished manuscript, which was supported by Newton's Royal Society. This great debate divided mathematicians into two factions: one was a British mathematician who defended Newton; The other group is mathematicians from continental Europe. The result was against the British mathematician. After careful verification in the future, Newton and Leibniz independently reached their own conclusions. Leibniz deduced from integral and Newton from differential. Today, Newton and Leibniz are considered as two independent authors who invented calculus. The name of "calculus" and its operation symbols were created by Leibniz, and Newton called it "flow number".

Calculus has been constantly improved by many people, and it is also inseparable from the contributions of Barrow, Descartes, Fermat, Huygens, Wallis and others. The earliest complete analytical works about finite and infinitesimal were summarized and edited by Maria Gaetana Agnesi in 1748. Newton and Leibniz systematized calculus, but it was not rigorous enough. However, when calculus was successfully used to solve many problems, mathematicians in the 18th century were more inclined to its application than to its rigor. At that time, the development of calculus was in the hands of several excellent mathematicians, such as Euler, Lagrange, Laplace, D'Alembert and Bernoulli family. The problems studied come from natural phenomena, so many inferences of calculus can be verified by data of natural phenomena, so that calculus will not imply errors because of unstable foundation. In the hands of these mathematicians, the scope of calculus quickly surpassed the calculus course taught in the early stage of university and moved towards a more advanced analytical science.

Source: Wikipedia.