The concepts of limit and calculus can be traced back to ancient times. In the second half of the 17th century, Newton and Leibniz completed the preparatory work that many mathematicians participated in, and independently established calculus. Their starting point of establishing calculus is intuitive infinitesimal, and their theoretical foundation is not solid. It was not until the19th century that Cauchy and Wilstrass established the limit theory, and Cantor and others established the strict real number theory that the discipline was rigorous.
Calculus is developed with practical application, and is widely used in various branches of natural science, social science and applied science such as astronomy, mechanics, chemistry, biology, engineering and economics. In particular, the invention of computers is more conducive to the continuous development of these applications.
Calculus is a general term for differential calculus and integral calculus.
Everything in the objective world, from particles to the universe, is always moving and changing. Therefore, after introducing the concept of variables into mathematics, it is possible to describe the movement phenomenon in mathematics.
Due to the emergence and application of the concept of function and the needs of the development of science and technology, a new branch of mathematics has emerged after analytic geometry, which is calculus. Calculus plays a very important role in the development of mathematics. It can be said that it is the greatest creation in all mathematics after Euclidean geometry.
The establishment of calculus
Calculus became a discipline in the seventeenth century, but the idea of differential and integral existed in ancient times.
In the third century BC, Archimedes of ancient Greece implied the idea of modern integral calculus when he studied and solved the problems of parabolic arch area, spherical surface and spherical cap area, area under spiral, and volume of hyperbola of rotation. As the basis of differential calculus, limit theory has been clearly discussed as early as ancient times. For example, the book Zhuangzi written by Zhuang Zhou in China records that "one foot of space is inexhaustible." Liu Hui in the Three Kingdoms period mentioned in his "Cutting Circle" that "if you cut it carefully, you will lose less, and if you cut it again, you will not even lose your circumference and body." These are simple and typical limit concepts.
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.
/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.
/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 intuitive infinitesimal, so this subject was also called infinitesimal analysis in the early days, which is also the source of the name of the big branch of mathematics now. Newton's research on calculus focused on kinematics, while Leibniz focused on geometry.
Newton wrote Flow Method 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, and denied 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 of Germany is a knowledgeable scholar. 1684, he published what is considered to be the earliest calculus literature in the world. This article has a long and strange name: a new method for finding minimax and tangents, which is also applicable to fractions and irrational numbers, and the wonderful types of calculation of this new method. It is such a vague article, but it has epoch-making significance. He is famous for containing 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. Leibniz carefully chose the universal symbol of calculus that we use now.
The establishment of calculus has greatly promoted the development of mathematics. In the past, many problems that elementary mathematics was helpless were often solved by calculus, which shows the extraordinary power of calculus.
As mentioned above, the establishment of a science is by no means a person's achievement. It must be completed by one person or several people through the efforts of many people and on the basis of accumulating many achievements. So is calculus.
Unfortunately, while people appreciate the magnificent function of calculus, when they put forward who is the founder of this subject, it actually caused an uproar, resulting in a long-term opposition between European continental mathematicians and British mathematicians. British mathematics was closed to the outside world for a period of time, limited by national prejudice, and too rigidly adhered to Newton's "flow counting", so the development of mathematics fell behind for a whole hundred years.
In fact, Newton and Leibniz studied independently, and completed them in roughly the same time. More specifically, Newton founded calculus about 10 years earlier than Leibniz, but Leibniz published all the theories of calculus three years earlier than Newton. Their research has both advantages and disadvantages. At that time, due to national prejudice, the debate about the priority of invention actually lasted from 1699 to 100 years.
It should be pointed out that this is the same as the completion of any major theory in history, and the work of Newton and Leibniz is also very imperfect. They have different views on infinity and infinitesimal, which is very vague. Newton's infinitesimal, sometimes zero, sometimes not zero but a finite small amount; Leibniz's can't justify himself. These basic defects eventually led to the second mathematical crisis.
Until the beginning of19th century, the scientists of French Academy of Sciences, led by Cauchy, made a serious study of the theory of calculus and established the limit theory, which was further tightened by the German mathematician Wilstrass, making the limit theory a solid foundation of calculus. Only in this way can calculus be further developed.
Any emerging and promising scientific achievements attract the vast number of scientific workers. In the history of calculus, there are also some stars: Swiss Jacques Bernoulli and his brothers johann bernoulli, Euler, French Lagrange, Cauchy …
Euclidean geometry and algebra in ancient and medieval times were constant mathematics, and calculus was the real variable mathematics, which was a great revolution in mathematics. Calculus is the main branch of higher mathematics, and it is not limited to solving the problem of variable speed in mechanics. It gallops in the garden of modern science and technology and has made countless great achievements.
Basic content of calculus
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.
Originally, mathematical analysis in a broad sense included many branches such as calculus and function theory, but now it is widely used to equate mathematical analysis with calculus, and mathematical analysis has become synonymous with calculus. When it comes to mathematical analysis, known refers to 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.
Calculus is developed in application. At first, Newton used calculus and differential equations to deduce Kepler's three laws of planetary motion from the law of universal gravitation. Since then, calculus has greatly promoted the development of mathematics, as well as astronomy, mechanics, physics, chemistry, biology, engineering, economics and other natural sciences, social sciences and applied sciences. And it is widely used in these disciplines, especially the appearance of computers is more conducive to the continuous development of these applications.