Ideological basis of early calculus
/kloc-in the 0/7th century, a series of discoveries by two mathematicians Galileo and Kepler led to the turning point of mathematics from classical mathematics to modern mathematics.
Before the age of 25, Galileo began to do a series of experiments and discovered many basic facts about the motion of objects in the earth's gravitational field, the most basic of which is the law of free fall. Kepler summed it up as the famous three laws of planetary motion around 16 19. These achievements have a great influence on most later branches of mathematics. Galileo's discovery led to the birth of modern dynamics, and Kepler's discovery led to the birth of modern celestial mechanics. In the process of establishing these disciplines, they all feel the need for a new mathematical tool, which is calculus to study the process of movement and transformation.
Interestingly, the origin of integral calculus can be traced back to ancient Greece, but it was not until the17th century that differential calculus made a major breakthrough.
The origin of holistic thinking
The quadrature problem is to find the area and volume of a graph. This problem has a long history, which can be traced back to the calculation of the area and volume of some simple figures by ancient civilizations, such as triangles, quadrangles, circles or spheres, cylinders and cones, and the calculation of the area of circles, spheres, curved triangles and curved quadrangles by Europeans in the17th century. It was not until Newton and Leibniz established calculus that these problems were fundamentally solved. Quadrature problem is one of the main factors to promote calculus.
In the development of integral thought, a group of great mathematicians have made outstanding contributions to it. Archimedes, the great mathematician and mechanic in ancient Greece, Liu Hui and Zu Chongzhi, the famous mathematicians in ancient China, made great contributions to the formation and development of integer thought.
16 and 17 centuries are the most active periods for the development of calculus, and its outstanding representative is Italian astronomer and mechanic Galileo.
There are German astronomers, mathematicians, physicists Kepler, cavalieri and so on. Their work laid the foundation for Newton and Leibniz to establish calculus theory.
The origin of differential calculus thought
Differential calculus mainly comes from the study of two problems, one is the tangent of a curve, and the other is the problem of finding the maximum and minimum of a function. Ancient Greece considered these two issues, but the discussion of these two issues in ancient Greece was far less extensive and in-depth than the discussion of area, volume and arc length.
It is Fermat who has done pioneering work in the study of these two issues. In 1629, Fermat gives the method of finding the maximum and minimum value of the function. However, this idea was not widely known until 1989.
Kepler observed that the increment of a function usually becomes infinite at the maximum and minimum of the function. Fermat took advantage of this.
In fact, the method of finding the maximum and minimum value of the function has been found. Its root is to minimize the function. Fermat also created a method to find the tangent of a curve. The essence of these methods is to find derivatives. Tangent of curve and minimax and minimax of function are basic problems of differential calculus. It is the study of these two problems that promotes the birth of differential calculus. Fermat has made important contributions to these two problems and is called the pioneer of calculus.
Fermat's approach to these two problems is the same, that is, take the increment first, and then make the increment tend to zero. This is the essence of differential calculus, and it is also the essence that distinguishes this method from the classical method. Fermat also discussed the solution of the area under the curve. This is the preliminary work of integral calculus. He divided the area under the curve into small-area elements, used the analytical equations of rectangle and curve to find the approximate values of these sums, and expressed the expression as the limit of the sum when the number of elements increased infinitely and the area of each element was infinite. But he didn't realize the significance of the operation itself, but stayed on the problem of finding the area itself and only answered a specific geometric question. Only Newton and Leibniz raised this problem to a general concept, thinking that it is a structural operation that does not depend on any geometry or physics, and gave it a special name-calculus.
In the process of establishing these disciplines, they all feel the need for a new mathematical tool, which is calculus to study the process of movement and change. Interestingly, the origin of integral calculus can be traced back to ancient Greece, but it was not until the17th century that differential calculus made a major breakthrough.
Fermat also created a method to find the tangent of a curve. The essence of these methods is to find derivatives. Tangent of curve and minimax and minimax of function are basic problems of differential calculus. It is the study of these two problems that promotes the birth of differential calculus. Fermat has made important contributions to these two problems and is called the pioneer of calculus.
Fermat's approach to these two problems is the same, that is, the increment is taken first, and then the increment tends to zero. This is the essence of differential calculus, and it is also the essence that distinguishes this method from the classical method. Fermat also discussed the solution of the area under the curve. This is the preliminary work of integral calculus. He divided the area under the curve into small-area elements, used the analytical equations of rectangle and curve to find the approximate values of these sums, and expressed the expression as the limit of the sum when the number of elements increased infinitely and the area of each element was infinite. But he didn't realize the significance of the operation itself, but stayed on the problem of finding the area itself and only answered a specific geometric question. Only Newton and Leibniz raised this problem to a general concept, thinking that it is a structural operation that does not depend on any geometry or physics, and gave it a special name-calculus.
The establishment of calculus
The seventeenth century is a period of transition from the Middle Ages to a new era. During this period, science and technology have made great progress. Precision science gained great impetus from the production and social life at that time; Navigation has aroused great interest in astronomy and optics; Shipbuilding, machine building and construction, the construction of dams and canals, ballistics and general military issues have all promoted the development of mechanics.
In the development and actual production of these disciplines, it is urgent to deal with the following four problems: 1. Knowing the relationship between the distance and time of an object's movement, we can find out the speed and acceleration of the object at any time. Then we can know the acceleration and speed of the object and find out the speed and distance of the object at any time. The average speed can be calculated by dividing the distance of movement by the time of movement, but the speed and acceleration involved in the17th century are always changing. For instantaneous speed, the distance and time of movement are zero, which leads to the problem of 0/0. This is the first time that mankind has encountered such a problem.
2. Find the tangent of the curve. This is a purely geometric problem, but it is of great significance to scientific application. For example, in optics, the design of lens uses the knowledge of tangent and normal of curve. In kinematics, it is also transmitted to the tangent of the curve. The direction of motion of a moving object at any point on its trajectory is the tangent direction of the trajectory.
3. Find the maximum and minimum values of the function. In ballistics, this involves the range of shells, and in astronomy, it involves the nearest and farthest distance from the planet to the sun.
4. Orthogonal problem. Find the arc length of the curve, the area enclosed by the curve, the volume enclosed by the surface and the center of gravity of the object. These problems began to be studied in ancient Greece, and some calculations are now only simple exercises of calculus, which was a headache for Greeks in the past. In fact, Archimedes' works are almost all about this kind of problem, and his results mark the climax of Greek mathematics.
It is these important problems in science and production that promote the birth and development of calculus.
During the birth and development of calculus, a group of great mathematicians made outstanding contributions, such as Galileo, Kepler, cavalieri, Fermat, Barrow, Newton and Leibniz.
Great progress in science is always based on the work of many people bit by bit. But it often takes one person to complete the "last step". This person needs to have keen insight, sort out the valuable thoughts of predecessors from chaotic speculation and explanation, have enough imagination to organize these isolated "fragments" and be able to boldly formulate a grand system. In the birth of calculus, Newton and Leibniz were giants to accomplish this mission.
In the18th century after the birth of calculus, mathematics ushered in unprecedented prosperity, and people called this era a heroic century in the history of mathematics. The main work of mathematicians in this period was to apply calculus to astronomy, mechanics, optics, heat and other fields, and achieved fruitful results.
166 1 year, Newton entered Trinity College of Cambridge University and studied the works of Galileo, Kepler, Descartes and Wallis under Barrow. Trinity College still retains Newton's reading notes. From these notes, we can see that Descartes' geometry and Wallis' arithmetica infinitorum have the deepest influence on him in terms of the formation of mathematical thoughts. It was these two works that made Newton embark on the road of establishing calculus.
1665 In August, he returned to his hometown, where he began his great work in mechanics, mathematics and optics. These two years became the golden age of Newton's scientific career. He founded calculus and discovered the theory of gravity and color ... It can be said that most of Newton's blueprints for scientific creation were conceived in these two years.
Creation of calculus
1in the autumn of 664, Newton began to study calculus. At that time, he repeatedly read Descartes' Geometry, became interested in Descartes' tangent "circle method" and tried to find a better method. At this time, Newton pioneered the small O number, which represents the increment of X and is an infinitesimal amount that tends to zero.
Newton continued to explore calculus and made a breakthrough while avoiding the plague in his hometown. According to his account,1June 665, 1 1 year, "downstream counting" (difference method) was invented, and "counter-current counting" (integral method) was established in May of the following year. 1666 10 Newton compiled the research results of the previous two years into a summary paper, which is now called "On Flow Number". Although it was not officially published at that time, it was circulated among colleagues. On the number of flows is the first systematic calculus document in history.
The number of streams reflects the kinematic background of Newton's calculus. In fact, this paper introduces the concept of "traffic number" (that is, WeChat service) in the form of speed. Although the basic term "flow number" is not used, it raises the basic problems of calculus, which can be expressed in the current mathematical language as follows:
1) Knowing the distance of an object solves the problem of finding the speed of the object.
2) The problem of finding the distance of an object when its speed is known.
Newton pointed out that the first problem is a differential problem, and the second problem is the inverse operation of the first problem, and gave the corresponding calculation method. On this basis, the "Basic Theorem of Calculus" is established, which reveals the "internal relationship between derivative and integral". Of course, the basic theorem of calculus has not been strictly proved in the modern sense. In later works, Newton gave a clear proof of the basic theorem of calculus, which had nothing to do with kinematics.
Before Newton, the area was always considered as the sum of infinitesimal components. Newton started with determining the rate of change of the area and calculated the area through inverse differentiation. In this way, Newton not only reveals the reciprocal relationship between area calculation and tangent problem, but also clearly reveals that it is a universal law, thus laying the foundation for the general algorithm of calculus.
As Newton himself said in On the Number of Flows: Once the anti-differential problem can be solved, many problems will be solved.
Since ancient Greece, people have acquired many special skills to solve infinitesimal problems. Newton unified these special skills into two general algorithms-downstream counting method and countercurrent counting method, that is, differential method and integral method, and proved the reciprocal relationship between them. Furthermore, he unified these two operations into a whole-the basic theorem of calculus.
This is his feat of surpassing his predecessors. It is in this sense that we say that Newton invented calculus. In the rest of On the Number of Flows, Newton discussed the problems of 16, such as finding the tangent, curvature and inflection point of the curve, finding the length of the curve, finding the area surrounded by the curve, and finding the gravity and center of gravity. Newton used his unified algorithm to solve these problems, which fully showed the great universality and systematicness of Newton's calculus algorithm.
Newton spent about a quarter of a century studying calculus from 1667 to 1693. Newton made unremitting efforts to improve and perfect his calculus theory, and wrote three calculus papers successively:
(1) 1669 completed "Analysis with Infinite Polynomial Equation", referred to as "Analysis";
(2) 167 1 year to complete "flow number method and infinite series", referred to as "flow number method";
(3) 169 1 year, and the curve quadrature is completed.
Newton was cautious about publishing his own scientific works, and most of his works were published under the repeated urging of his friends. The above three papers were published very late. The first one is the last one, curve quadrature. The analysis was published in1771; However, the law of flow number was not published until 1736, when Newton had died. 1687, Newton published his famous mechanical work "Mathematical Principles of Natural Philosophy", which expressed Newton's calculus theory for the first time. Therefore, Principle has also become an epoch-making work in the history of mathematics.
Title page of mathematical principles of natural philosophy
This principle was praised by Einstein as "incomparably brilliant deductive achievement". Starting from the three basic laws of mechanics, the book strictly deduces and proves a series of conclusions including Kepler's three laws of planetary motion and the law of universal gravitation by using calculus tools, and also applies calculus to fluid motion, sound, light, tides, comets and even cosmic systems, fully demonstrating the power of this mathematical tool.
Newton's scientific contributions are manifold. In mathematics, in addition to calculus, his algebraic masterpiece Universal Arithmetic contains many theoretical achievements of equations, such as the appearance of imaginary roots in pairs, the popularization of Descartes' sign law, the power sum formula of roots and coefficients and so on. His geometric masterpiece "The Count of Cubic Curves" initiated the research on the classification of cubic curves and reached a new peak in the development of analytic geometry. In the field of numerical analysis, Newton's name can't be omitted from any course nowadays.
Newton's historical achievements
Newton is a giant of science and one of the greatest mathematicians in human history. Like Newton, Leibniz, a mathematician who made outstanding contributions to mathematics, commented: "In all mathematics from the beginning of the world to the time when Newton lived, Newton's work exceeded half."
Leibniz and the Birth of Calculus
Gottfried Wilhelm Leibniz was born in Leipzig. 166 1 studied law at the university of Leipzig and geometry at the university of jena, 1666 received a doctorate in law. 1672, he went to Paris on business and was inspired by C. Huygens to study mathematics. After that, he stepped into the field of mathematics and started creative work. This effort has led to many mathematical discoveries, the most prominent of which is the theory of calculus. Newton founded calculus mainly from the viewpoint of kinematics, while Leibniz considered it from the perspective of geometry.
Since 1684, Leibniz has published many calculus papers. This year, his first article on differential calculus, A New Method for Finding Maximum, Minimum and Tangent, was published, which is the earliest published differential calculus literature in the world. In this paper, he explained his differential calculus concisely. In this paper, the definition of differential and basic differential rules are given.
1686, he published the first paper on integral calculus in Yi Xue magazine. Leibniz carefully designed a set of satisfactory calculus symbols. He introduced the modern integral symbol ∫ in 1675, and extended the initial letter s of the Latin word Summa to represent integral. However, the name "integral" appeared relatively late and was put forward by J. Bernoulli in 1696.
Leibniz is the greatest symbol scholar in the history of mathematics. In the process of creating calculus, he spent a lot of time choosing exquisite symbols. He realized that good symbols can accurately and profoundly express concepts, methods and logical relationships. He once said, "If you want to invent, you must choose the right symbol. To do this, it is necessary to express or faithfully describe the inner essence of things with a small number of symbols with simple meanings, thus minimizing people's thinking labor. " Now the symbols of calculus are basically created by him. These superior symbols have brought great convenience to the development of analytical science in the future.
Leibniz invented some other symbols and mathematical terms, such as "function" and "coordinate". Leibniz is versatile and unprecedented.