The basic theorem of calculus points out that finding indefinite integral and derivative function are reciprocal operations [the integral value can be obtained by substituting the upper and lower bounds into indefinite integral, while differentiation is the product of derivative value and increment of independent variable], which is why the two theories are unified into calculus. We can discuss calculus from either of them, but in teaching, differential calculus is usually introduced first. Calculus is a general term for differential calculus and integral calculus. It is a mathematical idea, in which' infinite subdivision' is differential and' infinite summation' is integral. /kloc-In the second half of the 7th century, Newton and Leibniz completed the preparatory work involving many mathematicians and independently established calculus. Their starting point of establishing calculus is intuitive infinitesimal, but their theoretical foundation is not solid. Because the concept of "infinity" can't be calculated by the existing algebraic formula, it was not until Cauchy and Wilstras established the limit theory and Cantor established the strict real number theory in the19th century that the subject was considered rigorous. The first step in learning calculus is to understand the necessity of introducing "limit", because algebra is a familiar concept, but algebra can't handle the concept of "infinity". So it is necessary to use algebra to deal with the quantity representing infinity, and then carefully construct the concept of "limit". In the definition of "limit", we can know that this concept bypasses the trouble of dividing a number by 0, but introduces an arbitrarily small process. In other words, the number divided by is not zero, so it makes sense. At the same time, this small number can be arbitrarily small. As long as the δ interval is less than this arbitrarily small quantity, we say that its limit is this number-you can think it is opportunistic, but its practicality proves that this definition is relatively perfect and gives the possibility of correct inference. This concept is successful. Mathematics can be used as an ideal tool for natural science, because it can deal with natural problems conveniently and quantitatively. Some natural problems can't be handled by constant mathematics, so calculus is necessary. But why can't constant mathematics and calculus work? Most people can't answer, even mathematicians can't answer well! Many beginners of calculus can't understand the methods of calculus. There is a reason for this, because their philosophical foundation is weak, even if they have studied it, they don't understand it. Calculus is not to understand the definition of limit δ. Calculus appeared at least 150 years earlier than the definition of limit δ! In fact, learners should reflect on how much calculus is better than constant mathematics; What kind of method to study nature is effective; What attitude should we take towards human consciousness and nature! A Brief Introduction to the General Thought of Calculus 1. Some Simple Contents of Marxist Philosophy 1. The Origin of Philosophical Philosophy In English, the word "philosophy" in ancient China means wisdom. After the translation of the Western Zhou Dynasty by a Japanese scholar, the knowledge that ancient Greece loved wisdom was called philosophy. In Aristotle's knowledge classification, philosophy is also called metaphysics. Aristotle divided human knowledge into two categories. The first kind of knowledge is the object of studying abstract transcendentalism and is called the first philosophy. The second kind of knowledge is the object of studying concrete experience, which is called the second philosophy, also called physics. After Aristotle's death, his students published the First Philosophy after the Second Philosophy while editing the teacher's works. When China people first translated Aristotle's philosophical works, they translated the first philosophy after physics. Later, philosophy was translated into metaphysics according to two sentences in China's ancient Book of Changes: metaphysics refers to Tao and metaphysics refers to utensil. Second, philosophy is a theoretical and systematic world outlook. 1. Defined from the perspective of philosophical research objects. 2. World outlook is people's fundamental view of the whole world. 3. Methodology is the fundamental way for people to understand and transform the world under the guidance of a certain world outlook. Thirdly, philosophy is a summary of knowledge of nature, social knowledge and thinking knowledge. Fourth, to understand the philosophical principles of calculus 1. Dialectical relationship between nature and man: nature exists before man and his consciousness; After the appearance of human beings, the existence and development of nature does not depend on human consciousness. Therefore, the existence and development of nature is objective. 2. Philosophical Material Marxist philosophy calls an objective entity that is independent of human consciousness and can be reflected by human consciousness as material, pointing out that the whole world is an objective material world and the essence of the world is material. 3. What is human consciousness is the reflection of objective existence in the human brain. Consciousness, whether right or wrong, is a reflection! 4. The dialectical relationship between matter and movement. Matter is the moving matter, and movement is the movement of matter. Movement is the fundamental attribute and existing mode of matter, and matter is the subject of movement, and matter and movement are inseparable. It is wrong to talk about sports without talking about matter, or to talk about matter without talking about sports. In later calculus, you will see the relativistic effect. )-Note: This is used for functions and dialectics of nature. Well, to understand the role of calculus and relativity (don't run after the light! ), these are enough. Those who have the ability can look at other philosophies. Second, the philosophical analysis of constant mathematics 1, the concept of constant mathematics The so-called constant mathematics refers to: elementary mathematics, that is, mathematics formed from the primitive society to the middle of17th century. The main research objects are constants, constants and invariant graphs. 2. The basic composition of constant mathematics Elementary mathematics can be divided into three stages according to the formation and development of the main disciplines: the budding stage, before the 6th century BC; Geometric priority stage, from the 5th century BC to the 2nd century AD; Algebra priority stage, from the 3rd century to the beginning of17th century. At this point, the main parts of elementary mathematics-arithmetic, algebra and geometry have been formed and matured. Therefore, the composition of constant mathematics can be considered as: arithmetic+elementary algebra+elementary geometry, plus a little restriction of primitive theory. For example, Liu Weichuang, an outstanding mathematician in the Wei and Jin Dynasties in China, founded "Circumcision" and once said, "If you cut it carefully, you can cut it again, so that it can't be cut, and you have nothing to lose with the circumference." He Zhuangzhou's book "Zhuangzi" in "The World" records that "a foot of pestle is inexhaustible." These are simple and typical limit concepts. Strange, since ancient Greece had a "limit theory", why couldn't calculus be born? The reason is that there is no concept of function! First of all, calculus is not the inevitable product of limit, but the inevitable product of function. Therefore, it is impossible to establish calculus theory in any part, and it can even be said that the limit is the derivative of the function. 3. Dialectical analysis of constant mathematics in philosophy Let me mention arithmetic first. It is actually an artificial agreement, which originated from the original labor inventory and collection. This is a unique consciousness of human beings and a reflection of natural activities. If there is a man and a dog next to you, you say:1+1= 3; People next to you will point out: no! 1+ 1=2, I believe there will be no problem with the dog next to you. Let's look at elementary geometry (Euclidean geometry). Geometry must learn graphics. So Euclid said: 1 A dot is something without parts. 2. A line is a length without width. 3. A straight line is a line aligned with all the points in it. 4. Face is the kind of thing that only has length and width. 5. The edge of the face is a line. 6. A plane is something that is aligned with a straight line on it. 15. A circle is a plane figure contained in a curve, so that all straight lines from one point to the line are equal to each other. Almost all of them are philosophical definitions, so we can't help asking, what are "things without parts" and "things with only length and width"? "No parts" exist? Can we see them or know them? Of course, the premise is that if this "thing" exists. Besides, can we say that the "point" is in our hearts? Is "point" fictional? In actual processing, we can see this. For example, you can get a dot pattern by pressing the pen tip gently on the paper. Of course, your teacher said that he would tell you that this figure has no length, no area and no volume. Just say this thing doesn't exist! Euphemistically called: "This is the abstraction of mathematics!" This is a sophistry of objective things in nature. The undefined "points", "lines" and "faces" are abstracted, and they are considered to have no natural properties but only geometric features. However, all "points", "lines" and "faces" in nature exist objectively and have natural attributes. Not to mention Einstein's view of time and space, even Newton's absolute view of time and space has: Theorem 1: All objects always occupy space and are unaffected, and can exchange space. It can be seen that nature really can't find points, lines and surfaces without volume (occupying no space). So there must be a contradiction between elementary geometry and nature. For example, any curve (function, equation) in the plane rectangular coordinate system is an objective entity of nature, and elements (locus of point-set) are real substances with length! But are there any dots or dots with length? Of course, at least it is not a line segment, but it exists unmeasurably! It can be seen that "points" in nature are not within the definition of human consciousness (unmeasurable). In other words, arithmetic is also contradictory to nature. So it can't be described by (arithmetic scale+elementary geometry), because it can't be described unless it is described (a point is a length without a length)! This is a typical Russell retort: a point is of length 0 or a point is not of length! What is this? This is only a superficial phenomenon, which fundamentally illustrates a dialectical relationship: nature is independent and consciousness is only a reflection of the human brain. How about algebra? There is no solution equation, x+1= 2; Then x=2- 1= 1. Or pure human consciousness, the objectivity of nature, is basically not mentioned! Do you think it is possible to study the real objective things in nature with elementary mathematics that is divorced from the objectivity of nature? Therefore, in the long river of history, mathematics has to wait for a new concept to recognize the objectivity of nature. Only in this way can mathematics burst into amazing power. Enlightenment: How to deal with the objective problems of nature by mathematical methods? Mathematical object is the objective entity of nature, so it is necessary to maintain the objectivity of nature in methods and finally return to nature instead of staying in consciousness! Third, the cry of17th century Descartes, the originator of modern science, now we once again open the magnificent historical picture in the history of mathematics in17th century and trace back to the Renaissance in16th century to see what problems Europeans encountered at this time. /kloc-since the 0/6th century, European capitalism has gradually developed and accumulated a lot of new experiences in production practice. The development of science has laid a new foundation for the renewal of technology. The invention and application of many new technologies provide more abundant materials for science, and also raise many new problems, many of which are in front of mathematicians. However, for many mathematical problems in the fields of machinery, architecture, water conservancy, navigation, shipbuilding, microscope and firearms manufacturing. /kloc-in the first half of the 7th century, a brand-new branch of mathematics, analytic geometry, was founded, which marked the beginning of modern mathematics and opened up a broad field for its application. "If a worker wants to do a good job, he must sharpen his tools first." First of all, we need to know the mathematical tools of analytic geometry. Let's invite Comrade Descartes to talk. Descartes said: "algebra, which was popular at that time, I think it is completely subordinate to laws and formulas and cannot be a science to improve intelligence." Therefore, we must combine the advantages of geometry and algebra to establish a real mathematics. " "Have you found a way?" (2) Descartes smiled and said: "The core of my thought is to simplify the geometric problem into an algebraic problem and calculate and prove it by algebraic method, so as to finally solve the geometric problem." . "Can you be more specific?" (3) Descartes went on to say: "I published geometry in 1637 and founded the rectangular coordinate system. Use the distance from a point on a plane to two fixed straight lines to determine the position of the point, and use coordinates to describe the point in space. In this way, the opposite "number" and "shape" can be unified, so geometric curves can be combined with algebraic equations; Therefore, geometric problems can be reduced to algebraic form, and geometric properties can be discovered and proved through algebraic transformation. " And from the point of view of motion, the curve can be regarded as the trajectory of point motion. Marxist philosophy points out that the dialectical relationship between matter and movement is that matter is a moving matter and movement is a material movement. Movement is the fundamental attribute and existing mode of matter, and matter is the subject of movement, and matter and movement are inseparable. It is wrong to talk about sports without talking about matter, or to talk about matter without talking about sports. Since the curve can be regarded as the trajectory of a point, we have to admit the materiality of the point. If there is no point, there is no movement! According to Marx's conclusion: Movement is the fundamental attribute and mode of existence of matter, and matter is the subject of movement, and matter and movement are inseparable. The theory of motion can be transformed into the theory of matter! The fact is the same: what kind of point sets are the definitions of modern analytic geometry pairs and curves talking about! Generally, a circle is a set of points whose distance from a fixed point is equal to a fixed length, P={M|MC=r}. As Engels said: "The turning point in mathematics is Cartesian variables. With variables, motion enters mathematics, with variables, dialectics enters mathematics, with variables, differentiation and integration become necessary immediately. "In fact, the greatest achievement of variables lies in the mathematical recognition of the objectivity of nature. In the long river of history, mathematics has to wait for a new idea, that is, to recognize the objectivity of nature. Only such mathematics can deal with the objective problems of nature. Fourth, the historical significance of the function The center of variable mathematics is actually the function. Elementary geometry denies the materiality of points, lines and surfaces, and only recognizes geometric characteristics, which is divorced from the objectivity of objective entities. /kloc-in the 7th century, Descartes established analytic geometry, paving the way for the establishment of functions. Because a curve can be regarded as the trajectory of point motion, that is, a straight line is a collection of points, and so on, a surface is a collection of straight lines, and so on. Their natural attribute-overall recognition is recognized. This view is logically embodied in the function. For example, a circle is a set of points whose distance from a fixed point is equal to a fixed length. The expression of P={M|MC=r} implicit function is: x 2+y 2 = r 2. Therefore, the function is an objective reflection of the mathematical object (materiality), and we can understand the natural attributes in the macro (whole); In this way, the objectivity of the microscopic part of the whole has also been recognized, which will be shown when studying the local properties of the function. This is the basic fact of calculus! Only mathematics that recognizes the objectivity of nature can have the ability to study nature. Constant mathematics denies the nature attribute-divorced from certain reality, which limits its ability to solve nature; This is the essence of constant mathematics and variable mathematics, which are just the external manifestations of a mathematical form. I think Herman Will asked well in Philosophy of Mathematics and Philosophy of Science: Why can natural events be predicted through the combination of observation and mathematical analysis (calculus)? Because of mathematical analysis, the objectivity of nature was admitted from the beginning! As Marx eloquently answered: "Consciousness can correctly reflect objective things". Calculus loses its soul without function. Descartes' analytic geometry introduced variables, deepened the concept of function and acknowledged the objectivity of nature. With functions, calculus can really be established. Newton-Leibniz formula profoundly reflects the objective relationship between the whole and the part of nature. The function itself is a natural micro-sculpture, and studying the function through mathematical analysis is to study the local properties of natural micro-sculpture. On the other hand, if we study the part of natural micro-sculpture, we can express nature (differential equation) as a whole and reduce it to a function. 5. Philosophical analysis of the locality of function We now know that function is the logical embodiment of understanding the objectivity of nature at the macro level. у = f (x), because it is a natural attribute recognized in the macro (whole); In this way, the objectivity of the microscopic part of the whole has also been recognized, which will be shown when studying the local properties of the function.
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. /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.
Significance of the establishment of calculus
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 calculus theory 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 ancient and medieval algebra are constant mathematics, and calculus is the real variable mathematics, which is 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. Calculus is developed in scientific application. At first, Newton used calculus and differential equations to analyze Tycho's massive astronomical observation data, and obtained the law of gravity, and further deduced Kepler's three laws of planetary motion. Since then, calculus has become a powerful engine to promote the development of modern mathematics, and has also greatly promoted the development of various branches of natural science, social science and applied science such as astronomy, physics, chemistry, biology, engineering and economics. And it is widely used in these disciplines, especially the appearance of computers is more conducive to the continuous development of these applications.
The birth of calculus is another milestone in the development of mathematics after the establishment of Euclid geometry. Before the birth of calculus, human beings were basically in the period of agricultural civilization. The birth of analytic geometry is a prelude to the arrival of a new era, but it is not the beginning of a new era. It summarizes the old mathematics, combines algebra and geometry, and leads to the concept of variables. Variable, a brand-new concept, provides a basis for the study of sports and deduces a lot of laws of the universe. We must wait for the arrival of such an era, make ideological preparations in this respect, and produce a group of leaders like Newton, Leibniz and Laplace who can create the future, provide methods and point out the direction for scientific activities, but we must also wait for the emergence of an indispensable tool-calculus. Without calculus, it is impossible to deduce the laws of the universe. Among all the treasures of knowledge developed by geniuses in the17th century, this field is the richest, and calculus provides a source for the establishment of many new disciplines.
I've been looking for you online for a long time ... and sorted out the above contents ... Take it if you are satisfied. ...