Engels said in Anti-Dühring: "The mathematics of variables, the most important part of which is calculus. von Neumann said: "Calculus is the greatest achievement in modern mathematics, and its importance cannot be overestimated." This shows the importance of calculus in mathematics. The "calculus" is usually composed of two parts, method and principle, and the principle part is the core of calculus. However, there is little awareness to distinguish between calculus methods and calculus principles, but the two are not the same thing after all. From the pedagogical point of view of calculus, high numbers focus on the learning of methods, while numbers focus more on the study of principles. The method serves practice and tests its effects through practice; the principle is to understand the "what" and "why" and serves to reveal the mechanism behind the method. The purpose of this article is to help readers better understand the calculus method and the principles of calculus, so as to have a deeper understanding of calculus itself.
Newton and Leibniz independently created calculus around the same time. This is a great invention with far-reaching impact. Unfortunately, Newton did not form a relatively mature calculus system in his lifetime, and his method of derivating numbers (i.e., the method of flow numbers or the method of first-last ratio) has been criticized, and the key lies in the understanding of "ο". Leibniz's calculus theory, while concise, is unclear about his differential definition. The reason for the confusion is that he believes that differentiation is the difference between two points of proximity. In fact, two points next to each other do not exist even in the current system of real numbers. So although both of them created calculus, it did not form a complete system of principles.
Figure 1 Portraits of Newton (left) and Leibniz (right).
The history of calculus tells us that although the principles of calculus were not well established at the time, it was already closely intertwined with a wide range of applications, stimulating and driving the establishment of many new branches of mathematics.
The 18th century can be said to be the era of analysis. Since the establishment of calculus, the mainstream of mathematics as a whole has entered the great development of calculus, and it is mainly manifested in the great development of calculus methods. In particular, mathematicians on this side of the European continent, such as the Bernoulli brothers, Euler, D'Alembert, Lagrange and others, have made a lot of contributions. At the same time, due to the lax calculus of Newton and Leibniz, some mathematicians made various attempts to overcome the difficulties of calculus foundations, although this was not mainstream at the time. After nearly a century of experimentation and brewing, mathematicians' efforts to reconstruct calculus on a strict basis began to bear fruit in the early 19th century, and it was curches, Weerstrass, Riemann, Cantor, Lebel and others who finally established the strict principle of calculus.
Cauchy was inspired by Newton's "the final ratio of vanishing quantities is not the ratio of the final quantity, but the limit of the ratio of these infinitely reduced quantities" and proposed the use of limit ideas to establish a calculus system; Weierstrass gave a strict formal description of limit theory, that is, the current ε-δ language; and then, through Riemann's idea of integration, Canto's set theory, Lebeg's idea of integration (based on measurement theory), and finally formed the current calculus system, also known as the second generation of calculus system. In essence, it is a system formed by developing Newton's calculus principle with the idea of limits and introducing Leibniz's marks. It should be noted that at this point the core of the calculus system has changed from differentiation to derivative.
When we talk about calculus today, we don't have a clear distinction between calculus methods and calculus principles. So what are calculus methods and principles of calculus? Why do we have to make a distinction?
The calculus method is the means and way for people to use calculus as a tool to solve problems. For example, before Newton and Leibniz created calculus, a large number of problems were accumulated, mainly boiled down to two types of problems: differential problems based on finding curve tangents, finding instantaneous rates of change, finding function extremums, etc., and integrating problems based on finding areas and volumes. Of course, after Niu Lai, many new problems have arisen, so we need to use calculus as a tool to solve these problems.
The 18th century was dominated by the development of calculus methods. The basic methods of calculus include the differential method, the derivative method, and the integral method. The integral methods include the commutative integral method and the partial integral method. Through the study of the integrals of irrational functions, it was found that the integrals of some functions could not be represented by known elementary functions, and finally a profound theory of elliptic functions was established in the 1820s. When we take the basic methods of calculus to apply, we can also produce something new.
Disciplines such as ordinary differential equations, partial differential equations, differential geometry, and functional analysis have arisen with the development of calculus. With differential equations, through the efforts of mathematicians such as Euler, D'Alembert, and Lagrange, methods for solving differential equations have emerged. When calculus is used to study curved surfaces, differential geometry methods for solving corresponding geometric problems also appear. The concept of functions can be generalized to functionals, hence the variational method. Taking ordinary differential equations as an example, people have gradually discovered methods such as the separation variable method, the variable substitution method, the parameter variation method, and the integral factor method. By around 1740, almost all elementary methods for solving first-order equations were known. An important breakthrough in the solution of higher-order ordinary differential equations was Euler's complete solution of the linear homogeneous equations of the nth-order constant coefficients in 1743, when Euler introduced the famous exponential transformation. The highest achievement of the 18th century ordinary differential equation solving was Lagrange's 1774-1775 solution of the general n-order variable coefficient heterogeneous ordinary differential equation using the parametric transmutation method.
Figure 2
So what is the principle of calculus? Before we do that, let's figure out what the principle is.
In the case of mechanics, for example, the phenomenon associated with "force" is being studied before the concept of "force" is revealed. For example, if we build a small cart, we have a round wheel, not a flat thing pushing forward, what does this mean? This shows that people do this with experience before they have the concept of "force", because it is more labor-saving to push. But why would it be more labor-intensive to do so? With the concept of "force", we know that friction can explain this problem. Because we can compare the friction between the two, it has the difference between sliding friction and rolling friction, and the rolling friction resistance is smaller, then it will be more labor-saving to make a round wheel.
Figure 3
In this way we reveal its principles. Once the principle is revealed, we can also use the principle to design something that better reduces the rolling friction. Taking car driving as an example, its rolling friction is used as resistance, and we can find ways to change the material and shape of the tire to minimize the rolling friction. However, the sliding friction also has a part of the effect, and the sliding friction force should cause it to be relatively displaced. In some cases, the sliding friction is as large as possible, such as snow days, because the sliding friction is not large enough, the tire will slip. Why engrave a pattern on the tire, the design of the pattern is to increase the sliding friction.
The importance of principles is that we can use them to optimize the original design. Then the same is true for calculus. The calculus method works well, and we need to distill its core concepts and form a mathematical deductive system on top of these concepts. This system, which needs to be able to explain why these existing methods are correct, reveal its essence clearly. Not only that, but it is also necessary to be able to optimize some mathematical expressions of existing methods, and even develop more calculus methods. The principle is to play such a role. Of course, we also need to require this principle to be logical and self-consistent and as concise as possible.
This is the principle of calculus that we are talking about, and its specific performance is a mathematical deductive system. Under the current calculus principle system with the limit concept as the core, it is mainly manifested in the basic concepts of limits, real numbers, sets, functions, continuums, derivatives, definite integrals, indefinite integrals, and differentiations, as well as the deductive system composed of these basic concepts.
We say that the calculus method is universal, because its correctness has been proven in practice. But why it is correct can not only stay on the test of practice, people also need to reveal the mechanism behind it from the perspective of principle, and even to optimize it or to discover more calculus methods. This is where the principle of calculus differs radically from the calculus method. In fact, we often confuse these two concepts. The correctness of the calculus method does not need to be proved by the calculus principle, because practice has proved it, but only the calculus principle needs to be able to explain it better. Similarly, we cannot say that because the calculus method is correct, there is no problem with the calculus principle. If so, we do not need the current calculus principle with the limit concept at its core, because Newton and Leibniz's incomplete calculus principle is sufficient. People's understanding of objective things is a process of deepening from vivid and intuitive to abstract thinking, from perceptual to rational understanding. The validity of the principle of calculus is entirely proved by its logical self-consistency and concise practicality. The principle did not come about overnight, but it wasn't dead either. When a principle does not play its due role, it has room for further development.
This article is reprinted with permission from the WeChat public account "Mathematical Graticule", the original title is "High Perspective! How to recognize calculus methods and principles? 》。
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