First, the historical evolution of the differential mean value theorem
The differential mean value theorem is the core theorem of differential calculus, an important tool for studying functions and a bridge between functions and derivatives, which has always been valued by people.
The differential mean value theorem has obvious geometric significance. Taking Lagrange mean value theorem as an example, it is explained that "the tangent of the point on the curve segment of differentiable function must be parallel to the chord at the end of the curve." In this sense, people's understanding of the differential mean value theorem can be traced back to ancient Greece and BC.
When studying geometry, ancient Greek mathematicians came to the following conclusion: "The tangent line passing through the vertex of parabolic bow must be parallel to the bottom of parabolic bow". This is a special case of Lagrange's mean value theorem. Archimedes, a famous Greek mathematician, skillfully used this conclusion to find out the area of parabolic arch.
Cavalieri, an Italian mathematician, gave an interesting lemma to deal with the tangents of plane and solid figures in Essential Geometry, in which Lemma 3 stated the same fact based on the geometric point of view: the tangent of a point on a curve segment must be parallel to the chord of the curve. This is a geometric differential mean value theorem, which is called "cavalieri theorem".
People's research on differential mean value theorem began at the beginning of the establishment of calculus. 1637, Fermat, a famous French mathematician, gave Fermat's theorem in the method of finding the maximum and minimum values. In textbooks, people usually regard it as the first theorem of differential mean value theorem. 169 1 year, French mathematician Rolle gave Rolle's theorem in polynomial form in the article "Solutions of Equations".
1797, the French mathematician Lagrange gave Lagrange's theorem and the initial proof in his book Analytic Function Theory. It is French mathematician Cauchy who systematically studies the differential mean value theorem.
He is an advocate of the strict movement of mathematical analysis. His three representative works, Analysis Course, Introduction to Calculus Course and Differential Calculation Course, reconstructed the theory of calculus with strict as the main goal. He first endowed the mean value theorem with an important role and made it the core theorem of differential calculus. In Introduction to Infinitesimal Computing, Cauchy first strictly proved Lagrange's theorem.
In the process of differential calculation, it is extended to the generalized mean value theorem-Cauchy theorem, and the last puzzle, the last differential mean value theorem, is found.
Second, the differential mean value theorem
1 and introduction of differential mean value theorem
1. 1 Fermat theorem
When Fermat studied the solution of minimax problem, he obtained the unified solution "virtual equality method", and thus obtained the original Fermat theorem.
Fermat's "virtual equation method" may be based on a very intuitive idea. At that time, calculus was still in its infancy, and there were no clear concepts such as derivative, limit and continuity. From a modern point of view, its judgment is not rigorous. The Fermat Theorem we see now was recreated by later generations according to the calculus theory and the essence of Fermat's discovery.
1.2 Rolle theorem
The original Rolle theorem and the modern Rolle theorem are not only different in content, but also very different in proof. Rolle proved it with pure algebraic theory, which has nothing to do with calculus. The Rolle theorem we see now was re-proved by later generations according to the calculus theory and extended to general functions. The name "Rolle Theorem" was given by Drobig in 1834, and was officially used by the Italian mathematician Belavi Titis in a paper published in 1846.
1.3 Lagrange theorem
Lagrange theorem is the most important theorem in differential mean value theorem.
The proof of Lagrange theorem in history was originally given by Lagrange in analytic function theory. This proof is largely intuitive and intuitive: "If variables change continuously, the function will change accordingly, but it will not transition from one value to another without going through all intermediate values."
/kloc-At the beginning of the 9th century, in the strict movement of calculus represented by Cauchy, people gave strict definitions of limit, continuity and derivative, and also gave a new proof of Lagrange's theorem.
French mathematician Bonet wrote the modern form of Lagrange's theorem. Instead of using the continuity of f'(x), he used Rolle's theorem to prove Lagrange's theorem again. Dabu used this conclusion to prove that when f'(x) is integrable. Therefore, the differential mean value theorem has become an important research tool of calculus.
1.4 Cauchy Theorem
Cauchy theorem is a generalization of Lagrange theorem, and Cauchy's proof is very similar to Lagrange's proof of Lagrange's mean value theorem. Differential mean value theorem plays an important role in Cauchy calculus theory system.
For example, he used the differential mean value theorem to give a strict proof of L'H?pital's law and studied the rest of Taylor's theorem. Since Cauchy, the differential mean value theorem has become an important part of differential calculus and an important tool for studying functions.
People have been studying the differential mean value theorem for about 200 years. Starting from Fermat's last theorem, it has gone through the development stages from special to general, from intuition to abstraction, from strong conditions to weak conditions. It is in this development process that people gradually realize their internal relations and essence.
When bonet used Rolle theorem to prove Lagrange theorem, and later generations used Lagrange theorem to prove Rolle theorem, the differential mean value theorem formed a centralized generalization, as American mathematician Kramer said, "The concept of people's contribution to mathematics at any time in the history of mathematics should be regarded as the most profound concept".
2. Differential mean value theorem of multivariate functions
The differential mean value theorems introduced earlier are all differential mean value theorems in one-dimensional differential calculus and plane domain, but in practical application, this limitation must be broken in many cases. In order to make full use of the important tool of differential mean value theorem, this paper extends it so that it can be used in n-dimensional differential calculus and n-dimensional space.
3. Higher order differential mean value theorem
Some practical problems involve higher derivative or higher differential of functions, and the differential mean value theorem is extended to the case of higher differential.
4. Differential mean value theorem of complex variable function
There are a set of important and widely used differential mean value theorems in the analysis. Similarly, the corresponding differential mean value theorem can be obtained in complex analysis, from which some differential mean value theorems of complex analysis similar to real analysis can be derived. However, the differential mean value theorem in analysis is generally not valid in complex variable functions, and it needs some constraints and improvements to make it applicable in complex variable functions, which is also a generalization of differential mean value theorem.
This paper mainly introduces the historical evolution of calculus and differential mean value theorem, and derives three forms of differential mean value theorem from it. This paper studies and discusses the generalization of differential mean value theorem from two aspects: differential mean value theorem of multivariate function and higher order differential mean value theorem. In addition, in the complex variable function, the differential mean value theorem corresponding to real analysis is given.