Cauchy

Cauchy (Augustin Louis 1789-1857) was born in Paris. His father, Louis Francois Cauchy, was an official of the French Bourbon Dynasty. He has held public office in France's tumultuous political maelstrom. Due to family reasons, Cauchy himself belonged to the orthodox faction that supported the Bourbon dynasty and was a devout Catholic. And in the field of mathematics, he has made great achievements and achievements. Many mathematical theorems and formulas are also named after him, such as Cauchy's inequality and Cauchy's integral formula.

Chinese name: Augustin Louis Cauchy

Foreign name: Augustin Louis Cauchy

Nationality: France

Birthplace: Paris

Date of birth: August 21, 1789

Date of death: May 23, 1857

Occupation: Mathematician, physicist, astronomer

Graduation school: Paris Bridges and Roads School

Belief: Orthodoxy supporting the Bourbon dynasty

Main achievements: Cauchy's maxim of existence

Cauchy sequence

Cauchy's inequality

Cauchy's integral formula

Representative works: Analysis tutorial, Introduction to infinitesimal analysis tutorial, Application of calculus in geometry

Character introduction

Cauchy (1789-1857) was a French mathematician, physicist, and astronomer. In the early 19th century, calculus had developed into a huge branch with rich content and wide applications. At the same time, its weaknesses are increasingly exposed, and the theoretical foundation of calculus is not strict. In order to solve new problems and clarify the concepts of calculus, mathematicians began to work on making mathematical analysis more rigorous. In laying the foundation for analysis, the great mathematician Cauchy was the first to make outstanding contributions.

Caucy was born in Paris on August 21, 1789. His father was a lawyer who was proficient in classical literature and had close contacts with Lagrange and Laplace, the great French mathematicians of the time. Cauchy's mathematical talent in his youth was greatly appreciated by these two mathematicians, who predicted that Cauchy would become a great person in the future. Lagrange suggested to his father that he "quickly give Cauchy a solid literary education" so that his hobbies would not lead him astray. As a result, his father strengthened Kexi's literary education, so that he also showed high talent in poetry.

Cauchy studied at the Engineering College from 1807 to 1810 and worked as a traffic and road engineer. Due to poor health, he accepted the advice of Lagrange and Laplace, gave up engineering and devoted himself to the study of pure mathematics. Cauchy's greatest contribution to mathematics was the introduction of the concept of limits in calculus and the establishment of a logically clear analytical system based on limits. This is the essence of the development history of calculus and Cauchy's great contribution to the development of human science.

In 1821, Cauchy proposed the method of limit definition, describing the limit process with inequalities. Later, it was improved by Weierstrass and became what is now known as the Cauchy limit definition or definition. All calculus textbooks today still (at least in essence) follow the definitions of concepts such as limits, continuity, derivatives, and convergence by Cauchy and others. His interpretation of calculus was generally adopted by later generations. Cauchy did the most systematic pioneering work on definite integrals. He defined definite integrals as the "limit" of sums. Before the operation of definite integrals, it is emphasized that the existence of integrals must be established. He first rigorously proved the fundamental theorem of calculus using the mean value theorem. Through the hard work of Cauchy and later Weierstrass, the basic concepts of mathematical analysis were rigorously discussed. This will end the two hundred years of ideological chaos in calculus, liberate calculus and its extension from complete dependence on geometric concepts, motion and intuitive understanding, and develop calculus into the most basic and complex mathematics of modern mathematics. discipline.

The work of rigorous mathematical analysis had a great impact from the beginning. Cauchy proposed the theory of series convergence at an academic conference.

After the meeting, Laplace hurried home and checked one by one whether the series used in his masterpiece "Celestial Mechanics" converged according to Cauchy's rigorous judgment method.

Cauchy's research results in other areas are also very rich. The calculus theory of complex functions was founded by him. He has also made outstanding contributions in algebra, theoretical physics, optics, and elasticity theory. Cauchy's mathematical achievements were not only brilliant, but also astonishing in number. The complete works of Cauchy have 27 volumes and more than 800 papers. He is the most prolific mathematician after Euler in the history of mathematics. His glorious name, along with many theorems and principles, are engraved in many textbooks today.

As a scholar, he has quick thinking and outstanding achievements. From Cochy's vast writings and achievements, it is not difficult to imagine how he worked tirelessly and diligently throughout his life. But Kesi was a man of complex character. He was a loyal royalist, an ardent Catholic, and a lonely scholar. Especially as a well-known scientific leader, he often ignores the creations of young scholars. For example, because Cauchy "lost" the seminal paper manuscripts of the talented young mathematicians Abel and Galois, group theory came out about half a century late.

Cauchy died of illness in Paris on May 23, 1857. One of his famous sayings before his death, "People are bound to die, but their achievements will last forever." has struck a chord in the hearts of generations of students for a long time.

Cauchy's skills in pure mathematics and applied mathematics are quite profound. In terms of mathematical writing, he is considered to be second only to Euler in quantity. He wrote a lot of books in his life. 789 papers and several books, some of which are classics, but not all of his creations are of high quality, so he has been criticized for being highly productive and rash, which is the opposite of the Prince of Mathematics. It is said that the French Academy of Sciences When the "Journal" was first published, there were so many works by Cauchy that the Academy of Sciences had to bear a large printing cost, which exceeded the budget of the Academy of Sciences. Therefore, the Academy of Sciences later stipulated that the longest paper could only be four pages. Therefore, Cauchy's longer paper had to be submitted elsewhere.

When Cauchy was a child, his father often took him to his office in the French Senate and guided him in his studies there, so he had the opportunity to meet Senators Laplace and Lagrand. Two great Japanese mathematicians. They appreciated his talent very much; Lagrange believed that he would become a great mathematician in the future, but suggested that his father not study mathematics until he could learn liberal arts well.

Biography

1811 and 1812

Cauchy entered high school in 1802. In middle school, he achieved excellent results in Latin and Greek and participated in many competitions and won awards; his math scores were also highly praised by his teachers. He was admitted to the Comprehensive Engineering School in 1805, where he mainly studied mathematics and mechanics; he was admitted to the Bridge and Highway School in 1807, graduated with honors in 1810, and went to Cherbourg to participate in the harbor construction project.

When Cauchy went to Cherbourg, he brought Lagrange's "Analytical Theory of Functions" and Laplace's "Celestial Mechanics". Later, he also received copies sent from Paris or borrowed from the local area. some mathematics books. In his spare time, he studied books on various branches of mathematics, from number theory to astronomy. According to Lagrange's suggestion, he conducted research on polyhedra and submitted two papers to the Academy of Sciences in 1811 and 1812. The main results were:

(1) Proved that convex regular polyhedra only have There are five kinds (the number of faces is 4, 6, 8, 12, and 20), and there are only four kinds of star-shaped regular polyhedrons (three kinds with the number of faces 12, and one kind with the number of faces 20).

(2) Obtained another proof of Euler’s relational expression regarding the number of vertices, faces and edges of a polyhedron and generalized it.

(3) It is proved that a polyhedron with fixed faces must be fixed. From this, a theorem of Euclid that has never been proved can be derived.

These two papers have had a great impact on the mathematical community. Cauchy fell ill due to overwork in Cherbourg and returned to Paris to recuperate at his parents' home in 1812.

1813

Cauchy was appointed as an engineer for the canal project in Paris in 1813. While he was recuperating and working as an engineer in Paris, he continued to study mathematics and participate in academic activities. His main contributions during this period were:

(1) Research on substitution theory and published basic papers on the history of substitution theory and group theory.

(2) Proved Fermat’s conjecture about polygonal numbers, that is, any positive integer is the sum of polygonal numbers. This conjecture had been proposed for more than a hundred years at that time, and after many studies by mathematicians, none of them could be solved. The above two studies were started by Cauchy while he was at Cherbourg.

(3) Use the integral of a function of a complex variable to calculate the real integral, which is the starting point of the Cauchy integral theorem in the theory of functions of a complex variable.

(4) He studied the propagation of liquid surface waves and obtained some classic results in fluid mechanics. He won the French Academy of Sciences Mathematics Prize in 1815.

The publication of the above outstanding results brought Cauchy a high reputation, and he became an internationally famous young mathematician at that time.

1815-1821

In 1815, Napoleon failed in France, the Bourbon dynasty was restored, and Louis XVIII became King of France. Cauchy was appointed as an academician of the French Academy of Sciences and a professor of the Polytechnic School in 1816. In 1821, he was appointed professor of mechanics at the University of Paris and also taught at the Collège de France. His main contributions during this period are:

(1) Teaching analysis courses in comprehensive engineering schools, establishing the basic limit theory of calculus, and also clarifying the limit theory. Before this, the concepts of calculus and series were vague. Because Cauchy's lecture method was different from the traditional way, teachers and students in the school raised many criticisms against him at that time.

The books Cauchy published during this period include "A Course in Algebraic Analysis", "A Course Outline in the Analysis of Infinitesimals" and "A Course in the Application of Calculus to Geometry". These works laid the foundation for calculus, promoted the development of mathematics, and became a model for mathematics tutorials.

(2) Cauchy returned to studying continuum mechanics after serving as a professor of mechanics at the University of Paris. In a paper in 1822, he established the basis of elasticity theory.

(3) Continue to study the calculation of integrals and residues on the complex plane, and apply relevant results to study partial differential equations in mathematical physics.

A large number of his papers were published in the Proceedings of the French Academy of Sciences and in his own journal "Mathematical Problems".

After 1830

In 1830, a revolution broke out in France to overthrow the Bourbon dynasty. King Charles X of France fled in panic, and Louis Philippe, Duke of Orleans, succeeded him as King of France. At that time, it was stipulated that those who held public office in France must swear allegiance to the new French king. Since Cauchy belonged to the orthodox faction that supported the Bourbon dynasty, he refused to swear allegiance and left France on his own. He first went to Switzerland, and then served as a professor of mathematical physics at the University of Turin in Italy from 1832 to 1833, and participated in academic activities of the local Academy of Sciences. At that time, he studied the series expansion and differential equations of complex functions (strong series method) and made important contributions to this.

From 1833 to 1838, Cauchy served as a teacher to the Duke of Bordeaux, the "Crown Prince" of the Bourbon Dynasty, first in Prague and then in Gorz, and was finally awarded the title of "Baron". During this period, his research work was relatively infrequent.

In 1838 Cauchy returned to Paris. Since he did not swear allegiance to the King, he could only participate in academic activities of the Academy of Sciences and could not hold teaching jobs. He published a large number of important papers on complex variable functions, celestial mechanics, elastic mechanics and other aspects in the report of the recently founded French Academy of Sciences "and in the journal analysis and mathematical physics exercises written by himself".

Another revolution broke out in France in 1848. Louis Philippe was overthrown, the French Republic was re-established, and the oath of allegiance taken by public officials to the King of France was abolished. Cauchy became professor of mathematical astronomy at the University of Paris in 1848, resuming his 18-year hiatus from teaching at French higher education institutions.

In 1852, Napoleon III launched a coup, France changed from a republic to an empire, and the oath of allegiance of public officials to the new regime was restored. Cauchy immediately resigned from the University of Paris.

Later, Napoleon III exempted him and the physicist Arago from the oath of loyalty. So Cauchy was able to continue his teaching work until his death in the suburbs of Paris in 1857. Cauchy continued to participate in academic activities and published scientific papers until his death.

On May 23, 1857, he died suddenly at the age of 68. He died of fever. Before his death, he was still talking to the Archbishop of Paris. The last words he said were:

“People are always going to die, but their achievements will last forever.”

Personal anecdotes

Nicknames

When Cauchy was a student, There is a nickname "Bitter Melon" because he is usually like a bitter melon, silent and silent. If he says anything, it is very brief and confusing. It is very painful to communicate with this kind of person. Kexi had no friends, only a group of people who were jealous of his intelligence. At that time, social philosophy was popular in France. The books that Cauchy often read after work were the mathematics books of Joseph Louis Lagrance (1736-1813) and the spiritual book "The Imitation of Christ", which won him another reputation. The name is "a person with a crackling brain", which means a psychopath.

Koshi’s mother heard the rumor and wrote to him asking him for the truth. Cauchy wrote back: "If Christians could become insane, lunatic asylums would be filled with philosophers. Dear mother, your child is like a windmill in the wilderness. Mathematics and faith are his wings. When the wind blows, the windmill will spin balancedly and generate motivation to help others. 』

In 1816, Cauchy returned to Paris and served as a professor of mathematics at his alma mater. Cauchy himself wrote: "I am as excited as a salmon who has found his own river." Soon he got married, and his happy married life helped his ability to communicate with others.

Famous

Mathematical master Bernoulli once said: "Only mathematics can explore "infinity", and "infinity" is one of the attributes of God." Physics, chemistry, and biology are all subjects within limits, and "infinity" can represent the limit that can never be measured. The concept of "infinity" drives philosophers crazy, makes theologians sigh, and makes many people deeply afraid. Cauchy, however, applied "infinity" to determine a more precise mathematical meaning. He regarded the differential calculus of mathematics as "changes in infinite hours" and expressed integrals as "the sum of infinite many infinitesimals". Cauchy redefined calculus using infinity, and it is still the opening chapter of every calculus textbook.

In 1821, Cauchy's reputation spread far and wide. Students from as far away as Berlin, Madrid, and St. Petersburg came to his classroom. He also published the very famous "eigenvalue" theory and wrote at the same time: "In the field of pure mathematics, there seems to be no actual physical phenomena. There is no natural thing to prove it, but it is the promised land that mathematicians see from afar. The theoretical mathematician is not a discoverer, but a reporter of the promised land."

Late life

After the age of forty, Cauchy was unwilling to be loyal to the new government. He believed that academics should be free from political influence. He gave up his job and his motherland and took his wife to travel and teach in Switzerland and Italy. Universities everywhere welcomed him. But he wrote: "The excitement for mathematics is a load that the body cannot bear for a long time. It's tiring!" "After Kexi turned forty, he stopped doing research after class.

His health gradually declined, and in 1838 he returned to teach at the University of Paris, but left again due to issues of political allegiance. Because of his persistence, in 1848 France passed the academic freedom of university professors, which was based on personal conscience and was not subject to political restrictions. Since then, universities around the world have followed this system, and universities have become places of academic freedom.

Paris paper is expensive

It is said that when Cauchy was young, he submitted papers to the Journal of the Paris Academy of Sciences. They were so fast and so large that the printers rushed to buy up all the papers in Paris in order to print these papers. The inventory of paper shops caused a shortage of paper on the market, causing a sharp increase in paper prices and rising costs for printing plants. Therefore, the Academy of Sciences passed a resolution that the length of each published paper in the future should not exceed 4 pages. Many of Cauchy's long papers were not allowed to be published in his own country and had to be resubmitted to journals in other countries.

Personal Achievements

Cauchy was a famous and prolific mathematician. His complete works were published from 1882 to 1974, when the last volume was published, for a total of 28 volumes. His main contributions are as follows;

Single complex function

Cauchy's most important and original work is on the theory of single complex function. Mathematicians in the 18th century used definite integrals whose upper and lower limits were imaginary numbers. But no clear definition is given. Cauchy first clarified the relevant concepts and used this kind of integral to study a variety of problems, such as the calculation of real definite integrals, the expansion of series and infinite products, the use of integrals containing parameter variables to express the solutions of differential equations, etc.

Fundamentals of Analysis

The analysis courses and related teaching materials taught by Cauchy at the Comprehensive Engineering School have had a great impact on the mathematics community. Ever since Newton and Leibniz invented calculus (i.e. analysis of infinitesimals, or analysis for short), the theoretical foundations of this discipline have been vague. For further development, rigorous theory must be established. Cauchy first successfully established the limit theory for this purpose.

The function of limit theory

Suppose the function f(x) is at point x. It is defined in a certain decentered neighborhood of that if there exists a constant A, for any given positive number ε (no matter how small it is), there is always a positive number δ such that when Then the constant A is called the function f(x) when x→x. time limit.

"Strictly speaking, there is no such thing as mathematical proof. In the end, we can do nothing but point and point; the proof is what Littlewood and I call Kamubuki. , is a rhetoric made up to impress people, a picture on the blackboard in class, and a way to stimulate students' imagination." - Hardy.

Mathematics is so important and has the same status as Chinese literature in China. The reason is that mathematics itself is a language, and it is a world language with universality. Therefore, it is very necessary to strictly distinguish the parts of speech of mathematical concepts, not only as a requirement of mathematics itself, but also as a requirement of language science.

When it comes to language and parts of speech, you need to understand some basic knowledge of Chinese language.

1. Noun: a word that expresses the name of a person or thing, place, location, etc.

2. Verb: a word that expresses action behavior, development and changes, psychological activities, etc.

Since the first day of its birth, calculus has never left contradictions and arguments. For example, Berkeley's refutation (refutation of infinitesimals), Zeno's paradox, etc. If, through these debates, it can be found that they are actually just discussing the final form of the issue in disguise! Just as Leibniz focused on the final fate of particles. Some people say that Cauchy-Weierstrass's limit definition has a phenomenon of "limit avoidance". This statement is one-sided and not objective, but it still points out some problems (it should be said that the final form avoids them). Cauchy-Weierstrass's definition of limit is very classic when translated into Chinese language. Cauchy-Weierstrass's definition of limit not only defines the limit, but also describes a movement phenomenon - movement closer to the limit (final form). Finally, to put the finishing touch, the final form a (if it exists, it is unclear how it came about) is called the limit.

From a grammatical analysis, this statement essentially gives the "final form" a title (name) - limit. Therefore, in the Cauchy-Weierstrass definition of limit, limit is a noun, not a verb.

Therefore, the movement approaching the limit is called the limit phenomenon. Many people confuse limit phenomena with limits when understanding Cauchy-Weierstrass's definition of limits, and generally refer to "limit phenomena" and "limits" as limits.

I briefly talked about the research on the final form in "Calculus Secret Report 4".

Since the modern function limit definition does not explain the final form (avoided)! So, what story does the limit definition of a function tell? What do the relevant mathematical proofs prove?

Actually, it is saying one thing: if there is a limit (final form), there must be a limit phenomenon; conversely, if there is a limit phenomenon, there must be a limit! To put it simply, the extreme phenomenon is a necessary and sufficient condition for the limit (final form). Therefore, to prove that the limit exists (you don’t have to study how it comes about), it is enough to prove that the limit phenomenon exists. It is indeed suspected of being opportunistic!

Because of this, the definition of modern limit cannot tell you where the limit comes from, it can only tell you that the limit exists (and can be proven). The ultimate phenomenon is essentially a movement phenomenon. What is the ideal tool to describe the movement phenomenon - function. Therefore, it is not surprising that the modern (professional) limit definitions of functions have some functional flavors (one-to-one correspondence, there is always ε and δ correspondence).

Some people are quite outrageous, saying that limit is a verb. The reason is that the essence of limit is: "A changing quantity is infinitely close to a fixed quantity." This is the essence of the limit phenomenon, not the limit.

However, it is necessary to describe the limiting phenomenon. Do you have to use the Cauchy-Weierstrass tongue-twisting model? Of course not, the model can be changed, and the elementaryization of calculus changes this model. Some complex mathematical proofs have been simplified, such as the uniqueness of limits, monotonicity of functions, etc.

In Cauchy's works, there is no common language, and his statement seems not to be precise enough, so there are sometimes errors, such as errors caused by not establishing the concepts of consistent continuity and consistent convergence. But regarding the principles of calculus, his concepts were mostly correct, and their clarity was unprecedented. For example, his definition of continuous functions and their integrals is exact, he first accurately proved Taylor's formula, and he gave the definition of series convergence and some discriminant methods.

Ordinary differential equations

Cauchy's most profound contribution in analysis is in the field of ordinary differential equations. He first proved the existence and uniqueness of the solution to the equation. No one had asked this question before him. It is generally considered to be the three main methods proposed by Cauchy, namely the Cauchy-Lipschitz method, the gradual approximation method and the strong series method. In fact, approximate calculations and estimates for solutions have also been scattered before. Cauchy's greatest contribution was to see that by computing strong series, it was possible to prove that the approximation step converged, the limit of which was the solution of the equation.

Mathematical Theory of Elastic Mechanics

Cauchy is the founder of the mathematical theory of elastic mechanics in mechanics. In his 1823 article "Study on the Equilibrium and Motion of Elastic Bodies and Fluids (Elastic or Inelastic)", he proposed a general equation for the equilibrium and motion of (isotropic) elastic bodies (he later extended this equation to various In the case of anisotropy), a strict definition of stress and strain is given, and it is proposed that they can be represented by six components respectively. This paper is also meaningful for the fluid motion equation, which is better than C. -L. -M. -H. Navier's results in 1821 were obtained later, but they used a continuum model, and the results were more general than those obtained by Navier. The fluid equation he proposed on this basis in 1828 only had one less static pressure term than the now commonly used Navier-Stokes equation (1848).

Others

Although Cauchy mainly studies analysis, he has made contributions in various fields of mathematics. Regarding other subjects that use mathematics, his achievements in astronomy and optics are minor, but he is one of the founders of the mathematical elasticity theory. In addition to the above, his other contributions in mathematics are as follows:

1. In terms of analysis: basic concepts of traveling characteristic lines in the theory of first-order partial differential equations; understanding the role of Fourier transform in solving differential equations, etc.

2. Geometry: He created integral geometry and obtained the formula to express the length of a plane convex curve by its orthogonal projection on a plane straight line.

3. In terms of algebra: first proved that matrices with orders exceeding 1 have eigenvalues; together with Binet, he discovered the formula for multiplying two determinants, first clearly proposed the concept of permutation groups, and obtained some non-trivial results in group theory; independently discovered The so-called "essentials of algebra" are Grassmann's principles of external algebra.