Portrait of leonhard euler (6) Euler 1707 15 was born in Switzerland in April and received education there. Euler is a mathematical genius. As a math professor, he taught in St. Petersburg and Berlin and then returned to St. Petersburg. Euler is the most prolific mathematician in history. His complete works have 75 volumes. Euler actually ruled the mathematics in the18th century, and he derived many results for the newly invented calculus at that time. In the last seven years of her life, Euler was completely blind. Nevertheless, he finished half his work at an amazing speed. Euler lived a pious life. Legend has it that Euler challenged Denis Derot in the court of Catherine II: "Sir, (a+b) n/n = x; So God exists, that's the answer! " Euler's death was also special: at a friend's party, he resigned and went to work, and finally left quietly on his desk. The asteroid Euler 2002 was named in memory of Euler.

contribution

"Euler's calculation seems effortless, just like a person breathing, like an eagle hovering in the wind." This sentence is not an exaggeration for Euler's unparalleled mathematical talent. He is the most prolific mathematician in history. His contemporaries called him "the embodiment of analysis". Euler wrote a long academic paper as easily as a quick-witted writer wrote a letter to a close friend. Even if he was completely blind in the last 17 years of his life, it didn't stop his great fertility. If blindness has any effect, it is to improve his inner thinking imagination. It was not until 1936 that people knew exactly how many works Euler wrote. However, it is estimated that it will take 60 to 80 volumes to publish Euler's anthology. From 65438 to 0909, the Swiss Federation of Natural Sciences began to collect and publish the academic papers of Euler's anthology. This work has been funded by many individuals and mathematical groups all over the world. This just shows that Euler belongs to the whole civilized world, not just Switzerland. The budget carefully prepared for this work (1909 coins about 80,000 dollars) was completely broken by the unexpected discovery of a large number of Euler manuscripts in St. Petersburg (Leningrad).

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The 300th anniversary of Euler's birth (8 photos) Euler's mathematical career began in the year of Newton's death. For a genius like Euler, it is impossible to choose a more favorable era. Analytic geometry (published in 1637) has been used for 90 years, calculus for about 50 years, and Newton's law of universal gravitation, the key of physical astronomy, has been used in mathematics for 40 years. In each of these fields, a large number of isolated problems have been solved, and obvious attempts have been made to unify them everywhere. However, the whole mathematics, pure mathematics and applied mathematics have not been systematically studied as later. In particular, the powerful analytical methods of De Kratos, Newton and Leibniz have not been fully utilized as they were later, especially in mechanics and geometry. Algebra and trigonometry at that time had been systematized and developed at a lower level. Especially the latter, has been basically improved. In Fermat's Diophantine analysis and general integer properties, there can be no such "temporary perfection" (even now). But in this respect, Euler also proved that he is indeed a master. In fact, one of the most striking features of Euler's versatility is that he has the same ability in two branches of mathematics-continuous mathematics and discrete mathematics. As an mathematician, Euler has never been surpassed by anyone. Perhaps no one can approach his level except jacoby. Algorithmists are mathematicians who design algorithms to solve various special problems. For a simple example, we can assume (or prove) that any positive real number has a real square root. But how can we work out this root? There are many known methods, and algorithm scientists should design practical concrete steps. For example, in Diophantine analysis and integral calculus, it is often impossible to solve the problem until one or more variables are skillfully (often simply) transformed by the functions of other variables. Algorithmists are mathematicians, and they will naturally find this trick. They don't have any identical procedures to follow. Algorithmists are like people who can write limericks at will-they are born, not made. At present, fashion despises "little arithmeticians". However, when a truly great mathematician, such as Lo Manu Kuo of India, suddenly appears from somewhere, even experienced analysts will hail him as a gift from heaven: his magical insight into seemingly unrelated formulas will reveal hidden clues from one field to another. So that analysts can find new topics for them to find these clues. Algorithmists are "formulists", and they like beautiful forms for the formula itself.

achievement

Euler and daniel bernoulli established the moment law of elastic body together: the moment acting on the elastic slender rod is proportional to the elasticity of matter and the inertia momentum passing through the center of mass axis and the cross section perpendicular to them. He also directly from Newton's law of motion, established the Euler equation in fluid mechanics. These equations are formally equivalent to Naville-Stokes equations with viscosity of 0. People's main interest in these equations is that they can be used to study shock waves. He made an important contribution to the theory of differential equations. He is also the founder of Euler approximation used in computational mechanics. The most famous one is called Euler method. In number theory, he introduced Euler function. The Euler function of natural numbers is defined as the number of natural numbers less than and coprime. For example, because there are four natural numbers 1, 3, 5, 7 and 8 are coprime. RSA public key cryptography algorithm widely used in computer field is also based on Euler function. In the field of analysis, Euler synthesized Leibniz's differential and Newton's flow number. He became famous in 1735 for solving the long-standing Bessel problem. Euler defines the power of imaginary number as the following formula: this is Euler formula, which becomes the center of exponential function. In elementary analysis, it is essentially either a variant of exponential function or a polynomial, and the two must be one of them. Richard feynman's "the most outstanding mathematical axiom" is a simple inference of Euler's formula (usually called Euler's identity): 1735, he defined Euler-Maceroni constant, which is very useful in differential equations:: He is one of the discoverers of Euler-Maceroni formula, which is very effective in calculating difficult integrals, sums and series. 1739, Euler wrote Tentamennovaetheoriaemusicae, trying to combine mathematics with music. A biographer wrote: This is a book "for musicians who are proficient in mathematics and mathematicians who are proficient in music". In economics, Euler proved that if every element of a product is used to pay for its marginal product, the total income and total output will be completely exhausted under the condition of constant return to scale. In geometry and algebraic topology, Euler formula gives edges, vertices and -(zh-hans: face; Zh-hant:Faces)- There is a relationship:: where F is the sum of faces of a given polyhedron, E is the sum of edges, and V is the sum of vertices. This theorem can also be used in plan view. For non-planar graphs, Euler's formula can be generalized as: If a graph can be embedded in a manifold, then:: where χ is the Euler eigenvalue of this manifold, which is invariant under the continuous deformation of the manifold. The Euler eigenvalue of a simply connected flow shape such as a sphere or a plane is 2. For any planar graph, Euler's formula can be generalized as:, where is the number of connected branches in the graph. 1736, Euler solved the problem of the Seven Bridges in Konigsberg and published the article "Solution to Geometric Problems", which expounded a problem and was the first model to use graph theory and topology. Sudoku is a Latin cube concept invented by Euler, which was not popular at that time until it was forged by ordinary Japanese office workers in the 20th century, bringing the most influential person-Euler.

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Euler was the best mathematician in18th century and one of the greatest mathematicians in history. /kloc-In the 8th century, Swiss mathematician and physicist Lennart Euler has been one of the most outstanding scientists in the world. All his creations are widely used in the whole physics and many engineering fields. Euler's achievements in mathematics and science are unbelievable. He wrote thirty-two long books, several of which were more than one volume, and also wrote many creative mathematical and scientific papers. His scientific works amount to more than seventy volumes. Euler's genius has enriched every field of pure mathematics and applied mathematics, and his achievements in mathematical physics have infinitely broad application fields. As early as last century, isaac newton put forward the basic laws of mechanics. Euler was particularly good at demonstrating how to apply these laws to some common physical phenomena. For example, he applied Newton's law to fluid motion and established fluid mechanics equations. Similarly, by carefully analyzing the possible motion of a rigid body and applying Newton's law, he established a set of equations that can completely determine the motion of a rigid body. Of course, in practice, nothing is completely rigid. Euler also contributed to elasticity, which is a theory to study how solids deform under external forces. Euler's genius also lies in his mathematical analysis of astronomical problems, especially how the three bodies, the sun, the moon and the earth, move under the interaction of gravity. This problem-a problem that still faces in 21century-has not been completely solved. By the way, Euler was a unique and outstanding scientist in the18th century. He supported the light wave theory, and it turned out that he was right. Euler's rich mind often opens the way for others to make famous discoveries. For example, the French mathematician and physicist Joseph-Louis Lagrange created a set of equations called "Lagrange Equation". This equation is very important in theory and can be used to solve many mechanical problems. However, because the basic equation was first proposed by Euler, it is usually called Euler-Lagrange equation. It is generally believed that another French mathematician, Joan Baptist Fourier, founded an important mathematical method called Fourier analysis, and its basic equation was originally founded by lennert Euler, so it is called Euler-Frish equation. These equations are widely used in many different physical fields, including acoustics and electromagnetism. In mathematics, he is particularly interested in two fields of calculus-differential equations and infinite series. He has made very important contributions in these two fields, but he is too professional to describe here. His contributions to variational calculus and complex mathematics laid the foundation for all the achievements made later. These two subjects are not only of great significance to pure mathematics, but also widely used in scientific work. Euler's formula eiq = cosθten isθ represents the relationship between trigonometric function and imaginary number, which can be used to find the logarithm of negative numbers. It is one of the most widely used formulas in all mathematical fields. Euler also wrote a textbook of analytic geometry, which made great contributions to differential geometry and general geometry. Euler is not only handy in making mathematical inventions that can be applied to science, but also has almost the same outstanding talent in the field of pure mathematics. But many of his contributions to number theory are too profound to be described here. Euler was also a pioneer in the field of topology, a branch of mathematics, which became very important in the twentieth century. Last but not least, Euler made an important contribution to the making of mathematical symbols used now. For example, he proposed the Greek letter π commonly used in pi. He also introduced many other simple symbols, which are often used in mathematics now. Euler was born in 1707 in Basel, Switzerland. He was admitted to university of basel at the age of thirteen. I studied theology at first, and soon changed to mathematics. He got his master's degree in university of basel at the age of seventeen, and at the age of twenty, he was invited by Catherine I to join St. Petersburg Academy of Sciences. At the age of 23, he became a professor of physics at the institute. At the age of 26, he succeeded the famous mathematician daniel bernoulli and became the director of the Institute of Mathematics. Two years later, he was blind in one eye, but he continued to work with great enthusiasm and wrote many excellent papers. 174 1 year, frederick the great of Prussia lured Euler from Russia to join the Berlin Academy of Sciences. After 25 years in Berlin, he returned to Russia on 1766. Soon, his other eye also lost its light. Even if such a disaster came, he didn't stop his research work. Euler has amazing mental arithmetic ability. He published first-class mathematical papers until his last breath. He died in St. Petersburg on 1783 at the age of 76. Euler was married twice and had thirteen children, but eight of them died in infancy. Even without Euler, all his discoveries will eventually be made by someone. But I think, as a measure of this situation, we should ask such a question: If no one can make his discovery at all, what will be the difference between science and the modern world? As far as lennert Euler is concerned, the answer seems clear: without Euler's formulas, equations and methods, the progress of modern science and technology will lag behind, which actually seems unimaginable. Looking through the index of mathematical physics textbooks, we will find the following photos: Euler angle (rigid body motion), Euler constant (infinite series), Euler equation (fluid mechanics), Euler formula (compound variable), Euler number (infinite series), Euler polygonal curve (differential equation), Euler function theorem (differential equation), Euler transformation (infinite series) and Bernoulli-Euler law. From all this, the reader may ask why Euler didn't rank high in this book. The main reason is that although Euler has made outstanding achievements in demonstrating how to apply Newton's laws, he has never found any original scientific laws himself, which is why people like William Conrad, Roentgen and Mendel are ahead of him. Each of them mainly discovered new scientific phenomena or laws. Nevertheless, Euler made great contributions to science, engineering and mathematics.

In the name of Euler

Euler formula

Euler formula refers to many formulas named after Euler. Among them, the most famous is Euler's formula of amplitude and angle in complex variable function-linked complex number, exponential function and trigonometric function; Euler polyhedron formula in topology: Euler function formula in elementary number theory. In addition, it also includes some other Euler formulas, such as fractional formula and so on.

The ultimate function of rulers

Euler function, in number theory, for a positive integer n, Euler function is the number of numbers less than or equal to n that are coprime with n. This function is named after its first researcher, Euler's congruence function, φ function, Euler quotient and so on. For example, φ(8)=4, because 1, 3, 5 and 7 are all coprime with 8. Euler theorem's proof includes facts in ring theory and Lagrange theorem derived from Euler function.

euler theorem

Many constants, formulas and theorems named after Euler can be seen in mathematics and many branches. In number theory, euler theorem (also called Fermat-euler theorem or Euler function theorem) is about the property of congruence. Euler theorem is named after Swiss mathematician leonhard euler, and this theorem is considered as one of the most beautiful theorems in mathematics. Euler theorem is actually a generalization of Fermat's theorem. There are also euler theorem in plane geometry and euler theorem of polyhedron (in a convex polyhedron, the number of vertices-number of edges+number of faces =2). In western economics, euler theorem is also known as the net theorem of output distribution, which means that under the condition of perfect competition, all products are just enough to be distributed to all factors, assuming that the long-term medium-scale income remains unchanged.

Euler pt

Three independent angular parameters used to determine the position of a fixed-point rotating rigid body are composed of nutation angle θ, precession angle ψ and self-rotation angle J, which were first proposed by Euler.

euler equation

1755, the Swiss mathematician L. Euler first proposed this equation in his book General Principles of Fluid Motion. When studying some physical problems, such as heat conduction, vibration of circular membrane and electromagnetic wave propagation, the following equations are often encountered: (ax 2d 2+bxd+c) y = f (x), where A, B and C are constants, which is a second-order linear differential equation with variable coefficients. Its coefficients have certain rules: the coefficient of the second derivative D^2y is a quadratic function ax^2, the coefficient of the first derivative Dy is a linear function bx, and the coefficient of Y is constant. Such an equation is called Euler equation.