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Euler's series of achievements laid the foundation for number theory as an independent branch of mathematics. A large part of Euler's works is related to the theory of divisibility of the same number. Euler's most important discovery in number theory is the quadratic anti-law.
2. Algebra
Euler's Introduction to Algebra is a systematic summary of algebra that began to develop in the middle of16th century.
3. Infinite series
Euler's Introduction to Calculus is (1755) the first treatise on finite difference calculus, and he was the first person to introduce difference operator. Euler not only widely used power series, but also introduced a new class of extremely important Fourier trigonometric series. 1777, in order to expand the given function into cosine series in the interval of (0, "180", Euler derived the Fourier coefficient formula. Euler also introduced function expansion into infinite product and found the sum of elementary fractions, which played an important role in the later general theory of analytic functions. He put forward a new and broader definition of the concept of sum of series. He also proposed two summation methods. These rich ideas had a profound influence on two topics in divergent series theory from the end of 19 to the beginning of the 20th century, namely, asymptotic series theory and the concept of summability.
4. Functional concept
/kloc-in the mid-8th century, there were many new discoveries in the field of analysis, many of which were Euler's own work. They are systematically summarized in Euler's analytic trilogy, which includes the introduction of infinite analysis, the principle of differential calculus and the principle of integral calculus. These three books are works of four styles, which are milestones in the development of analytical science.
5. Elementary function
Introduction to Infinite Analysis Volume I *** 18, mainly studying elementary function theory. Among them, the eighth chapter studies circular function, expounds the analytic theory of trigonometric function for the first time, and gives a derivation of de Morville formula. Euler studied exponential function and logarithmic function in Introduction to Infinite Analysis, and he gave a famous expression-Euler identity (in the expression, a number approaching infinity was used; After 1777, Euler used to represent imaginary unit), but only the logarithmic function of positive independent variables was considered. 175 1 year, Euler published a complete complex number theory.
6. Simple complex function
Through the study of elementary functions, D'Alembert and Euler successively came to the conclusion that algebraic operation and transcendental operation are closed in complex number field (expressed by modern numbers) during the period of 1747- 1 year. The two of them have also made initial progress in the general theory of analytic functions.
7. Stones
Euler's two books, Principles of Differential Calculus and Principles of Integral Calculus, elaborated the calculus method in the most detailed and systematic way at that time. He enriched these two branches of infinitesimal analysis with his many discoveries.
8. Differential equations
The principle of integration also shows many discoveries of Euler in the theory of ordinary differential equations and partial differential equations. He and other mathematicians created the subject of differential equations in the process of solving mechanical and physical problems.
In terms of ordinary differential equations, Euler gave the classical solution of linear homogeneous equations with constant coefficients of any order by method of substitution in his paper published in 1743, in which the terms "general solution" and "special solution" were introduced for the first time. 1753, he published the solution of non-homogeneous linear equation with constant coefficients, that is, gradually reducing the order of the equation.
Euler began to study partial differential equations in 1930s. His most important work in this field is about second-order linear equations.
9. Variational method
1734 He generalized the steepest descent line problem. Then, set out to find a more general method about this problem. 1744, Euler's book The Method of Finding Curves with Some Maximal or Minimal Properties was published. This is a milestone in the history of variational science and marks the birth of variational method as a new mathematical analysis method.
10.geometry
In coordinate geometry, Euler's main contribution is to apply Euler angle to the corresponding transformation for the first time, and to deeply study the general equation of quadric surface.
In differential geometry, Euler first introduced the concept of intrinsic coordinates of plane curve in 1736, that is, the geometric quantity of curve arc length was taken as the coordinates of points on the curve, and thus the study of curve intrinsic geometry began. 1760, Euler established the theory of surface in the study of curves on surfaces. This book is Euler's most important contribution to differential geometry and a milestone in the history of differential geometry.
Euler's research on topology is also first-class. 1735, Euler solved the famous game problem of Koenigsberg Seven Bridges with simplified (or idealized) representation, and obtained the decision rule of the river bridge graph with topological significance, that is, the euler theorem in today's network theory.
1 1. Mechanics
Euler applied mathematical analysis methods to mechanics and made outstanding contributions in various fields of mechanics. He is the founder of rigid body dynamics and fluid mechanics, and the pioneer of elastic system pin theory. In the two-volume Analysis and Interpretation of Mechanics or Sports Science published by 1736, he considered the differential equations of motion of free particles and constrained particles and their solutions. In his book, Euler interprets mechanics as "the science of motion" and excludes "the science of balance", that is, statics. As far as mechanical principles are concerned, Euler agrees with Maboti's principle of minimum action. In the research of rigid body kinematics and dynamics, he obtained the most basic results, including: the finite rotation of a fixed point of a rigid body is equivalent to the rotation of a fixed point around an axis, and the fixed point motion of a rigid body can be described by the changes of three angles (called Euler angles); The relationship between angular velocity change and external torque when a rigid body rotates at a fixed point; The law of motion of a fixed-point rigid body without external torque (called Euler case of fixed-point motion, explained geometrically by L. Panso in 1834) and the differential equation of motion of a free rigid body. These achievements are all contained in his monograph "On the Motion of Rigid Bodies" (1765). Euler thinks that the differential equation of particle dynamics can be applied to liquids (1750). He used two methods to describe the motion of fluid, that is, he described the fluid velocity field according to the fixed point in space (1755) and the determined fluid particle (1759) respectively. These two methods are usually called Euler representation and Lagrange representation. Euler laid a theoretical foundation for the motion of an ideal fluid (assuming that the fluid is incompressible and its viscosity can be ignored), and gave a continuity equation reflecting the conservation of mass halo (1752) and a hydrodynamic equation reflecting the law of momentum change (1755). Euler studied the vibration of elastic systems such as strings and rods. Together with Daniel I Bernoulli, he analyzed the vibration of the heavy chain suspended at the upper end and the vibration of the corresponding discrete model (with a string of mass lines suspended). With the help of Daniel I Bernoulli, he obtained the exact solution of the elastic curve of the elastic compression thin rod after instability. The minimum pressure that can make the thin rod produce this deflection is called the Euler critical load of the thin rod. Euler also studied applied mechanics such as ballistics, ship theory and moon motion theory. Euler's life is a life of struggle for the development of mathematics. His outstanding wisdom, tenacious perseverance, tireless spirit of struggle and noble scientific ethics are always worth learning. Euler also created many mathematical symbols, such as π( 1736), i( 1777) and e( 1748). Tg( 1753), △x( 1755), σ (1755), f(x)( 1734), etc.
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, φ(8)=4, because there are four natural numbers, 1, 3,5 and 7,8, which are prime numbers.
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 defined the power of imaginary number as Euler formula and became 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 "most outstanding mathematical formula" 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: The relationship between face and face.
1736, Euler solved the problem of the Seven Bridges in Konigsberg, and published the article "Solving the Problem of Location Geometry", which expounded a problem and was the earliest model to apply graph theory and topology.
Sudoku is a Latin cube concept invented by Euler, which was not popular at that time and was not forged by ordinary Japanese office workers until the 20th century.