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. The Swiss Federation of Natural Sciences began to collect and publish the academic papers of Euler's anthology on 1909. This work is carried out with the support of many individuals and mathematical groups all over the world. This just shows that Euler belongs to the whole civilized world, not just Switzerland. The carefully prepared budget of this work (1909 coin is about 80,000 dollars) was completely broken by the unexpected discovery of a large number of Euler manuscripts in St. Petersburg (Leningrad).
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 established Euler equation in fluid mechanics directly from Newton's law of motion. These equations are formally equivalent to Naville-Stokes equations with viscosity of 0. People are interested in these equations mainly because 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 method 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 and 7,8 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:
Where is the Riemann function?
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 "most outstanding mathematical generality" is a simple inference of Euler's formula (usually called Euler's identity):
In 1735, he defined Euler-Maceroni constant which is 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, then 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; The relationship between zh-hant:surface)- is:
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 applied to plane drawings. 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 square concept invented by Euler, which was not popular at that time and was not forged by ordinary Japanese office workers until the 20th century.