More general equations of fluid flow-the Euler equations-were published by Leonhard Euler in 1757.

The reason why Euler and some other authors relate the Cauchy – Riemann equations with analyticity is that a major theorem in complex analysis says that holomorphic functions are analytic and viceversa.

The equation was eventually solved by Euler in the early 18th century, who also solved a number of other Diophantine equations.

The standard equations of inviscid flow are the Euler equations.

Another often used model, especially in computational fluid dynamics, is to use the Euler equations away from the body and the boundary layer equations, which incorporates viscosity, in a region close to the body.

The Euler equations can be integrated along a streamline to get Bernoulli's equation.

Variation of the pressure around an airfoil as obtained by a solution of the Euler equations.

In large parts of the flow viscosity may be neglected ; such an inviscid flow can be described mathematically through the Euler equations, resulting from the Navier-Stokes equations when the viscosity is neglected.

Neither the Navier-Stokes equations nor the Euler equations lend themselves to exact analytic solutions ; usually engineers have to resort to numerical solutions to solve them, however Euler's equation can be solved by making further simplifying assumptions.

) A vortex flow of any strength may be added to this uniform flow and the equation is solved, thus there are many flows that solve the Euler equations.

These types of integrals seem first to have attracted Laplace's attention in 1782 where he was following in the spirit of Euler in using the integrals themselves as solutions of equations.

While Maxwell's equations are consistent within special and general relativity, there are some quantum mechanical situations in which Maxwell's equations are significantly inaccurate: including extremely strong fields ( see Euler – Heisenberg Lagrangian ) and extremely short distances ( see vacuum polarization ).

If the generalized coordinates are represented as a vector and time differentiation is represented by a dot over the variable, then the equations of motion ( known as the Lagrange or Euler – Lagrange equations ) are a set of equations:

Brahmagupta ( 628 CE ) started the systematic study of indefinite quadratic equations — in particular, the misnamed Pell equation, in which Archimedes may have first been interested, and which did not start to be solved in the West until the time of Fermat and Euler.

This algebra is quotiented over by the ideal generated by the Euler – Lagrange equations.

They were first studied by Giulio Fagnano and Leonhard Euler.

The nobility were suspicious of the academy's foreign scientists, and thus cut funding and caused other difficulties for Euler and his colleagues.

Euler wrote over 200 letters to her in the early 1760s, which were later compiled into a best-selling volume entitled Letters of Euler on different Subjects in Natural Philosophy Addressed to a German Princess.

The development of infinitesimal calculus was at the forefront of 18th Century mathematical research, and the Bernoullis — family friends of Euler — were responsible for much of the early progress in the field.

Starting around the 15th century, new algorithms based on infinite series revolutionized the computation of, and were used by mathematicians including Madhava of Sangamagrama, Isaac Newton, Leonhard Euler, Carl Friedrich Gauss, and Srinivasa Ramanujan.

The values of the Riemann zeta function at even positive integers were computed by Euler.

Other attempts were made by Euler ( 1749 ), de Foncenex ( 1759 ), Lagrange ( 1772 ), and Laplace ( 1795 ).

These last four attempts assumed implicitly Girard's assertion ; to be more precise, the existence of solutions was assumed and all that remained to be proved was that their form was a + bi for some real numbers a and b. In modern terms, Euler, de Foncenex, Lagrange, and Laplace were assuming the existence of a splitting field of the polynomial p ( z ).

Venn diagrams are very similar to Euler diagrams, which were invented by Leonhard Euler ( 1708 – 1783 ) in the 18th century.

Venn diagrams and Euler diagrams were incorporated as part of instruction in set theory as part of the new math movement in the 1960s.

The components of the angular velocity pseudovector were first calculated by Leonhard Euler using his Euler angles and an intermediate frame made out of the intermediate frames of the construction:

Under this heading, the Board made many lesser awards, including some awards in total £ 5, 000 made to John Harrison before he received his main prize, an award of £ 3, 000 to the widow of Tobias Mayer, whose lunar tables were the basis of the lunar data in the early decades of the Nautical Almanac, £ 300 to Leonhard Euler for his ( assumed ) contribution to the work of Mayer, £ 50 each to Richard Dunthorne and Israel Lyons for contributing methods to shorten the calculations connected with lunar distances, and awards made to the designers of improvements in chronometers.

In solving it, he developed new methods that were refined by Leonhard Euler into what the latter called ( in 1766 ) the calculus of variations.

These were developed intensively from the second half of the eighteenth century ( by, for example, D ' Alembert, Euler, and Lagrange ) until the 1930s.

In the eighteenth century, two of the innovators of mathematical physics were Swiss: Daniel Bernoulli ( for contributions to fluid dynamics, and vibrating strings ), and, more especially, Leonhard Euler, ( for his work in variational calculus, dynamics, fluid dynamics, and many other things ).

Amongst the fruits of his industry may be mentioned a laborious investigation of the disturbances of Jupiter by Saturn, the results of which were employed and confirmed by Euler in his prize essay of 1748 ; a series of lunar observations extending over fifty years ; some interesting researches in terrestrial magnetism and atmospheric electricity, in the latter of which he detected a regular diurnal period ; and the determination of the places of a great number of stars, including at least twelve separate observations of Uranus, between 1750 and its discovery as a planet.

However, the concept was developed in 1727 by Leonhard Euler, and the first experiments that used the concept of Young's modulus in its current form were performed by the Italian scientist Giordano Riccati in 1782, predating Young's work by 25 years.

Euler noted that if there were only a finite number of primes, then the product on the right would clearly converge, contradicting the divergence of the harmonic series.

They were first discussed by Leonhard Euler while solving the famous Seven Bridges of Königsberg problem in 1736.

Some coordinate systems in mathematics behave as if there were real gimbals used to measure the angles, notably Euler angles.

The digital forms of Euler characteristic theorem and Gauss – Bonnet theorem were obtained by Chen et al.

Thābit's formula was rediscovered by Fermat ( 1601 – 1665 ) and Descartes ( 1596 – 1650 ), to whom it is sometimes ascribed, and extended by Euler ( 1707 – 1783 ).

Bernhard Riemann in his memoir " On the Number of Primes Less Than a Given Magnitude " published in 1859 extended the Euler definition to a complex variable, proved its meromorphic continuation and functional equation and established a relation between its zeros and the distribution of prime numbers.

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1.

In Tractatus de proportionibus ( 1328 ), Bradwardine extended the theory of proportions of Eudoxus of Cnidus to anticipate the concept of exponential growth, later developed by the Bernoulli and Euler, with compound interest as a special case.

These four lines now represent the Euler lines of the four possible triangles where the extended line HN is the Euler line of triangle ABC and the extended line AN is the Euler line of triangle BCH etc.

In particular, it can be defined as a Dirichlet series, it has an Euler product expansion, it satisfies a functional equation, it has an analytic continuation to a meromorphic function on the complex plane C with only a simple pole at s = 1, and its values encode arithmetic data of K. The extended Riemann hypothesis states that if ζ < sub > K </ sub >( s ) = 0 and 0 < Re ( s ) < 1, then Re ( s ) = 1 / 2.

There are two main groups of q-analogs, the " classical " q-analogs, with beginnings in the work of Leonhard Euler and extended by F. H. Jackson and others.