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

The Euler equations were extended to incorporate the effects of viscosity in the first half of the 1800s, resulting in the Navier-Stokes equations.

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.

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.

The additional terms on the force side of the equation can be recognized as, reading from left to right, the Euler force, the Coriolis force, and the centrifugal force, respectively.

When the angular velocity of this co-rotating frame is not constant, that is, for non-circular orbits, other fictitious forces — the Coriolis force and the Euler force — will arise, but can be ignored since they will cancel each other, yielding a net zero acceleration transverse to the moving radial vector, as required by the starting assumption that the vector co-rotates with the planet.

From a qualitative standpoint, the path can be approximated by an arc of a circle for a limited time, and for the limited time a particular radius of curvature applies, the centrifugal and Euler forces can be analyzed on the basis of circular motion with that radius.

Euler also suggested that the complex logarithms can have infinitely many values.

The Euler – MacLaurin formula can be understood as a curious application of some ideas from Banach spaces and functional analysis.

This way one can obtain expressions for ƒ ( n ), n = 0, 1, 2, ..., N, and adding them up gives the Euler – MacLaurin formula.

The Euler – MacLaurin summation formula can thus be seen to be an outcome of the representation of functions on the unit interval by the direct product of the Bernoulli polynomials and their duals.

For positive integer m the derivative of gamma function can be calculated as follows ( here γ is the Euler – Mascheroni constant ):

The motion of the top can be described by three Euler angles: the tilt angle between the symmetry axis of the top and the vertical ; the azimuth of the top about the vertical ; and the rotation angle of the top about its own axis.

It can be defined as a change in direction of the rotation axis in which the second Euler angle ( nutation ) is constant.

The Euler product formula can be used to calculate the asymptotic probability that s randomly selected integers are set-wise coprime.

#: Consider a unit sphere placed at the origin, a rotation around the x, y or z axis will map the sphere onto itself, indeed any rotation about a line through the origin can be expressed as a combination of rotations around the three-coordinate axis, see Euler angles.

Mean performance for the stage can be calculated from the velocity triangles, at this radius, using the Euler equation:

* The proof that every Haefliger structure on a manifold can be integrated to a foliation ( this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one ).

The Euler number E < sub > 2n </ sub > can be expressed as a sum over the even partitions of 2n,

Furthermore, the idea can be traced back to a paper by Leonhard Euler published in 1727, some 80 years before Thomas Young's 1807 paper.

Alternatively, it can be defined in terms of the Euler characteristic χ, via the relationship χ = 2 − 2g for closed surfaces, where g is the genus.

Alternatively, it can be defined for a closed surface in terms of the Euler characteristic χ, via the relationship χ = 2 − k, where k is the non-orientable genus.

The Euler genus is the minimal integer n such that the graph can be drawn without crossing itself on a sphere with n cross-caps or on a sphere with n / 2 handles.

Components can be calculated from the derivatives of the parameters defining the moving frame ( Euler angles or rotation matrices )

The angles of inclination, longitude of the ascending node, and argument of periapsis can also be described as the Euler angles defining the orientation of the orbit relative to the reference coordinate system.

Following the classical dynamics of Newton and Euler, the motion of a material body is produced by the action of externally applied forces which are assumed to be of two kinds: surface forces and body forces.

Paul Euler was a friend of the Bernoulli family — Johann Bernoulli, who was then regarded as Europe's foremost mathematician, would eventually be the most important influence on young Leonhard.

On July 10, 1726, Nicolas died of appendicitis after spending a year in Russia, and when Daniel assumed his brother's position in the mathematics / physics division, he recommended that the post in physiology that he had vacated be filled by his friend Euler.

After nearly two centuries, M < sub > 31 </ sub > was verified to be prime by Euler in 1772.

The Earth's axis of rotation – and hence the position of the North Pole – was commonly believed to be fixed ( relative to the surface of the Earth ) until, in the 18th century, the mathematician Leonhard Euler predicted that the axis might " wobble " slightly.

* The Euler – Mascheroni constant γ ( which has not even been proven to be irrational ).