Euler needed it to compute slowly converging infinite series while Maclaurin used it to calculate integrals.

The Euler – Maclaurin formula is also used for detailed error analysis in numerical quadrature.

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.

Sometimes is used, which is unfortunate since this is also used for the Euler characteristic

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

In 1821, Cauchy began to put calculus on a firm logical foundation by rejecting the principle of the generality of algebra widely used in earlier work, particularly by Euler.

The Euler – Tricomi equation is used in the investigation of transonic flow.

In the shortest of them ( 43 pages as of 2009 ), which he titles " Apology for the Proof of the Riemann Hypothesis " ( using the word " apology " in the rarely used sense of apologia ), he claims to use his tools on the theory of Hilbert spaces of entire functions to prove the Riemann Hypothesis for Dirichlet L-functions ( thus proving GRH ) and a similar statement for the Euler zeta function, and even to be able to assert that zeros are simple.

If the Euler criterion formula is used modulo a composite number, the result may or may not be the value of the Jacobi symbol.

In the case of 3 × 3 matrices, three such rotations suffice ; and by fixing the sequence we can thus describe all 3 × 3 rotation matrices ( though not uniquely ) in terms of the three angles used, often called Euler angles.

The Euler characteristic was originally defined for polyhedra and used to prove various theorems about them, including the classification of the Platonic solids.

The Herbrand quotient is a type of Euler characteristic, used in homological algebra.

In algebraic topology, Chi is used to represent the Euler characteristic of a surface.

Euler wrote about this identity in a letter dated May 4, 1748 to Goldbach ( but he used a different sign convention from the above ).

( a / n ) ( mod n ), where ( a, n ) = 1 and ( a / n ) is the Jacobi symbol, is sometimes used for a definition of an Euler pseudoprime.

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.

The typical Euler diagram used to represent the propinquity effect is shown below where U

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

It can be used to model motion constrained by a hinges and Axles, and so is easily visualised, perhaps even more so than Euler angles.

But like Euler angles it is usually converted to another representation before being used.

Sometimes " Euler angles " is used for all of them.

When the intrinsic rotations equivalence is used to name the possible conventions of Euler Angles the names are like Z-X ’- Z ’’.

Whereas optimization methods are nearly as old as calculus, dating back to Isaac Newton, Leonhard Euler, Daniel Bernoulli, and Joseph Louis Lagrange, who used them to solve problems such as the shape of the catenary curve, numerical optimization reached prominence in the digital age.

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 finishing his studies he went on long educational voyages from 1710 to 1724 through Europe, visiting other German states, England, Holland, Italy, and France, meeting with many famous mathematicians, such as Gottfried Leibniz, Leonhard Euler, and Nicholas I Bernoulli.

He also studied and proved some theorems on perfect powers, such as the Goldbach – Euler theorem, and made several notable contributions to analysis.

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.

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.

* In the plane ( d = 2 ), if there are b vertices on the convex hull, then any triangulation of the points has at most 2n − 2 − b triangles, plus one exterior face ( see Euler characteristic ).

In this way we get a proof of the Euler – Maclaurin summation formula by mathematical induction, in which the induction step relies on integration by parts and on the identities for periodic Bernoulli functions.

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 closed ( orientable or non-orientable ) surfaces with positive genus, the maximum number p of colors needed depends on the surface's Euler characteristic χ according to the formula

This view was further elaborated by Belidor ( representation of rough surfaces with spherical asperities, 1737 ) and Leonhard Euler ( 1750 ) who derived the angle of repose of a weight on an inclined plane and first distinguished between static and kinetic friction ..

The paper written by Leonhard Euler on the Seven Bridges of Königsberg and published in 1736 is regarded as the first paper in the history of graph theory.

A statement attributed to Pierre-Simon Laplace expresses Euler's influence on mathematics: " Read Euler, read Euler, he is the master of us all.

Euler was born on April 15, 1707, in Basel to Paul Euler, a pastor of the Reformed Church.

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.

In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono.

Euler arrived in the Russian capital on 17 May 1727.

Stamp of the former German Democratic Republic honoring Euler on the 200th anniversary of his death.

Concerned about the continuing turmoil in Russia, Euler left St. Petersburg on 19 June 1741 to take up a post at the Berlin Academy, which he had been offered by Frederick the Great of Prussia.

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.

Three years after suffering a near-fatal fever in 1735 he became nearly blind in his right eye, but Euler rather blamed his condition on the painstaking work on cartography he performed for the St. Petersburg Academy.

In St. Petersburg on 18 September 1783, after a lunch with his family, during a conversation with a fellow academician Anders Johan Lexell about the newly discovered Uranus and its orbit, Euler suffered a brain hemorrhage and died a few hours later.