In mathematics, the Euler – Maclaurin formula provides a powerful connection between integrals ( see calculus ) and sums.

The Euler – Maclaurin formula provides expressions for the difference between the sum and the integral in terms of the higher derivatives ƒ < sup >( k )</ sup > at the end points of the interval m and n. Explicitly, for any natural number p, we have

The Gauss-Bonnet theorem links the total curvature of a surface to its Euler characteristic and provides an important link between local geometric properties and global topological properties.

Hierholzer's 1873 paper provides a different method for finding Euler cycles that is more efficient than Fleury's algorithm:

The Euler – Lotka equation provides a means of identifying the intrinsic growth rate.

Euler – Bernoulli beam theory ( also known as engineer's beam theory or classical beam theory ) is a simplification of the linear theory of elasticity which provides a means of calculating the load-carrying and deflection characteristics of beams.

Named after Leonhard Euler, it is a prototypical example of a q-series, a modular form, and provides the prototypical example of a relation between combinatorics and complex analysis.

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

Euler discussed a generalization of Euclidean geometry called affine geometry, which retains the fifth postulate unmodified while weakening postulates three and four in a way that eliminates the notions of angle ( whence right triangles become meaningless ) and of equality of length of line segments in general ( whence circles become meaningless ) while retaining the notions of parallelism as an equivalence relation between lines, and equality of length of parallel line segments ( so line segments continue to have a midpoint ).

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 ).

Euler was aware of the work of Lord Brouncker, the first European mathematician to find a general solution of the equation, but apparently confused Brouncker with Pell.

They are not the three dimensional instance of a general approach, like matrices ; nor are they easily related to real world models, like Euler angles or axis angles.

* Leonhard Euler solves the general homogeneous linear ordinary differential equation with constant coefficients.

For a correspondence of curves, there is a more general formula, Zeuthen's theorem, which gives the ramification correction to the first approximation that the Euler characteristics are in the inverse ratio to the degrees of the correspondence.

Euler employs a general type concept.

This is the case for the Euler characteristic, and a general method for defining and computing invariants is to define them for a given presentation and then show that they are independent of the choice of presentation.

For more general rotations, see Euler angles.

This operator has the general form ( summation convention: sum over repeated indices — in this case over the three Euler angles ):

The young Ampère, however, soon resumed his Latin lessons, which enable him to master the works of Leonhard Euler and Daniel Bernoulli.

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

Euler also discovered dozens of new pairs.

The Bernoulli numbers appear in the Taylor series expansions of the tangent and hyperbolic tangent functions, in formulas for the sum of powers of the first positive integers, in the Euler – Maclaurin formula, and in expressions for certain values of the Riemann zeta function.

Works of Pascal, Newton, Jacob Bernoulli and Euler became foundational in the emerging field.

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.

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.

Goldbach is most noted for his correspondence with Leibniz, Euler, and Bernoulli, especially in his 1742 letter to Euler stating his Goldbach's conjecture.

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

Euler proved in 1744 that the catenary is the curve which, when rotated about the x-axis, gives the surface of minimum surface area ( the catenoid ) for the given bounding circles.

If objects are seen as moving within a rotating frame, this movement results in another fictitious force, the Coriolis force ; and if the rate of rotation of the frame is changing, a third fictitious force, the Euler force is experienced.

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.

If this acceleration is multiplied by the particle mass, the leading term is the centripetal force and the negative of the second term related to angular acceleration is sometimes called the Euler force.

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.

Later, Leonhard Euler connected this system to the analytic functions.

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.

Euler diagram representing a definition of knowledge.

For Euler's formula in algebraic topology and polyhedral combinatorics see Euler characteristic.

Euler's formula, named after Leonhard Euler, is a mathematical formula in complex analysis that establishes the deep relationship between the trigonometric functions and the complex exponential function.

His correspondence with Euler ( who also knew the above equation ) shows that he didn't fully understand logarithms.

In following this general principle, Mason provides the observer with a natural eye progression from foreground to background, and the illusion of depth is instantly created.

Classics Professor G. P. Goold at University College, wrote of Housman's scholarly accomplishments: " The legacy of Housman's scholarship is a thing of permanent value ; and that value consists less in obvious results, the establishment of general propositions about Latin and the removal of scribal mistakes, than in the shining example he provides of a wonderful mind at work ....

The study of group homomorphisms then provides a tool for studying general properties of groups and consequences of the group axioms.

The introduction provides some general notes about Paul.

It provides a general chronology from Jesus ' baptism to the empty tomb.

In comparison, scheduled flights operate from around 560 airports in the U. S. According to the U. S. Aircraft Owners and Pilots Association, general aviation provides more than one percent of the United States ' GDP, accounting for 1. 3 million jobs in professional services and manufacturing.

In How to Solve It, Pólya provides general heuristics for solving problems of all kinds, not only mathematical ones.

Eight of the lines provide general metro services, whereas the Airport Express provides a direct link from the Hong Kong International Airport into the city centre, while the Disneyland Resort Line exclusively takes passengers to Hong Kong Disneyland.

A public library provides services to the general public and usually makes at least some of its books available for borrowing.

Leo Clubs are required to have a Leo Club Advisor, a member of the sponsoring Lions Club who attends meetings and provides general advice to the club.

Its main value, however, is seen in the opportunities it offers for the research in the physiology and behaviour of the moose, as well as in the insights it provides into the general principles of animal domestication.

Medical school provides a general medical education and grants students a Doctor of Medicine ( MD ), Doctor of Osteopathic Medicine ( DO ), or Bachelor of Medicine / Bachelor of Surgery ( MBBS, MBChB ) upon successful completion.

According to the current definition of open content on the OpenContent website, any general, royalty-free copyright license would qualify as an open license because it ' provides users with the right to make more kinds of uses than those normally permitted under the law-at no cost to the user.

As a general purpose language, Prolog also provides various built-in predicates to perform routine activities like input / output, using graphics and otherwise communicating with the operating system.

In the United States, for example, the general perjury statute under Federal law classifies perjury as a felony and provides for a prison sentence of up to five years.

Cox Field provides general aviation services.

In general, people will choose the object that provides the greatest reward at the lowest cost.

* Radiosity, by Hugo Elias ( also provides a general overview of lighting algorithms, along with programming examples )

The reporter should delay notifying the general community about the bug if the vendor provides feasible reasons for requiring so.

Lufttransport provides regular corporate charter services from Longyearbyen to Ny-Ålesund Airport and Svea Airport for Kings Bay and Store Norske ; these flights are in general not available to the public.

Experimental tests of general relativity have confirmed that non-Euclidean space provides a better model for the shape of space.

Also located at the SFU Library is the Electronic Document Centre, which provides internet access to digitized documents from a number of archival collections, such as Harrison Brown's Xi ' an Incident collection, and the history of British Columbia and Western Canada in general, including documents from the Doukhobor migration from the Russian Empire to Saskatchewan and then to British Columbia assembled for donation to the university by John Keenlyside

It groups English words into sets of synonyms called synsets, provides short, general definitions, and records the various semantic relations between these synonym sets.

The general theory provides the law of gravitation, and its relation to other forces of nature.

This theory provides an attempt of identifying general relativity and the standard model within the Lie group E8.