* 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 ).

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

Euler was at this point studying theology, Greek, and Hebrew at his father's urging, in order to become a pastor, but Bernoulli convinced Paul Euler that Leonhard was destined to become a great mathematician.

In November 1726 Euler eagerly accepted the offer, but delayed making the trip to St Petersburg while he unsuccessfully applied for a physics professorship at the University of Basel.

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.

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.

The most valuable result of her labours was the Instituzioni analitiche ad uso della gioventù italiana, a work of great merit, which was published at Milan in 1748 and " was regarded as the best introduction extant to the works of Euler.

His work is notable for the use of the zeta function ζ ( s ) ( for real values of the argument " s ", as are works of Leonhard Euler, as early as 1737 ) predating Riemann's celebrated memoir of 1859, and he succeeded in proving a slightly weaker form of the asymptotic law, namely, that if the limit of π ( x )/( x / ln ( x )) as x goes to infinity exists at all, then it is necessarily equal to one.

This function as a function of a real argument was introduced and studied by Leonhard Euler in the first half of the eighteenth century without using complex analysis, which was not available at that time.

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

The values at negative integer points, also found by Euler, are rational numbers and play an important role in the theory of modular forms.

#: 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:

* Euler Function and Theorem at cut-the-knot

Alternatively, it is possible to show that any bridgeless bipartite planar graph with n vertices and m edges has by combining the Euler formula ( where f is the number of faces of a planar embedding ) with the observation that the number of faces is at most half the number of edges ( because each face has at least four edges and each edge belongs to exactly two faces ).

Solovay and Strassen showed that for every composite n, for at least n / 2 bases less than n, n is not an Euler – Jacobi pseudoprime.

If one bends and deforms the surface, its Euler characteristic, being a topological invariant, will not change, while the curvatures at some points will.

* The center of any nine-point circle ( the nine-point center ) lies on the corresponding triangle's Euler line, at the midpoint between that triangle's orthocenter and circumcenter.

* Nine-point conic and Euler line generalization at Dynamic Geometry Sketches Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.

Beginning with the equation of motion for a fluid ( say, the Euler equations or the Navier-Stokes equations without viscosity ) and taking the curl, one arrives at the equation of motion for the curl of the fluid velocity, that is to say, the vorticity.

* Nine-point conic and Euler line generalization at Dynamic Geometry Sketches Generalizes nine-point circle to a nine-point conic with an associated generalization of the Euler line.

* A further Euler line generalization at Dynamic Geometry Sketches Generalizes the Euler line further by disassociating it from the nine-point conic ( see above ).

* An interactive applet showing several triangle centers that lies on the Euler line.

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 ).

Euler worked in almost all areas of mathematics: geometry, infinitesimal calculus, trigonometry, algebra, and number theory, as well as continuum physics, lunar theory and other areas of physics.

The Gauss – Bonnet theorem or Gauss – Bonnet formula in differential geometry is an important statement about surfaces which connects their geometry ( in the sense of curvature ) to their topology ( in the sense of the Euler characteristic ).

In geometry, the Euler line, named after Leonhard Euler, is a line determined from any triangle that is not equilateral ; it passes through several important points determined from the triangle.

Issues that cause deviation from the pure Euler strut behaviour include imperfections in geometry in combination with plasticity / non-linear stress strain behaviour of the column's material.

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

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.

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.