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

M. E. Baron has noted that Leibniz ( 1646 – 1716 ) in the 17th century produced similar diagrams before Euler, but much of it was unpublished.

Ulf S. von Euler was born in Stockholm, the son of two noted scientists, Dr. Hans von Euler-Chelpin, a professor of chemistry, and Dr. Astrid Cleve, a professor of botany and geology.

( It should be noted that Euler and Lagrange applied this method to nonlinear differential equations and that, instead of varying the coefficients of linear combinations of solutions to homogeneous equations, they varied the constants of the unperturbed motions of the celestial bodies.

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.

* 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 ).

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 means that as the number of contours increases, Euler diagrams are typically less visually complex than the equivalent Venn diagram, particularly if the number of non-empty intersections is small.

In number theory, Euler's theorem ( also known as the Fermat – Euler theorem or Euler's totient theorem ) states that if n and a are coprime positive integers, then

In number theory, an odd composite integer n is called an Euler – Jacobi pseudoprime to base a, if a and n are coprime, and

So for instance if you have a sphere with a " dent ", then its total curvature is 4π ( the Euler characteristic of a sphere being 2 ), no matter how big or deep the dent.

More generally, if the polyhedron has Euler characteristic ( where g is the genus, meaning " number of holes "), then the sum of the defect is

Bridge 8: Euler walks are possible if exactly zero or two nodes have an odd number of edges.

In arithmetic, an odd composite integer n is called an Euler pseudoprime to base a, if a and n are coprime, and

The fallacy in the argument can be illustrated through the use of an Euler Diagram: " A " satisfies the requirement that it is part of both sets " B " and " C ", but if one represents this as an Euler diagram, it can clearly be seen that it is possible that a part of set " B " is not part of set " C ", refuting the conclusion that " all Bs are Cs ".

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

However, it only admits a nowhere vanishing section if its Euler class is zero.

The Gauss map can be defined ( globally ) if and only if the surface is orientable, in which case its degree is half the Euler characteristic.

In fact, if we consider these as formal generating functions, the existence of such a formal Euler product expansion is a necessary and sufficient condition that be multiplicative: this says exactly that is the product of the whenever factors as the product of the powers of distinct primes.

For example if we use three angles ( Euler angles ), such parameterization is degenerate at some points on the hypersphere, leading to the problem of gimbal lock.

Euler angles between two frames are defined only if both frames have the same handedness.

That is because each simplex of S should be covered by exactly N in S ′ — at least if we use a fine enough triangulation of S, as we are entitled to do since the Euler characteristic is a topological invariant.

A quantitative form of Dirichlet's theorem states that if N ≥ 2 is an integer and a is coprime to N, then the proportion of the primes p congruent to a mod N is asymptotic to 1 / n, where n = φ ( N ) is the Euler totient function.

* If a graph G has an Euler cycle, that is, if G is connected and has an even number of edges at each vertex, then the line graph of G is Hamiltonian.

As such, if spoke tension is increased beyond a safe level, the wheel spontaneously fails into a characteristic saddle shape ( sometimes called a " taco " or a " pringle ") like a three-dimensional Euler column.

Additionally, if the acceleration indeed results from the forces in a conservative mechanical or Hamiltonian system, the energy of the approximation essentially oscillates around the constant energy of the exactly solved system, with a global error bound again of order one for semi-explicit Euler and order two for Verlet-leapfrog.

De Camp and Ley have claimed ( in their Lands Beyond ) that Leonhard Euler also proposed a hollow-Earth idea, getting rid of multiple shells and postulating an interior sun across to provide light to advanced inner-Earth civilization but they provide no references ; indeed, Euler did not propose a hollow-Earth, but there is a slightly related thought experiment.

) 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.

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 ).

Assuming that in the context cheese means some type of dairy product, the Euler diagram has the cheese zone entirely contained within the dairy-product zone — there is no zone for ( non-existent ) non-dairy cheese.

Along a different line of study, there is a combinatorially defined cohomology theory of knots called Khovanov homology whose Euler characteristic is the Jones polynomial.

It is not possible to produce a definite test of primality based on whether a number is an Euler pseudoprime because there exist absolute Euler pseudoprimes, numbers which are Euler pseudoprimes to every base relatively prime to themselves.

* The eigenvalues of Frobenius on H ( U, E ) can now be estimated as they are the zeros of the zeta function of the sheaf E. This zeta function can be written as an Euler product of zeta functions of the stalks of E, and using the estimate for the eigenvalues on these stalks shows that this product converges for | T |< q < sup >− d / 2 − 1 / 2 </ sup >, so that there are no zeros of the zeta function in this region.

In mathematics, there are two types of Euler integral:

This was proved by Leonhard Euler in 1737, and strengthens Euclid's 3rd-century-BC result that there are infinitely many prime numbers.

However, scholarship indicates that this claim of priority is not so clear ; Leonhard Euler discussed the principle in 1744, and there is evidence that Gottfried Leibniz preceded both by 39 years.

Euler angles provide a means for giving a numerical description of any rotation in three dimensional space using three numbers, but not only is this description not unique, but there are some points where not every change in the target space ( rotations ) can be realized by a change in the source space ( Euler angles ).

One can choose another convention for representing a rotation with a matrix using Euler angles than the Z-X-Z convention above, and also choose other variation intervals for the angles, but at the end there is always at least one value for which a degree of freedom is lost.

This is not the case for the Euler class, as detailed there, not least because the Euler class of a k-dimensional bundle lives in ( hence pulls back from, so it can ’ t pull back from a class in, as the dimensions differ.

When Euler Angles are defined as a sequence of rotations all the solutions can be valid, but there will be only one inside the angle ranges.

The fact that there are two logarithms ( log of a log ) in the limit for the Meissel – Mertens constant may be thought of as a consequence of the combination of the prime number theorem and the limit of the Euler – Mascheroni constant.

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.

According to this theorem, any such measure can be expressed as a linear combination of d + 1 fundamental measures ; for instance, in two dimensions, there are three possible measures of this type, one corresponding to area, a second corresponding to perimeter, and a third corresponding to the Euler characteristic.