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

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.

The non-orientable genus, demigenus, or Euler genus of a connected, non-orientable closed surface is a positive integer representing the number of cross-caps attached to a sphere.

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.

Note that for orientable compact surfaces without boundary, the Euler characteristic equals, where is the genus of the surface: Any orientable compact surface without boundary is topologically equivalent to a sphere with some handles attached, and counts the number of handles.

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

As another example, it shows immediately that a curve of genus 0 has no cover with N > 1 that is unramified everywhere: because that would give rise to an Euler characteristic > 2.

The exponential of its vacuum expectation value determines the coupling constant g, as for compact worldsheets by the Gauss-Bonnet theorem and the Euler characteristic, where g is the genus that counts the number of handles and thus the number of loops or string interactions described by a specific worldsheet.

Replacing the genus by the Euler characteristic, we obtain a formula that covers both the orientable and non-orientable cases,

In higher-dimensional manifolds genus is replaced by the notion of Euler characteristic, and more generally Betti numbers and homology and cohomology.

Thābit's formula was rediscovered by Fermat ( 1601 – 1665 ) and Descartes ( 1596 – 1650 ), to whom it is sometimes ascribed, and extended by Euler ( 1707 – 1783 ).

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.

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

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.

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.

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.

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.

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

Clenshaw – Curtis quadrature is essentially a change of variables to cast an arbitrary integral in terms of integrals of periodic functions where the Euler – Maclaurin approach is very accurate ( in that particular case the Euler – Maclaurin formula takes the form of a discrete cosine transform ).

In the context of computing asymptotic expansions of sums and series, usually the most useful form of the Euler – Maclaurin formula is

Euler's conjecture is a disproved conjecture in mathematics related to Fermat's last theorem which was proposed by Leonhard Euler in 1769.

The only exception to the formula is the Klein bottle, which has Euler characteristic 0 ( hence the formula gives p = 7 ) and requires 6 colors, as shown by P. Franklin in 1934 ( Weisstein ).

If the graph G is connected, then the rank of the free group is equal to 1 − χ ( G ): one minus the Euler characteristic of G.

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.

It was found and proved to be minimal by Leonhard Euler in 1744.

* Leonhard Euler discovers the catenoid and proves it to be a minimal surface.

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

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

Leonhard Euler considered the above series in 1740 for positive integer values of s, and later Chebyshev extended the definition to real s > 1.

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

An Euler probable prime to base a is an integer that is indicated prime by the somewhat stronger theorem that for any prime p, a < sup >( p − 1 )/ 2 </ sup > equals modulo p, where is the Legendre symbol.

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

For integer values of ν, these may be expressed in terms of the Euler polynomials.

For each primorial n, the fraction is smaller than for any lesser integer, where is the Euler totient function.

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.

Euler proved that for an odd prime number p and any integer a,

In mathematics, an Euler brick, named after Leonhard Euler, is a cuboid whose edges and face diagonals all have integer lengths.

A perfect cuboid ( also called a perfect box ) is an Euler brick whose space diagonal is also an integer.

Euler proved in the eighteenth century that if n is a positive integer then we have

Exploration Lab topics have included relationships between Pick's theorem and the Euler characteristic, linear fractional transformations, Linear Diophantine equations, integer partitions and compositions, finite differences, and Chebyshev polynomials.