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Euler and computed
The values of the Riemann zeta function at even positive integers were computed by Euler.
It turns out that the Euler characteristic can also be computed on the level of homology:
for Here, is the step size — a small positive number, and is the computed approximate value of The midpoint method is also known as the modified Euler method.
where is the mean curvature, is the Gaussian curvature, and dA is the area form of S. For a closed surface, by the Gauss Bonnet theorem, the integral of the Gaussian curvature may be computed in terms of the Euler characteristic of the surface, so

Euler and sum
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.
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 Riemann zeta function or Euler Riemann zeta function, ζ ( s ), is a function of a complex variable s that analytically continues the sum of the infinite series which converges when the real part of s is greater than 1.
The Meissel Mertens constant is analogous to the Euler Mascheroni constant, but the harmonic series sum in its definition is only over the primes rather than over all integers and the logarithm is taken twice, not just once.
The Euler number E < sub > 2n </ sub > can be expressed as a sum over the even partitions of 2n,
Note that unlike the other two formulae ( the Euler product and the divisor sum ) this one does not require knowing the factors of n.
The geodesic curvature of geodesics being zero, and the Euler characteristic of T being 1, the theorem then states that the sum of the turning angles of the geodesic triangle is equal to 2π minus the total curvature within the triangle.
More generally, if the polyhedron has Euler characteristic ( where g is the genus, meaning " number of holes "), then the sum of the defect is
where the first sum ranges over the vertices in the interior of, the second sum is over the boundary vertices, and is the Euler characteristic of.
If we take E to be the sum of the even exterior powers of the cotangent bundle, and F to be the sum of the odd powers, define D = d + d *, considered as a map from E to F. Then the topological index of D is the Euler characteristic of the Hodge cohomology of M, and the analytical index is the Euler class of the manifold.
The Euler product attached to the Riemann zeta function, using also the sum of the geometric series, is
Here γ is the famous Euler Mascheroni constant, which has a similar definition involving a sum over all integers ( not just the primes ).
where ƒ is a smooth function, you could use the Euler Maclaurin formula to convert the series into an integral, plus some corrections involving derivatives of S ( x ), then for large values of a you could use " stationary phase " method to calculate the integral and give an approximate evaluation of the sum.
Euler found the exact sum to be and announced this discovery in 1735.
Some six months later he gave a new proof that the sum of the indices of the zeros of a vector field on a manifold is independent of the choice of vector field and equal to the Euler characteristic of the manifold.
Thus the Lefschetz number of the identity map is equal to the alternating sum of the Betti numbers of the space, which in turn is equal to the Euler characteristic χ ( X ).
Euler conjectured a fourth power cannot be written as the sum of 3 smaller fourth powers, but 200 years later this was disproven with:
To make sense of this expression, which generalises the Euler characteristic as alternating sum of Betti numbers, two conditions must be fulfilled.
Secondly, a function may be defined in terms of some presentation or decomposition of a mathematical object ; for instance, the Euler characteristic of a cell complex is defined as the alternating sum of the number of cells in each dimension.

Euler and 20
The existence of a free nutation of the Earth was predicted by Isaac Newton in Corollaries 20 to 22 of Proposition 66, Book 1 of the Philosophiæ Naturalis Principia Mathematica, and by Leonhard Euler in 1755 as part of his studies of the dynamics of rotating bodies.

Euler and decimal
He calculated the Euler Mascheroni constant, perhaps somewhat eccentrically, to 236 decimal places and evaluated the Bernoulli numbers up to the 62nd.

Euler and places
Amongst the fruits of his industry may be mentioned a laborious investigation of the disturbances of Jupiter by Saturn, the results of which were employed and confirmed by Euler in his prize essay of 1748 ; a series of lunar observations extending over fifty years ; some interesting researches in terrestrial magnetism and atmospheric electricity, in the latter of which he detected a regular diurnal period ; and the determination of the places of a great number of stars, including at least twelve separate observations of Uranus, between 1750 and its discovery as a planet.

Euler and with
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.
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.
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.
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.
His correspondence with Euler ( who also knew the above equation ) shows that he didn't fully understand logarithms.
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 ..
In 1726, Euler completed a dissertation on the propagation of sound with the title De Sono.
This was partly because of a conflict of personality with Frederick, who came to regard Euler as unsophisticated, especially in comparison to the circle of philosophers the German king brought to the 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.
The use of the Greek letter π to denote the ratio of a circle's circumference to its diameter was also popularized by Euler, although it did not originate with him.
The component with a period of about 435 days is identified with the 8 month wandering predicted by Euler and is now called the Chandler wobble after its discoverer.
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.
Sophie had derived the correct differential equation, but her method did not predict experimental results with great accuracy, as she had relied on an incorrect equation from Euler, which led to incorrect boundary conditions.
Segner worked with Euler on some of the early mathematical theories of turbine design.
Note that, with the opposite sign convention, this is the potential generated by a pointlike sink ( see point particle ), which is the solution of the Euler equations in two-dimensional incompressible flow.
Condorcet was recognized worldwide and worked with such famous scientists as Leonhard Euler and Benjamin Franklin.
* The proof that every Haefliger structure on a manifold can be integrated to a foliation ( this implies, in particular that every manifold with zero Euler characteristic admits a foliation of codimension one ).
* Half loop ( also known as a " Euler " or " Thoren "), a full-rotation jump with a loop entry but landed on the back inside edge of the opposite foot.

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