## Riemann Surface Wikipedia

### Riemann surface - Wikipedia.

There are several equivalent definitions of a Riemann surface. A Riemann surface X is a connected complex manifold of complex dimension one. This means that X is a connected Hausdorff space that is endowed with an atlas of charts to the open unit disk of the complex plane: for every point x ? X there is a neighbourhood of x that is homeomorphic to the open unit disk of ....

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### Riemann sphere - Wikipedia.

In mathematics, the Riemann sphere, named after Bernhard Riemann, is a model of the extended complex plane: the complex plane plus one point at infinity.This extended plane represents the extended complex numbers, that is, the complex numbers plus a value for infinity.With the Riemann model, the point is near to very large numbers, just as the point is near to very small numbers..

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### Bernhard Riemann - Wikipedia.

Georg Friedrich Bernhard Riemann (German: ['ge:??k 'f?i:d?Ic 'be?nha?t '?i:man] (); 17 September 1826 - 20 July 1866) was a German mathematician who made contributions to analysis, number theory, and differential geometry.In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series..

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### Riemann–Liouville integral - Wikipedia.

In mathematics, the Riemann-Liouville integral associates with a real function: another function I ? f of the same kind for each value of the parameter ? > 0.The integral is a manner of generalization of the repeated antiderivative of f in the sense that for positive integer values of ?, I ? f is an iterated antiderivative of f of order ?.The Riemann-Liouville integral is named for ....

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### Riemann integral - Wikipedia.

In the branch of mathematics known as real analysis, the Riemann integral, created by Bernhard Riemann, was the first rigorous definition of the integral of a function on an interval.It was presented to the faculty at the University of Gottingen in 1854, but not published in a journal until 1868. For many functions and practical applications, the Riemann integral can be evaluated by ....

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### Complex logarithm - Wikipedia.

In mathematics, a complex logarithm is a generalization of the natural logarithm to nonzero complex numbers.The term refers to one of the following, which are strongly related: A complex logarithm of a nonzero complex number z, defined to be any complex number w for which e w = z. Such a number w is denoted by log z. If z is given in polar form as z = re i?, where r and ? are real ....

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### Curvature - Wikipedia.

In mathematics, curvature is any of several strongly related concepts in geometry.Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane.. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius.Smaller circles bend more sharply, and hence have higher ....

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### Complex analysis - Wikipedia.

Complex analysis, traditionally known as the theory of functions of a complex variable, is the branch of mathematical analysis that investigates functions of complex numbers.It is helpful in many branches of mathematics, including algebraic geometry, number theory, analytic combinatorics, applied mathematics; as well as in physics, including the branches of hydrodynamics, ....

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### Minimal surface - Wikipedia.

Mean curvature definition: A surface M ? R 3 is minimal if and only if its mean curvature is equal to zero at all points.. A direct implication of this definition is that every point on the surface is a saddle point with equal and opposite principal curvatures.Additionally, this makes minimal surfaces into the static solutions of mean curvature flow.By the Young-Laplace equation, the mean ....

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### Felix Klein - Wikipedia.

Felix Klein was born on 25 April 1849 in Dusseldorf, to Prussian parents. His father, Caspar Klein (1809-1889), was a Prussian government official's secretary stationed in the Rhine Province.His mother was Sophie Elise Klein (1819-1890, nee Kayser). He attended the Gymnasium in Dusseldorf, then studied mathematics and physics at the University of Bonn, 1865-1866, intending to ....

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### Christoffel symbols - Wikipedia.

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection. The metric connection is a specialization of the affine connection to surfaces or other manifolds endowed with a metric, allowing distances to be measured on that surface.In differential geometry, an affine connection can be defined without reference to a metric, and many ....

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### Bernhard Riemann — Wikipédia.

Georg Friedrich Bernhard Riemann, ne le 17 septembre 1826 a Breselenz, Royaume de Hanovre, mort le 20 juillet 1866 a Selasca, hameau de la commune de Verbania, Italie, est un mathematicien allemand.Influent sur le plan theorique, il a apporte de nombreuses contributions importantes a la topologie, l'analyse, la geometrie differentielle et le calcul, certaines d'entre elles ayant ....

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### Riemannian geometry - Wikipedia.

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume.From those, some other global quantities can be derived by ....

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### Multiple integral - Wikipedia.

In mathematics (specifically multivariable calculus), a multiple integral is a definite integral of a function of several real variables, for instance, f(x, y) or f(x, y, z).Integrals of a function of two variables over a region in (the real-number plane) are called double integrals, and integrals of a function of three variables over a region in (real-number 3D space) are called triple integrals..

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### Intégrale de Riemann — Wikipédia.

En analyse reelle, l'integrale de Riemann est une facon de definir l'integrale, sur un segment, d'une fonction reelle bornee et presque partout continue.En termes geometriques, cette integrale s'interprete comme l'aire du domaine sous la courbe representative de la fonction, comptee algebriquement. Le procede general utilise pour definir l'integrale de Riemann est l ....

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### Algebraic geometry and analytic geometry - Wikipedia.

Riemann surface theory shows that a compact Riemann surface has enough meromorphic functions on it, making it an algebraic curve. Under the name Riemann's existence theorem [3] [4] a deeper result on ramified coverings of a compact Riemann surface was known: such finite coverings as topological spaces are classified by permutation ....

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### Circle - Wikipedia.

A circle is a shape consisting of all points in a plane that are at a given distance from a given point, the centre.Equivalently, it is the curve traced out by a point that moves in a plane so that its distance from a given point is constant.The distance between any point of the circle and the centre is called the radius.Usually, the radius is required to be a positive number..

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### Lüneburg - Wikipedia.

Luneburg (officially the Hanseatic City of Luneburg, German: Hansestadt Luneburg, pronounced ['hanz??tat 'ly:n?b??k], Low German Lumborg, Latin Luneburgum or Lunaburgum, Old High German Luneburc, Old Saxon Hliuni, Polabian Glain), also called Lunenburg (/ ' lj u: n ? n b ?:r g / LEW-n?n-burg) in English, is a town in the German state of Lower Saxony..

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### Coordonnées polaires — Wikipédia.

Les coordonnees polaires [1] sont, en mathematiques, un systeme de coordonnees curvilignes [2] a deux dimensions, dans lequel chaque point du plan est entierement determine par un angle et une distance.Ce systeme est particulierement utile dans les situations ou la relation entre deux points est plus facile a exprimer en termes d'angle et de distance, comme dans le cas du pendule..

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### Jean Metzinger - Wikipedia.

Jean Dominique Antony Metzinger (French: [metse?e]; 24 June 1883 - 3 November 1956) was a major 20th-century French painter, theorist, writer, critic and poet, who along with Albert Gleizes wrote the first theoretical work on Cubism. His earliest works, from 1900 to 1904, were influenced by the neo-Impressionism of Georges Seurat and Henri-Edmond Cross..

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### Tangent - Wikipedia.

The intuitive notion that a tangent line "touches" a curve can be made more explicit by considering the sequence of straight lines (secant lines) passing through two points, A and B, those that lie on the function curve.The tangent at A is the limit when point B approximates or tends to A.The existence and uniqueness of the tangent line depends on a certain type of mathematical ....