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Manifold
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a manifold is a topological space In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no gener ... that locally resembles Euclidean space Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimens ... near each point. More precisely, an -dimensional manifold, or ''-manifold'' for short, is a topological space with the property that each poin ...
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Differentiable Manifold
In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su ... that is locally similar enough to a vector space In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of such topics as quantity (number theory), mathematical structure, structure (algebra), space (geometry), and calculus, change (mathematical analysis, analysis). ... to allow one to do calculus Calculus, originally called infinitesimal calculus or "the calculus of infinitesimals", is the mathematics, mathematical study of continuous change, in the same way that geometry is the study of shape and algebra is the study of generalizations .... Any manifold can be described by a collection of charts, also ...
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Symplectic Manifold
In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ..., a subject of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., a symplectic manifold is a smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's s ..., M , equipped with a closed nondegenerate differential 2-form \omega , called the symplectic form In mathematics, a symplectic vector s ...
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Riemannian Metric
In differential geometry Differential geometry is a Mathematics, mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds, using the techniques of differential calculus, integral calculus, linear algebra a ..., a Riemannian manifold or Riemannian space is a real Real may refer to: * Reality Reality is the sum or aggregate of all that is real or existent within a system, as opposed to that which is only Object of the mind, imaginary. The term is also used to refer to the ontological status of things, ind ..., smooth manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a vector space to allow one to do calculus. Any manifold can be described by a collection of charts, also known as an ... ''M'' equipped with a positive-definite inner product In mathematics, an inner product space or a Hausdorff space, Hausdorff pre-Hilbe ...
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Lorentzian Manifold
In differential geometry Differential geometry is a mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast E ..., a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold In mathematics, a differentiable manifold (also differential manifold) is a type of manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surfa ... with a metric tensor In the mathematical Mathematics (from Greek Greek may refer to: Greece Anything of, from, or related to Greece Greece ( el, Ελλάδα, , ), officially the Hellenic Republic, is a country located in Southeast Europe. Its population ... that is everywhere nondegenerate. This is a generalization of a Riemannian manifold ...
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Differentiable Structure
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., an ''n''-dimensional File:Dimension levels.svg, thumb , 236px , The first four spatial dimensions, represented in a two-dimensional picture. In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum numb ... differential structure (or differentiable structure) on a set ''M'' makes ''M'' into an ''n''-dimensional differential manifold Differential may refer to: Mathematics * Differential (mathematics) In mathematics, differential refers to infinitesimal differences or to the derivatives of functions. The term is used in various branches of mathematics such as calculus, differe ..., which is a topological manifold In topology, a branch of mathematics, ...
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Surface (topology)
In the part of mathematics referred to as topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ..., a surface is a two-dimensional manifold The real projective plane is a two-dimensional manifold that cannot be realized in three dimensions without self-intersection, shown here as Boy's surface. In mathematics Mathematics (from Ancient Greek, Greek: ) includes the study of su .... Some surfaces arise as the boundaries of three-dimensional solids; for example, the sphere is the boundary of the solid ball. Other surfaces arise as graphs of functions Function or functionality may refer to: Computing * Function key A function key is a key on a computer A computer is a machine that can be programmed to carry out sequences of arithmetic or logical operations ...
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Hamiltonian Mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics Introduced by the Italian-French mathematician and astronomer Joseph-Louis Lagrange Joseph-Louis Lagrange (born Giuseppe Luigi Lagrangia
. Introduced by
Sir William Rowan Hamilton Sir William Rowan Hamilton MRIA (3 August 1805 – 2 September 1865) was an Irish mathematician, Andrews Professor of Astronomy at Trinity College Dublin , name_Latin = Collegium Sanctae et Individuae Trinitatis Reginae Elizabeth ...
, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide ...
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Euclidean Space
Euclidean space is the fundamental space of classical geometry. Originally, it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are Euclidean spaces of any nonnegative integer dimension (mathematics), dimension, including the three-dimensional space and the ''Euclidean plane'' (dimension two). It was introduced by the Greek mathematics, Ancient Greek mathematician Euclid, Euclid of Alexandria, and the qualifier ''Euclidean'' is used to distinguish it from other spaces that were later discovered in physics and modern mathematics. Ancient History of geometry#Greek geometry, Greek geometers introduced Euclidean space for modeling the physical universe. Their great innovation was to ''mathematical proof, prove'' all properties of the space as theorems by starting from a few fundamental properties, called ''postulates'', which either were considered as evident (for example, there is exactly one straight line passing through two points), or seemed ...
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Geometry
Geometry (from the grc, γεωμετρία; '' geo-'' "earth", '' -metron'' "measurement") is, with arithmetic Arithmetic (from the Ancient Greek, Greek wikt:en:ἀριθμός#Ancient Greek, ἀριθμός ''arithmos'', 'number' and wikt:en:τική#Ancient Greek, τική wikt:en:τέχνη#Ancient Greek, έχνη ''tiké échne', 'art' or 'cra ..., one of the oldest branches of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal .... It is concerned with properties of space that are related with distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a geometer A geometer is a mathematician A mathematician is someone who uses an extensive knowledge of mathematics Mathematics (from Ancie ...
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Distance
Distance is a numerical measurement Measurement is the quantification (science), quantification of variable and attribute (research), attributes of an object or event, which can be used to compare with other objects or events. The scope and application of measurement are dependen ... of how far apart objects or points are. In physics Physics is the natural science that studies matter, its Elementary particle, fundamental constituents, its Motion (physics), motion and behavior through Spacetime, space and time, and the related entities of energy and force. "Physical scie ... or everyday usage, distance may refer to a physical length Length is a measure of distance Distance is a numerical measurement ' Measurement is the number, numerical quantification (science), quantification of the variable and attribute (research), attributes of an object or event, which can be us ... or an estimation based on other criteria (e.g. "two counties over"). The distance from ...
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Plane (geometry)
In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers ( and ), formulas and related structures (), shapes and spaces in which they are contained (), and quantities and their changes ( and ). There is no general consensus abo ..., a plane is a flat, two-dimension In and , the dimension of a (or object) is informally defined as the minimum number of needed to specify any within it. Thus a has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point ...al surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating below the textile. A surface, as the term is most generally used, is the outermost or uppermost layer of a physical obje ... that extends infinitely far. A plane is the two-dimensional analogue of a point Point or points may refer to: Places * Point, LewisImage:Point Western Isles NASA World ...
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Klein Bottle
In topology In mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities ..., a branch of mathematics Mathematics (from Greek: ) includes the study of such topics as numbers (arithmetic and number theory), formulas and related structures (algebra), shapes and spaces in which they are contained (geometry), and quantities and their changes (cal ..., the Klein bottle () is an example of a non-orientable is non-orientable In mathematics, orientability is a property of Surface (topology), surfaces in Euclidean space that measures whether it is possible to make a consistent choice of surface normal vector (mathematics), vector at every point. A ... surface File:Water droplet lying on a damask.jpg, Water droplet lying on a damask. Surface tension is high enough to prevent floating belo ...
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