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Grassmannian
In mathematics, the Grassmannian \mathbf_k(V) (named in honour of Hermann Grassmann) is a differentiable manifold that parameterizes the set of all k-dimension (vector space), dimensional linear subspaces of an n-dimensional vector space V over a field (mathematics), field K that has a differentiable structure. For example, the Grassmannian \mathbf_1(V) is the space of lines through the origin in V, so it is the same as the projective space \mathbf(V) of one dimension lower than V. When V is a real number, real or complex number, complex vector space, Grassmannians are compact space, compact smooth manifolds, of dimension k(n-k). In general they have the structure of a nonsingular projective algebraic variety. The earliest work on a non-trivial Grassmannian is due to Julius Plücker, who studied the set of projective lines in real projective 3-space, which is equivalent to \mathbf_2(\mathbf^4), parameterizing them by what are now called Plücker coordinates. (See below.) Herma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Julius Plücker
Julius Plücker (16 June 1801 – 22 May 1868) was a German mathematician and physicist. He made fundamental contributions to the field of analytical geometry and was a pioneer in the investigations of cathode rays that led eventually to the discovery of the electron. He also vastly extended the study of Lamé curves. Biography Early years Plücker was born at Elberfeld (now part of Wuppertal). After being educated at Düsseldorf and at the universities of Bonn, Heidelberg and Berlin he went to Paris in 1823, where he came under the influence of the great school of French geometers, whose founder, Gaspard Monge, had only recently died. In 1825 he returned to Bonn, and in 1828 was made professor of mathematics. In the same year he published the first volume of his ''Analytisch-geometrische Entwicklungen'', which introduced the method of "abridged notation". In 1831 he published the second volume, in which he clearly established on a firm and independent basis projecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Space
In mathematics, the concept of a projective space originated from the visual effect of perspective, where parallel lines seem to meet ''at infinity''. A projective space may thus be viewed as the extension of a Euclidean space, or, more generally, an affine space with points at infinity, in such a way that there is one point at infinity of each direction of parallel lines. This definition of a projective space has the disadvantage of not being isotropic, having two different sorts of points, which must be considered separately in proofs. Therefore, other definitions are generally preferred. There are two classes of definitions. In synthetic geometry, ''point'' and ''line'' are primitive entities that are related by the incidence relation "a point is on a line" or "a line passes through a point", which is subject to the axioms of projective geometry. For some such set of axioms, the projective spaces that are defined have been shown to be equivalent to those resulting from the f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Algebraic Variety
In algebraic geometry, a projective variety is an algebraic variety that is a closed subvariety of a projective space. That is, it is the zero-locus in \mathbb^n of some finite family of homogeneous polynomials that generate a prime ideal, the defining ideal of the variety. A projective variety is a projective curve if its dimension is one; it is a projective surface if its dimension is two; it is a projective hypersurface if its dimension is one less than the dimension of the containing projective space; in this case it is the set of zeros of a single homogeneous polynomial. If ''X'' is a projective variety defined by a homogeneous prime ideal ''I'', then the quotient ring :k _0, \ldots, x_nI is called the homogeneous coordinate ring of ''X''. Basic invariants of ''X'' such as the degree and the dimension can be read off the Hilbert polynomial of this graded ring. Projective varieties arise in many ways. They are complete, which roughly can be expressed by saying that there ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gauss Map
In differential geometry, the Gauss map of a surface is a function that maps each point in the surface to its normal direction, a unit vector that is orthogonal to the surface at that point. Namely, given a surface ''X'' in Euclidean space R3, the Gauss map is a map ''N'': ''X'' → ''S''2 (where ''S''2 is the unit sphere) such that for each ''p'' in ''X'', the function value ''N''(''p'') is a unit vector orthogonal to ''X'' at ''p''. The Gauss map is named after Carl F. Gauss. 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. The Gauss map can always be defined locally (i.e. on a small piece of the surface). The Jacobian determinant of the Gauss map is equal to Gaussian curvature, and the differential of the Gauss map is called the shape operator. Gauss first wrote a draft on the topic in 1825 and published in 1827. There is also a Gauss map for a link, which computes linking ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projective Plane
In mathematics, a projective plane is a geometric structure that extends the concept of a plane (geometry), plane. In the ordinary Euclidean plane, two lines typically intersect at a single point, but there are some pairs of lines (namely, parallel lines) that do not intersect. A projective plane can be thought of as an ordinary plane equipped with additional "points at infinity" where parallel lines intersect. Thus ''any'' two distinct lines in a projective plane intersect at exactly one point. Renaissance artists, in developing the techniques of drawing in Perspective (graphical)#Renaissance, perspective, laid the groundwork for this mathematical topic. The archetypical example is the real projective plane, also known as the extended Euclidean plane. This example, in slightly different guises, is important in algebraic geometry, topology and projective geometry where it may be denoted variously by , RP2, or P2(R), among other notations. There are many other projective planes, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hermann Grassmann
Hermann Günther Grassmann (, ; 15 April 1809 – 26 September 1877) was a German polymath known in his day as a linguist and now also as a mathematician. He was also a physicist, general scholar, and publisher. His mathematical work was little noted until he was in his sixties. His work preceded and exceeded the concept which is now known as a vector space. He introduced the Grassmannian, the space which parameterizes all ''k''-dimensional linear subspaces of an ''n''-dimensional vector space ''V''. In linguistics he helped free language history and structure from each other. Biography Hermann Grassmann was the third of 12 children of Justus Günter Grassmann, an ordained minister who taught mathematics and physics at the Stettin Gymnasium, where Hermann was educated. Grassmann was an undistinguished student until he obtained a high mark on the examinations for admission to Prussian universities. Beginning in 1827, he studied theology at the University of Berlin, also ta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Space
In mathematics and physics, a vector space (also called a linear space) is a set (mathematics), set whose elements, often called vector (mathematics and physics), ''vectors'', can be added together and multiplied ("scaled") by numbers called scalar (mathematics), ''scalars''. The operations of vector addition and scalar multiplication must satisfy certain requirements, called ''vector axioms''. Real vector spaces and complex vector spaces are kinds of vector spaces based on different kinds of scalars: real numbers and complex numbers. Scalars can also be, more generally, elements of any field (mathematics), field. Vector spaces generalize Euclidean vectors, which allow modeling of Physical quantity, physical quantities (such as forces and velocity) that have not only a Magnitude (mathematics), magnitude, but also a Orientation (geometry), direction. The concept of vector spaces is fundamental for linear algebra, together with the concept of matrix (mathematics), matrices, which ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector Bundle
In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X. The simplest example is the case that the family of vector spaces is constant, i.e., there is a fixed vector space V such that V(x)=V for all x in X: in this case there is a copy of V for each x in X and these copies fit together to form the vector bundle X\times V over X. Such vector bundles are said to be ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of smooth (or differentiable) manifolds: to every point of such a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is a field of study that discovers and organizes methods, Mathematical theory, theories and theorems that are developed and Mathematical proof, proved for the needs of empirical sciences and mathematics itself. There are many areas of mathematics, which include number theory (the study of numbers), algebra (the study of formulas and related structures), geometry (the study of shapes and spaces that contain them), Mathematical analysis, analysis (the study of continuous changes), and set theory (presently used as a foundation for all mathematics). Mathematics involves the description and manipulation of mathematical object, abstract objects that consist of either abstraction (mathematics), abstractions from nature orin modern mathematicspurely abstract entities that are stipulated to have certain properties, called axioms. Mathematics uses pure reason to proof (mathematics), prove properties of objects, a ''proof'' consisting of a succession of applications of in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the tangent space at a point ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Function (mathematics)
In mathematics, a function from a set (mathematics), set to a set assigns to each element of exactly one element of .; the words ''map'', ''mapping'', ''transformation'', ''correspondence'', and ''operator'' are sometimes used synonymously. The set is called the Domain of a function, domain of the function and the set is called the codomain of the function. Functions were originally the idealization of how a varying quantity depends on another quantity. For example, the position of a planet is a ''function'' of time. History of the function concept, Historically, the concept was elaborated with the infinitesimal calculus at the end of the 17th century, and, until the 19th century, the functions that were considered were differentiable function, differentiable (that is, they had a high degree of regularity). The concept of a function was formalized at the end of the 19th century in terms of set theory, and this greatly increased the possible applications of the concept. A f ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theorem
In mathematics and formal logic, a theorem is a statement (logic), statement that has been Mathematical proof, proven, or can be proven. The ''proof'' of a theorem is a logical argument that uses the inference rules of a deductive system to establish that the theorem is a logical consequence of the axioms and previously proved theorems. In mainstream mathematics, the axioms and the inference rules are commonly left implicit, and, in this case, they are almost always those of Zermelo–Fraenkel set theory with the axiom of choice (ZFC), or of a less powerful theory, such as Peano arithmetic. Generally, an assertion that is explicitly called a theorem is a proved result that is not an immediate consequence of other known theorems. Moreover, many authors qualify as ''theorems'' only the most important results, and use the terms ''lemma'', ''proposition'' and ''corollary'' for less important theorems. In mathematical logic, the concepts of theorems and proofs have been formal system ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
