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Universal Geometric Algebra
In mathematics, a universal geometric algebra is a type of geometric algebra generated by real number, real vector spaces endowed with an indefinite quadratic form. Some authors restrict this to the infinite-dimensional case. The universal geometric algebra \mathcal(n,n) of order is defined as the Clifford algebra of -dimension (vector space), dimensional pseudo-Euclidean space . This algebra is also called the "mother algebra". It has a nondegenerate signature. The vectors in this space generate the algebra through the geometric algebra, geometric product. This product makes the manipulation of vectors more similar to the familiar algebraic rules, although non-abelian algebra, commutative. When , i.e. there are countably many dimensions, then \mathcal(\infty,\infty) is called simply the ''universal geometric algebra'' (UGA), which contains vector spaces such as and their respective geometric algebras \mathcal(p,q). UGA contains all finite-dimensional geometric algebras (GA). Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematics
Mathematics is an area of knowledge that includes the topics of numbers, formulas and related structures, shapes and the spaces in which they are contained, and quantities and their changes. These topics are represented in modern mathematics with the major subdisciplines of number theory, algebra, geometry, and analysis, respectively. There is no general consensus among mathematicians about a common definition for their academic discipline. Most mathematical activity involves the discovery of properties of abstract objects and the use of pure reason to prove them. These objects consist of either abstractions from nature orin modern mathematicsentities that are stipulated to have certain properties, called axioms. A ''proof'' consists of a succession of applications of deductive rules to already established results. These results include previously proved theorems, axioms, andin case of abstraction from naturesome basic properties that are considered true starting points of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Up To
Two Mathematical object, mathematical objects ''a'' and ''b'' are called equal up to an equivalence relation ''R'' * if ''a'' and ''b'' are related by ''R'', that is, * if ''aRb'' holds, that is, * if the equivalence classes of ''a'' and ''b'' with respect to ''R'' are equal. This figure of speech is mostly used in connection with expressions derived from equality, such as uniqueness or count. For example, ''x'' is unique up to ''R'' means that all objects ''x'' under consideration are in the same equivalence class with respect to the relation ''R''. Moreover, the equivalence relation ''R'' is often designated rather implicitly by a generating condition or transformation. For example, the statement "an integer's prime factorization is unique up to ordering" is a concise way to say that any two lists of prime factors of a given integer are equivalent with respect to the relation ''R'' that relates two lists if one can be obtained by reordering (permutation) from the other. As anot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Calculus
In mathematics, geometric calculus extends the geometric algebra to include differentiation and integration. The formalism is powerful and can be shown to encompass other mathematical theories including differential geometry and differential forms. Differentiation With a geometric algebra given, let a and b be vectors and let F be a multivector-valued function of a vector. The directional derivative of F along b at a is defined as :(\nabla_b F)(a) = \lim_, provided that the limit exists for all b, where the limit is taken for scalar \epsilon. This is similar to the usual definition of a directional derivative but extends it to functions that are not necessarily scalar-valued. Next, choose a set of basis vectors \ and consider the operators, denoted \partial_i, that perform directional derivatives in the directions of e_i: :\partial_i : F \mapsto (x\mapsto (\nabla_ F)(x)). Then, using the Einstein summation notation, consider the operator: :e^i\partial_i, which means :F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle (or, in Commonwealth English: fibre bundle) is a space that is a product space, but may have a different topological structure. Specifically, the similarity between a space E and a product space B \times F is defined using a continuous surjective map, \pi : E \to B, that in small regions of E behaves just like a projection from corresponding regions of B \times F to B. The map \pi, called the projection or submersion of the bundle, is regarded as part of the structure of the bundle. The space E is known as the total space of the fiber bundle, B as the base space, and F the fiber. In the ''trivial'' case, E is just B \times F, and the map \pi is just the projection from the product space to the first factor. This is called a trivial bundle. Examples of non-trivial fiber bundles include the Möbius strip and Klein bottle, as well as nontrivial covering spaces. Fiber bundles, such as the tangent bundle of a mani ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Affine Connection
In differential geometry, an affine connection is a geometric object on a smooth manifold which ''connects'' nearby tangent spaces, so it permits tangent vector fields to be differentiated as if they were functions on the manifold with values in a fixed vector space. Connections are among the simplest methods of defining differentiation of the sections of vector bundles. The notion of an affine connection has its roots in 19th-century geometry and tensor calculus, but was not fully developed until the early 1920s, by Élie Cartan (as part of his general theory of connections) and Hermann Weyl (who used the notion as a part of his foundations for general relativity). The terminology is due to Cartan and has its origins in the identification of tangent spaces in Euclidean space by translation: the idea is that a choice of affine connection makes a manifold look infinitesimally like Euclidean space not just smoothly, but as an affine space. On any manifold of positive dimension ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Geometry
Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds. It uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra. The field has its origins in the study of spherical geometry as far back as antiquity. It also relates to astronomy, the geodesy of the Earth, and later the study of hyperbolic geometry by Lobachevsky. The simplest examples of smooth spaces are the plane and space curves and surfaces in the three-dimensional Euclidean space, and the study of these shapes formed the basis for development of modern differential geometry during the 18th and 19th centuries. Since the late 19th century, differential geometry has grown into a field concerned more generally with geometric structures on differentiable manifolds. A geometric structure is one which defines some notion of size, distance, shape, volume, or other rigidifying structu ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orientation (vector Space)
The orientation of a real vector space or simply orientation of a vector space is the arbitrary choice of which ordered bases are "positively" oriented and which are "negatively" oriented. In the three-dimensional Euclidean space, right-handed bases are typically declared to be positively oriented, but the choice is arbitrary, as they may also be assigned a negative orientation. A vector space with an orientation selected is called an oriented vector space, while one not having an orientation selected, is called . In mathematics, orientability is a broader notion that, in two dimensions, allows one to say when a cycle goes around clockwise or counterclockwise, and in three dimensions when a figure is left-handed or right-handed. In linear algebra over the real numbers, the notion of orientation makes sense in arbitrary finite dimension, and is a kind of asymmetry that makes a reflection impossible to replicate by means of a simple displacement. Thus, in three dimensions, it is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Real Coordinate Space
In mathematics, the real coordinate space of dimension , denoted ( ) or is the set of the -tuples of real numbers, that is the set of all sequences of real numbers. With component-wise addition and scalar multiplication, it is a real vector space, and its elements are called coordinate vectors. The coordinates over any basis of the elements of a real vector space form a ''real coordinate space'' of the same dimension as that of the vector space. Similarly, the Cartesian coordinates of the points of a Euclidean space of dimension form a ''real coordinate space'' of dimension . These one to one correspondences between vectors, points and coordinate vectors explain the names of ''coordinate space'' and ''coordinate vector''. It allows using geometric terms and methods for studying real coordinate spaces, and, conversely, to use methods of calculus in geometry. This approach of geometry was introduced by René Descartes in the 17th century. It is widely used, as it allows loc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudoscalar
In linear algebra, a pseudoscalar is a quantity that behaves like a scalar, except that it changes sign under a parity inversion while a true scalar does not. Any scalar product between a pseudovector and an ordinary vector is a pseudoscalar. The prototypical example of a pseudoscalar is the scalar triple product, which can be written as the scalar product between one of the vectors in the triple product and the cross product between the two other vectors, where the latter is a pseudovector. A pseudoscalar, when multiplied by an ordinary vector, becomes a pseudovector (axial vector); a similar construction creates the pseudotensor. Mathematically, a pseudoscalar is an element of the top exterior power of a vector space, or the top power of a Clifford algebra; see pseudoscalar (Clifford algebra). More generally, it is an element of the canonical bundle of a differentiable manifold. In physics In physics, a pseudoscalar denotes a physical quantity analogous to a scalar. Both ar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Manifold
In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point. More precisely, an n-dimensional manifold, or ''n-manifold'' for short, is a topological space with the property that each point has a neighborhood that is homeomorphic to an open subset of n-dimensional Euclidean space. One-dimensional manifolds include lines and circles, but not lemniscates. Two-dimensional manifolds are also called surfaces. Examples include the plane, the sphere, and the torus, and also the Klein bottle and real projective plane. The concept of a manifold is central to many parts of geometry and modern mathematical physics because it allows complicated structures to be described in terms of well-understood topological properties of simpler spaces. Manifolds naturally arise as solution sets of systems of equations and as graphs of functions. The concept has applications in computer-graphics given the need to associate pictures with coordinates (e.g ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
