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Tensor Fields
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a ''scalar field'' and a ''vector field'' that assigns, respectively, a scalar or vector to each point of space. If a tensor is defined on a vector fields set over a module , we call a tensor field on . A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the ''Riemann curvature tensor'' refers a tensor ''field'', as it associates a tensor to each p ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product Of Fields
In mathematics, the tensor product of two field (mathematics), fields is their tensor product of algebras, tensor product as algebra over a field, algebras over a common subfield (mathematics), subfield. If no subfield is explicitly specified, the two fields must have the same characteristic (algebra), characteristic and the common subfield is their prime field, prime subfield. The tensor product of two fields is sometimes a field, and often a product ring, direct product of fields; In some cases, it can contain non-zero nilpotent elements. The tensor product of two fields expresses in a single structure the different way to embed the two fields in a common extension field. Compositum of fields First, one defines the notion of the compositum of fields. This construction occurs frequently in field theory (mathematics), field theory. The idea behind the compositum is to make the smallest field containing two other fields. In order to formally define the compositum, one must firs ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vector (physics)
In mathematics and physics, vector is a term that refers to quantities that cannot be expressed by a single number (a scalar), or to elements of some vector spaces. Historically, vectors were introduced in geometry and physics (typically in mechanics) for quantities that have both a magnitude and a direction, such as displacements, forces and velocity. Such quantities are represented by geometric vectors in the same way as distances, masses and time are represented by real numbers. The term ''vector'' is also used, in some contexts, for tuples, which are finite sequences (of numbers or other objects) of a fixed length. Both geometric vectors and tuples can be added and scaled, and these vector operations led to the concept of a vector space, which is a set equipped with a vector addition and a scalar multiplication that satisfy some axioms generalizing the main properties of operations on the above sorts of vectors. A vector space formed by geometric vectors is called ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate System
In geometry, a coordinate system is a system that uses one or more numbers, or coordinates, to uniquely determine and standardize the position of the points or other geometric elements on a manifold such as Euclidean space. The coordinates are not interchangeable; they are commonly distinguished by their position in an ordered tuple, or by a label, such as in "the ''x''-coordinate". The coordinates are taken to be real numbers in elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows problems in geometry to be translated into problems about numbers and ''vice versa''; this is the basis of analytic geometry. Common coordinate systems Number line The simplest example of a coordinate system is the identification of points on a line with real numbers using the '' number line''. In this system, an arbitrary point ''O'' (the ''origin'') is chosen on a given line. The coordinate o ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tissot's Indicatrix
In cartography, a Tissot's indicatrix (Tissot indicatrix, Tissot's ellipse, Tissot ellipse, ellipse of distortion) (plural: "Tissot's indicatrices") is a mathematical contrivance presented by French mathematician Nicolas Auguste Tissot in 1859 and 1871 in order to characterize local distortions due to map projection. It is the geometry that results from Projection (linear algebra), projecting a circle of infinitesimal radius from a curved geometric model, such as a globe, onto a map. Tissot proved that the resulting diagram is an ellipse whose axes indicate the two Principal curvature, principal directions along which scale is maximal and minimal at that point on the map. A single indicatrix describes the distortion at a single point. Because distortion varies across a map, generally Tissot's indicatrices are placed across a map to illustrate the spatial change in distortion. A common scheme places them at each intersection of displayed meridians and parallels. These schematics ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curved Space
Curved space often refers to a spatial geometry which is not "flat", where a '' flat space'' has zero curvature, as described by Euclidean geometry. Curved spaces can generally be described by Riemannian geometry, though some simple cases can be described in other ways. Curved spaces play an essential role in general relativity, where gravity is often visualized as '' curved spacetime''. The Friedmann–Lemaître–Robertson–Walker metric is a curved metric which forms the current foundation for the description of the expansion of the universe and the shape of the universe. The fact that photons have no mass yet are distorted by gravity, means that the explanation would have to be something besides photonic mass. Hence, the belief that large bodies curve space and so light, traveling on the curved space will, appear as being subject to gravity. It is not, but it is subject to the curvature of space. Simple two-dimensional example A very familiar example of a curved space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Bundle
In mathematics, the dual bundle is an operation on vector bundles extending the operation of duality for vector spaces. Definition The dual bundle of a vector bundle \pi: E \to X is the vector bundle \pi^*: E^* \to X whose fibers are the dual spaces to the fibers of E. Equivalently, E^* can be defined as the Hom bundle ''\mathrm(E,\mathbb \times X),'' that is, the vector bundle of morphisms from ''E'' to the trivial line bundle ''\R \times X \to X.'' Constructions and examples Given a local trivialization of ''E'' with transition functions t_, a local trivialization of E^* is given by the same open cover of ''X'' with transition functions t_^* = (t_^T)^ (the inverse of the transpose). The dual bundle E^* is then constructed using the fiber bundle construction theorem. As particular cases: * The dual bundle of an associated bundle is the bundle associated to the dual representation of the structure group. * The dual bundle of the tangent bundle of a differentiable manifold i ... [...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] |
Tensor Product Bundle
In differential geometry, the tensor product of vector bundles , (over the same space ) is a vector bundle, denoted by , whose fiber over each point is the tensor product of vector spaces .To construct a tensor-product bundle over a paracompact base, first note the construction is clear for trivial bundles. For the general case, if the base is compact, choose such that is trivial. Choose in the same way. Then let be the subbundle of with the desired fibers. Finally, use the approximation argument to handle a non-compact base. See Hatcher for a general direct approach. Example: If is a trivial line bundle, then for any . Example: is canonically isomorphic to the endomorphism bundle , where is the dual bundle of . Example: A line bundle has a tensor inverse: in fact, is (isomorphic to) a trivial bundle by the previous example, as is trivial. Thus, the set of the isomorphism classes of all line bundles on some topological space forms an abelian group called the Pica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Bundle
A tangent bundle is the collection of all of the tangent spaces for all points on a manifold, structured in a way that it forms a new manifold itself. Formally, in differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M . As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points and of manifold the tangent spaces and have no common vector. This is graphically illustrated in the accompanying picture for tangent bundle of circle , see Examples section: all tangents to a circle lie in the plane of the circle. In order to make them disjoint it is necessary to align them in a plane perpendicular to the plane of the circle. of the tangent spaces of M . That is, : \begin TM &= \bigsqcup_ T_xM \\ &= \bigcup_ \left\ \times T_xM \\ &= \bigcup_ \left\ \\ &= \left\ \end where T_x M denotes the tangent space to M at the point x . So, an el ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a ''scalar field'' and a ''vector field'' that assigns, respectively, a scalar or vector to each point of space. If a tensor is defined on a vector fields set over a module , we call a tensor field on . A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the ''Riemann curvature tensor'' refers a tensor ''field'', as it associates a tensor to each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Topological Space
In mathematics, a topological space is, roughly speaking, a Geometry, geometrical space in which Closeness (mathematics), closeness is defined but cannot necessarily be measured by a numeric Distance (mathematics), distance. More specifically, a topological space is a Set (mathematics), set whose elements are called Point (geometry), points, along with an additional structure called a topology, which can be defined as a set of Neighbourhood (mathematics), neighbourhoods for each point that satisfy some Axiom#Non-logical axioms, axioms formalizing the concept of closeness. There are several equivalent definitions of a topology, the most commonly used of which is the definition through open sets, which is easier than the others to manipulate. A topological space is the most general type of a space (mathematics), mathematical space that allows for the definition of Limit (mathematics), limits, Continuous function (topology), continuity, and Connected space, connectedness. Common types ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Manifold
In differential geometry, a Riemannian manifold is a geometric space on which many geometric notions such as distance, angles, length, volume, and curvature are defined. Euclidean space, the N-sphere, n-sphere, hyperbolic space, and smooth surfaces in three-dimensional space, such as ellipsoids and paraboloids, are all examples of Riemannian manifold, manifolds. Riemannian manifolds are named after German mathematician Bernhard Riemann, who first conceptualized them. Formally, a Riemannian metric (or just a metric) on a smooth manifold is a choice of inner product for each tangent space of the manifold. A Riemannian manifold is a smooth manifold together with a Riemannian metric. The techniques of differential and integral calculus are used to pull geometric data out of the Riemannian metric. For example, integration leads to the Riemannian distance function, whereas differentiation is used to define curvature and parallel transport. Any smooth surface in three-dimensional Eucl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
