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Subbundle
In mathematics, a subbundle L of a vector bundle E over a topological space M is a collection of linear subspaces L_xof the fibers E_x of E at x in M, that make up a vector bundle in their own right. In connection with foliation theory, a subbundle of the tangent bundle of a smooth manifold may be called a distribution (of tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ...s). If locally, in a neighborhood N_x of x \in M , a set of vector fields Y_k span the vector spaces L_y, y \in N_x, and all Lie commutators \left _i, Y_j\right/math> are linear combinations of Y_1, \dots, Y_n then one says that L is an involutive distribution. See also * * {{Manifolds Fiber bundles ... [...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] |
Distribution (differential Geometry)
In differential geometry, a discipline within mathematics, a distribution on a differentiable manifold, manifold M is an assignment x \mapsto \Delta_x \subseteq T_x M of vector subspaces satisfying certain properties. In the most common situations, a distribution is asked to be a vector subbundle of the tangent bundle TM. Distributions satisfying a further integrability condition give rise to foliations, i.e. partitions of the manifold into smaller submanifolds. These notions have several applications in many fields of mathematics, including integrable systems, Poisson geometry, Noncommutative geometry, non-commutative geometry, sub-Riemannian geometry, differential topology. Even though they share the same name, distributions presented in this article have nothing to do with distribution (mathematics), distributions in the sense of analysis. Definition Let M be a smooth manifold; a (smooth) distribution \Delta assigns to any point x \in M a vector subspace \Delta_x \subset T_xM ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Subbundle
In mathematics, a subbundle L of a vector bundle E over a topological space M is a collection of linear subspaces L_xof the fibers E_x of E at x in M, that make up a vector bundle in their own right. In connection with foliation theory, a subbundle of the tangent bundle of a smooth manifold may be called a distribution (of tangent vector In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are ...s). If locally, in a neighborhood N_x of x \in M , a set of vector fields Y_k span the vector spaces L_y, y \in N_x, and all Lie commutators \left _i, Y_j\right/math> are linear combinations of Y_1, \dots, Y_n then one says that L is an involutive distribution. See also * * {{Manifolds Fiber bundles ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the manifold decomposition, decomposition of the real coordinate space R''n'' into the cosets ''x'' + R''p'' of the standardly embedding, embedded subspace topology, subspace R''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear manifold, piecewise-linear, differentiable manifold, differentiable (of class ''Cr''), or analytic manifold, analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''Cr'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''0 is a topological foliation). The number ''p'' (the dime ... [...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] |
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] |
Linear Subspace
In mathematics, the term ''linear'' is used in two distinct senses for two different properties: * linearity of a ''function (mathematics), function'' (or ''mapping (mathematics), mapping''); * linearity of a ''polynomial''. An example of a linear function is the function defined by f(x)=(ax,bx) that maps the real line to a line in the Euclidean plane R2 that passes through the origin. An example of a linear polynomial in the variables X, Y and Z is aX+bY+cZ+d. Linearity of a mapping is closely related to ''Proportionality (mathematics), proportionality''. Examples in physics include the linear relationship of voltage and Electric current, current in an electrical conductor (Ohm's law), and the relationship of mass and weight. By contrast, more complicated relationships, such as between velocity and kinetic energy, are ''Nonlinear system, nonlinear''. Generalized for functions in more than one dimension (mathematics), dimension, linearity means the property of a function of b ... [...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] |
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 apply calculus. Any manifold can be described by a collection of charts (atlas). One may then apply ideas from calculus while working within the individual charts, since each chart lies within a vector space to which the usual rules of calculus apply. If the charts are suitably compatible (namely, the transition from one chart to another is differentiable), then computations done in one chart are valid in any other differentiable chart. In formal terms, a differentiable manifold is a topological manifold with a globally defined differential structure. Any topological manifold can be given a differential structure locally by using the homeomorphisms in its atlas and the standard differential structure on a vector space. To induce a global differential structure on the local coordinate systems induced by the homeomorphisms, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Vector
In mathematics, a tangent vector is a vector that is tangent to a curve or surface at a given point. Tangent vectors are described in the differential geometry of curves in the context of curves in R''n''. More generally, tangent vectors are elements of a '' tangent space'' of a differentiable manifold. Tangent vectors can also be described in terms of germs. Formally, a tangent vector at the point x is a linear derivation of the algebra defined by the set of germs at x. Motivation Before proceeding to a general definition of the tangent vector, we discuss its use in calculus and its tensor In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other ... properties. Calculus Let \mathbf(t) be a parametric smooth curve. The tangent vector is given by \mathbf'(t) provided it exists and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Linear Span
In mathematics, the linear span (also called the linear hull or just span) of a set S of elements of a vector space V is the smallest linear subspace of V that contains S. It is the set of all finite linear combinations of the elements of , and the intersection of all linear subspaces that contain S. It is often denoted pp. 29-30, §§ 2.5, 2.8 or \langle S \rangle. For example, in geometry, two linearly independent vectors span a plane. To express that a vector space is a linear span of a subset , one commonly uses one of the following phrases: spans ; is a spanning set of ; is spanned or generated by ; is a generator set or a generating set of . Spans can be generalized to many mathematical structures, in which case, the smallest substructure containing S is generally called the substructure ''generated'' by S. Definition Given a vector space over a field , the span of a set of vectors (not necessarily finite) is defined to be the intersection of all subsp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Commutator
In differential geometry, the Lie derivative ( ), named after Sophus Lie by Władysław Ślebodziński, evaluates the change of a tensor field (including scalar functions, vector fields and one-forms), along the flow defined by another vector field. This change is coordinate invariant and therefore the Lie derivative is defined on any differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If ''T'' is a tensor field and ''X'' is a vector field, then the Lie derivative of ''T'' with respect to ''X'' is denoted \mathcal_X T. The differential operator T \mapsto \mathcal_X T is a derivation of the algebra of tensor fields of the underlying manifold. The Lie derivative commutes with contraction and the exterior derivative on differential forms. Although there are many concepts of taking a derivative in differential geometry, they all agree when the expression being differentiated is a function or scalar field. Thus in this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
