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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] |
Cotangent Bundle
In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold. It may be described also as the dual bundle to the tangent bundle. This may be generalized to categories with more structure than smooth manifolds, such as complex manifolds, or (in the form of cotangent sheaf) algebraic varieties or schemes. In the smooth case, any Riemannian metric or symplectic form gives an isomorphism between the cotangent bundle and the tangent bundle, but they are not in general isomorphic in other categories. Formal definition via diagonal morphism There are several equivalent ways to define the cotangent bundle. One way is through a diagonal mapping Δ and germs. Let ''M'' be a smooth manifold and let ''M''×''M'' be the Cartesian product of ''M'' with itself. The diagonal mapping Δ sends a point ''p'' in ''M'' to the point (''p'',''p'') of ''M''×''M'' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Space
Euclidean space is the fundamental space of geometry, intended to represent physical space. Originally, in Euclid's ''Elements'', it was the three-dimensional space of Euclidean geometry, but in modern mathematics there are ''Euclidean spaces'' of any positive integer dimension ''n'', which are called Euclidean ''n''-spaces when one wants to specify their dimension. For ''n'' equal to one or two, they are commonly called respectively Euclidean lines and Euclidean planes. The qualifier "Euclidean" is used to distinguish Euclidean spaces from other spaces that were later considered in physics and modern mathematics. Ancient Greek geometers introduced Euclidean space for modeling the physical space. Their work was collected by the ancient Greek mathematician Euclid in his ''Elements'', with the great innovation of '' proving'' all properties of the space as theorems, by starting from a few fundamental properties, called '' postulates'', which either were considered as evid ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Parallelizable
In mathematics, a differentiable manifold M of dimension ''n'' is called parallelizable if there exist Smooth function, smooth vector fields \ on the manifold, such that at every point p of M the tangent vectors \ provide a Basis of a vector space, basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of frame bundle, linear frames has a global section on M. A particular choice of such a basis of vector fields on M is called a Parallelization (mathematics), parallelization (or an absolute parallelism) of M. Examples *An example with n = 1 is the circle: we can take ''V''1 to be the unit tangent vector field, say pointing in the anti-clockwise direction. The torus of dimension n is also parallelizable, as can be seen by expressing it as a cartesian product of circles. For example, take n = 2, and construct a torus from a square of graph paper with opposite edges glued together, to get an idea of the two tangen ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lie Group
In mathematics, a Lie group (pronounced ) is a group (mathematics), group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable. A manifold is a space that locally resembles Euclidean space, whereas groups define the abstract concept of a binary operation along with the additional properties it must have to be thought of as a "transformation" in the abstract sense, for instance multiplication and the taking of inverses (to allow division), or equivalently, the concept of addition and subtraction. Combining these two ideas, one obtains a continuous group where multiplying points and their inverses is continuous. If the multiplication and taking of inverses are smoothness, smooth (differentiable) as well, one obtains a Lie group. Lie groups provide a natural model for the concept of continuous symmetry, a celebrated example of which is the circle group. Rotating a circle is an example of a continuous symmetry. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are continuously differentiable. Definition Given two differentiable manifolds M and N, a Differentiable manifold#Differentiability of mappings between manifolds, continuously differentiable map f \colon M \rightarrow N is a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. Two C^r-differentiable manifolds are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contractible Space
In mathematics, a topological space ''X'' is contractible if the identity map on ''X'' is null-homotopic, i.e. if it is homotopic to some constant map. Intuitively, a contractible space is one that can be continuously shrunk to a point within that space. Properties A contractible space is precisely one with the homotopy type of a point. It follows that all the homotopy groups of a contractible space are trivial. Therefore any space with a nontrivial homotopy group cannot be contractible. Similarly, since singular homology is a homotopy invariant, the reduced homology groups of a contractible space are all trivial. For a nonempty topological space ''X'' the following are all equivalent: *''X'' is contractible (i.e. the identity map is null-homotopic). *''X'' is homotopy equivalent to a one-point space. *''X'' deformation retracts onto a point. (However, there exist contractible spaces which do not ''strongly'' deformation retract to a point.) *For any path-connected space ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Structure
In mathematics, a smooth structure on a manifold allows for an unambiguous notion of smooth function. In particular, a smooth structure allows mathematical analysis to be performed on the manifold. Definition A smooth structure on a manifold M is a collection of smoothly equivalent smooth atlases. Here, a smooth atlas for a topological manifold M is an atlas for M such that each transition function is a smooth map, and two smooth atlases for M are smoothly equivalent provided their union is again a smooth atlas for M. This gives a natural equivalence relation on the set of smooth atlases. A smooth manifold is a topological manifold M together with a smooth structure on M. Maximal smooth atlases By taking the union of all atlases belonging to a smooth structure, we obtain a maximal smooth atlas. This atlas contains every chart that is compatible with the smooth structure. There is a natural one-to-one correspondence between smooth structures and maximal smooth atlases. ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Disjoint Union Topology
In general topology and related areas of mathematics, the disjoint union (also called the direct sum, free union, free sum, topological sum, or coproduct) of a family of topological spaces is a space formed by equipping the disjoint union of the underlying sets with a natural topology called the disjoint union topology. Roughly speaking, in the disjoint union the given spaces are considered as part of a single new space where each looks as it would alone and they are isolated from each other. The name ''coproduct'' originates from the fact that the disjoint union is the categorical dual of the product space construction. Definition Let be a family of topological spaces indexed by ''I''. Let :X = \coprod_i X_i be the disjoint union of the underlying sets. For each ''i'' in ''I'', let :\varphi_i : X_i \to X\, be the canonical injection (defined by \varphi_i(x)=(x,i)). The disjoint union topology on ''X'' is defined as the finest topology on ''X'' for which all the canonical inj ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Derivative (generalizations)
In mathematics, the derivative is a fundamental construction of differential calculus and admits many possible generalizations within the fields of mathematical analysis, combinatorics, algebra, geometry, etc. Fréchet derivative The Fréchet derivative defines the derivative for general normed vector spaces V, W. Briefly, a function f : U \to W, where U is an open subset of V, is called ''Fréchet differentiable'' at x \in U if there exists a bounded linear operator A:V\to W such that \lim_ \frac = 0. Functions are defined as being differentiable in some open neighbourhood of x, rather than at individual points, as not doing so tends to lead to many pathological counterexamples. The Fréchet derivative is quite similar to the formula for the derivative found in elementary one-variable calculus, \lim_\frac = A, and simply moves ''A'' to the left hand side. However, the Fréchet derivative ''A'' denotes the function t \mapsto f'(x) \cdot t. In multivariable calculus, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Whitney Sum
In mathematics, a vector bundle is a topological construction that makes precise the idea of a Family of sets, family of vector spaces parameterized by another space (mathematics), 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 Fiber bundle#Trivial bundle, ''trivial''. A more complicated (and prototypical) class of examples are the tangent bundles of manifold, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Framed (mathematics)
Framed may refer to: Common meanings *A painting or photograph that has been placed within a picture frame *Someone falsely shown to be guilty of a crime as part of a frameup Film and television * ''Framed'' (1930 film), a pre-code crime action starring Evelyn Brent, Regis Toomey and Ralf Harolde * ''Framed'' (1940 film), an American crime film * ''Framed'' (1947 film), a film noir starring Glenn Ford and Janis Carter * ''Framed'' (1975 film), a crime drama based on a novel starring Joe Don Baker and Conny Van Dyke * ''Framed'' (1990 film), HBO made-for-television film starring Jeff Goldblum * ''Framed'' (TV series), a 1992 drama series of 4 x 1hr episodes, and cut to a 2hr TV film for the US (and elsewhere) in 1993 * "Framed" (''Spider-Man'': 1994 TV series), a 1996 episode of the animated series * "Framed", a 1998 episode of ''Dexter's Laboratory'' * ''Framed'' (2002 film), TNT made-for-television film remake of the 1992 TV series starring Rob Lowe * ''Framed'' (U.S. TV progr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
