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Pullback (differential Geometry)
Let \phi:M\to N be a smooth map between smooth manifolds M and N. Then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by \phi), and is frequently denoted by \phi^*. More generally, any covariant tensor field – in particular any differential form – on N may be pulled back to M using \phi. When the map \phi is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from N to M or vice versa. In particular, if \phi is a diffeomorphism between open subsets of \R^n and \R^n, viewed as a change of coordinates (perhaps between different charts on a manifold M), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches to the subject. The idea behind the pullba ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Differential Form
In mathematics, differential forms provide a unified approach to define integrands over curves, surfaces, solids, and higher-dimensional manifolds. The modern notion of differential forms was pioneered by Élie Cartan. It has many applications, especially in geometry, topology and physics. For instance, the expression f(x) \, dx is an example of a -form, and can be integrated over an interval ,b/math> contained in the domain of f: \int_a^b f(x)\,dx. Similarly, the expression f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz is a -form that can be integrated over a surface S: \int_S \left(f(x,y,z) \, dx \wedge dy + g(x,y,z) \, dz \wedge dx + h(x,y,z) \, dy \wedge dz\right). The symbol \wedge denotes the exterior product, sometimes called the ''wedge product'', of two differential forms. Likewise, a -form f(x,y,z) \, dx \wedge dy \wedge dz represents a volume element that can be integrated over a region of space. In general, a -form is an object ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Contravariant Functor
In mathematics, specifically category theory, a functor is a mapping between categories. Functors were first considered in algebraic topology, where algebraic objects (such as the fundamental group) are associated to topological spaces, and maps between these algebraic objects are associated to continuous maps between spaces. Nowadays, functors are used throughout modern mathematics to relate various categories. Thus, functors are important in all areas within mathematics to which category theory is applied. The words ''category'' and ''functor'' were borrowed by mathematicians from the philosophers Aristotle and Rudolf Carnap, respectively. The latter used ''functor'' in a linguistic context; see function word. Definition Let ''C'' and ''D'' be categories. A functor ''F'' from ''C'' to ''D'' is a mapping that * associates each object X in ''C'' to an object F(X) in ''D'', * associates each morphism f \colon X \to Y in ''C'' to a morphism F(f) \colon F(X) \to F ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that \varphi\colon M\to N is a smooth map between smooth manifolds; then the differential of \varphi at a point x, denoted \mathrm d\varphi_x, is, in some sense, the best linear approximation of \varphi near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, the differential is a linear map from the tangent space of M at x to the tangent space of N at \varphi(x), \mathrm d\varphi_x\colon T_xM \to T_N. Hence it can be used to ''push'' tangent vectors on M ''forward'' to tangent vectors on N. The differential of a map \varphi is also called, by various authors, the derivative or total derivative of \varphi. Motivation Let \varphi: U \to V be a Smooth function#Smooth functions on and between manifolds, smooth map from an Open subset#Euclidean space, open subset U of \R^m to an open subset V of \R^n. For an ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Manifolds
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 (topology), 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 function, 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 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Product
In mathematics, the tensor product V \otimes W of two vector spaces V and W (over the same field) is a vector space to which is associated a bilinear map V\times W \rightarrow V\otimes W that maps a pair (v,w),\ v\in V, w\in W to an element of V \otimes W denoted . An element of the form v \otimes w is called the tensor product of v and w. An element of V \otimes W is a tensor, and the tensor product of two vectors is sometimes called an ''elementary tensor'' or a ''decomposable tensor''. The elementary tensors span V \otimes W in the sense that every element of V \otimes W is a sum of elementary tensors. If bases are given for V and W, a basis of V \otimes W is formed by all tensor products of a basis element of V and a basis element of W. The tensor product of two vector spaces captures the properties of all bilinear maps in the sense that a bilinear map from V\times W into another vector space Z factors uniquely through a linear map V\otimes W\to Z (see the section below ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dual Space
In mathematics, any vector space ''V'' has a corresponding dual vector space (or just dual space for short) consisting of all linear forms on ''V,'' together with the vector space structure of pointwise addition and scalar multiplication by constants. The dual space as defined above is defined for all vector spaces, and to avoid ambiguity may also be called the . When defined for a topological vector space, there is a subspace of the dual space, corresponding to continuous linear functionals, called the continuous dual space. Dual vector spaces find application in many branches of mathematics that use vector spaces, such as in tensor analysis with finite-dimensional vector spaces. When applied to vector spaces of functions (which are typically infinite-dimensional), dual spaces are used to describe measures, distributions, and Hilbert spaces. Consequently, the dual space is an important concept in functional analysis. Early terms for ''dual'' include ''polarer Raum'' ahn 1 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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 tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics ( electromagnetic ten ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber (mathematics)
In mathematics, the fiber (American English, US English) or fibre (British English) of an element (mathematics), element y under a function (mathematics), function f is the preimage of the singleton (mathematics), singleton set \, that is :f^(y) = \. Properties and applications In elementary set theory If X and Y are the domain of a function, domain and image of a function, image of f, respectively, then the fibers of f are the sets in :\left\\quad=\quad \left\ which is a partition (mathematics), partition of the domain set X. Note that y must be restricted to the image set Y of f, since otherwise f^(y) would be the empty set which is not allowed in a partition. The fiber containing an element x\in X is the set f^(f(x)). For example, let f be the function from \R^2 to \R that sends point (a,b) to a+b. The fiber of 5 under f are all the points on the straight line with equation (mathematics), equation a+b=5. The fibers of f are that line and all the straight lines parallel ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pullback Bundle
In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space. Given a fiber bundle and a continuous map one can define a "pullback" of by as a bundle over . The fiber of over a point in is just the fiber of over . Thus is the disjoint union of all these fibers equipped with a suitable topology. Formal definition Let be a fiber bundle with abstract fiber and let be a continuous map. Define the pullback bundle by :f^E = \\subseteq B'\times E and equip it with the subspace topology and the projection map given by the projection onto the first factor, i.e., :\pi'(b',e) = b'.\, The projection onto the second factor gives a map :h \colon f^E \to E such that the following diagram commutes: :\begin f^E & \stackrel & E\\ ' \downarrow & & \downarrow \pi\\ B' & \stackrel f & B \end If is a local trivialization of then is a local trivialization of where :\psi(b',e) = (b', \mbox_2(\varphi(e))).\, It then fol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fiber Bundle
In mathematics, and particularly topology, a fiber bundle ( ''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 manifol ... [...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] |
Direct Image Sheaf
In mathematics, the direct image functor is a construction in sheaf theory that generalizes the global sections functor to the relative case. It is of fundamental importance in topology and algebraic geometry. Given a sheaf ''F'' defined on a topological space ''X'' and a continuous map ''f'': ''X'' → ''Y'', we can define a new sheaf ''f''∗''F'' on ''Y'', called the direct image sheaf or the pushforward sheaf of ''F'' along ''f'', such that the global sections of ''f''∗''F'' is given by the global sections of ''F''. This assignment gives rise to a functor ''f''∗ from the category of sheaves on ''X'' to the category of sheaves on ''Y'', which is known as the direct image functor. Similar constructions exist in many other algebraic and geometric contexts, including that of quasi-coherent sheaves and étale sheaves on a scheme. Definition Let ''f'': ''X'' → ''Y'' be a continuous map of topological spaces, and let Sh(–) denote the category of sheaves of abelian groups on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
