pullback

Smooth sections of the cotangent bundle are differential one-forms. Cotangent bundle - Definition of the cotangent sheaf. 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. The image of Δ is called the diagonal. Let be the sheaf of germs of smooth functions on M×M which vanish on the diagonal. Then the quotient sheaf consists of equivalence classes of fu ...

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Cotangent bundle, Cotangent bundle - One-forms the cotangent sheaf, Cotangent bundle - Definition of the cotangent sheaf, Cotangent bundle - The cotangent bundle as phase space, Cotangent bundle - The canonical one-form, Cotangent bundle - Symplectic form, Cotangent bundle - Phase space

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The connection form for the vector bundle is the form on the total space of the associated principal bundle, but it can also be completely described by the following form (on the base in a not invariant way). This subsection can be considered as a smoother but somewhat inaccurate introduction to connection forms. A covariant derivative on a vector bundle is a way to "differentiate" bundle sections along tangent vectors; it is also sometimes called a connection. Let be a vector bundle over a smooth manifold See also:

Connection form, Connection form - Principal bundles, Connection form - Related definitions, Connection form - Vector bundles, Connection form - Related definitions

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Let M be a smooth manifold and let f ∈ C∞(M) be a smooth function. The differential of f at a point p is the map dfp(Xp) = Xp(f) where Xp is a tangent vector at p, thought of as a derivation. That is is the Lie derivative of f in the direction X, and one has df(X) = X(f). Equi ...

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Cotangent space, Cotangent space - Formal definitions, Cotangent space - Definition as linear functionals, Cotangent space - Alternate definition, Cotangent space - The differential of a function, Cotangent space - The pullback of a smooth map, Cotangent space - Exterior powers, Cotangent space - Reference

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Just as every differentiable map f : M → N between manifolds induces a linear map (called the pushforward or derivative) between the tangent spaces every such map induces a linear map (called the pullback) between the cotangent spaces, only this time in the reverse direction: The pullback is naturally defined as the dual (or transpose) of the pushforward. Unravelin ...

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Cotangent space, Cotangent space - Formal definitions, Cotangent space - Definition as linear functionals, Cotangent space - Alternate definition, Cotangent space - The differential of a function, Cotangent space - The pullback of a smooth map, Cotangent space - Exterior powers, Cotangent space - Reference

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The logarithmic derivative idea is closely connected to the integrating factor method, for first order differential equations. In operator terms, write D = d/dx and let M denote the operator of multiplication by some given function G(x). Then M−1DM can be written (by the product rule) as D + M* where M* now denotes the multiplication operator by the logarithmic derivative G′/G. In practice we are given an opera ...

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Logarithmic derivative, Logarithmic derivative - Formulae, Logarithmic derivative - Integrating factors, Logarithmic derivative - Complex analysis, Logarithmic derivative - The multiplicative group

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In category theory, there are covariant functors and contravariant functors. The dual space of a vector space is a standard example of a contravariant functor. Some constructions of multilinear algebra are of 'mixed' variance, which prevents them from being functors. The distinction between homology theory and cohomology theory in topology is that homology is a covariant functor, while cohomology is a contravariant functor (it was suggested in a book, Hilton & Wylie, that contrahomology was therefore a better term for cohomolog ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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In tensor analysis, a covariant vector varies more or less reciprocally to a corresponding contravariant vector. Expressions for lengths, areas and volumes of objects in the vector space can then be given in terms of tensors with covariant and contravariant indices. Under simple expansions and contractions of the coordinates, the reciprocity is exact; under affine transformations the components of a v ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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Representations of finite-dimensional subgroups of the group of symplectomorphisms (after -deformations, in general) on Hilbert spaces are called quantizations. When the Lie group is the one defined by a Hamiltonian, it is called a "quantization by energy". The corresponding operator from the Lie algebra to the Lie algebra of continuous linear operators is also sometimes called the quantization; this is a more common way of looking at it in physics. Locally, symplectomorphisms can be generated by a generating function over a (local) ...

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Symplectomorphism, Symplectomorphism - Formal definition, Symplectomorphism - Flows, Symplectomorphism - Comparison with Riemannian geometry, Symplectomorphism - Quantizations, Symplectomorphism - The group of Hamiltonian symplectomorphisms, Symplectomorphism - Arnold conjecture

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Let G be a group, and for a topological space X, write bG(X) for the set of isomorphism classes of principal G-bundles. This is a functor from Top to Set, sending a map f to the pullback operation f*. A characteristic class c of principal G-bundles is then a natural transformation from bG to a cohomology functor H*See also:

Characteristic class, Characteristic class - Definition, Characteristic class - Motivation

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In common physics usage, the adjective covariant may sometimes be used informally as a synonym for invariant (or equivariant, in mathematicians' terms). For example, the Schrödinger equation does not keep its written form under the coordinate transformations of special relativity; thus one might say that the Schrödinger equation is not covariant. By contrast, the Klein-Gordon equation and the Dirac equation take the same form in any coordinate frame of special relativity: thus, one might say that these equations are covari ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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If e1, e2, e3 are contravariant basis vectors of R3 (not necessarily orthogonal nor of unit norm) then the covariant basis vectors of their reciprocal system are: Note that even if the ei and ei are not orthonormal, they are still by this definition mutually orthonormal: Then the contravariant coordinates of any vector v< ...

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Covariance and contravariance, Covariance and contravariance - Informal usage, Covariance and contravariance - Example: covariant basis vectors in Euclidean R³, Covariance and contravariance - What 'contravariant' means, Covariance and contravariance - Usage in tensor analysis, Covariance and contravariance - Algebra and geometry

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The formula as given can be applied more widely; for example if f(z) is a meromorphic function, it makes sense at all complex values of z at which f has neither a zero nor a pole. Further, at a zero or a pole the logarithmic derivative behaves in a way that is easily analysed in terms of the particular case zn with n an integer, n≠0. The logarithmi ...

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Logarithmic derivative, Logarithmic derivative - Formulae, Logarithmic derivative - Integrating factors, Logarithmic derivative - Complex analysis, Logarithmic derivative - The multiplicative group

Read more here: » Logarithmic derivative: Encyclopedia II - Logarithmic derivative - Complex analysis