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In mathematics, a change of variables is a basic technique used to simplify problems in which the original variables are replaced with functions of other variables. The intent is that when expressed in new variables, the problem may become simpler, or equivalent to a better understood problem. Change of variables is an operation that is related ...
Pullback (differential geometry) Let be a smooth map between smooth manifolds and . Then there is an associated linear map from the space of 1-forms on (the linear space of sections of the cotangent bundle) to the space of 1-forms on . This linear map is known as the pullback (by ), and is frequently denoted by .
The pullback bundle is an example that bridges the notion of a pullback as precomposition, and the notion of a pullback as a Cartesian square. In that example, the base space of a fiber bundle is pulled back, in the sense of precomposition, above. The fibers then travel along with the points in the base space at which they are anchored: the ...
Fibred categories (or fibered categories) are abstract entities in mathematics used to provide a general framework for descent theory. They formalise the various situations in geometry and algebra in which inverse images (or pull-backs) of objects such as vector bundles can be defined. As an example, for each topological space there is the ...
Base change theorems. In mathematics, the base change theorems relate the direct image and the inverse image of sheaves. More precisely, they are about the base change map, given by the following natural transformation of sheaves: where. is a Cartesian square of topological spaces and is a sheaf on X . Such theorems exist in different branches ...
Multilinear form. In abstract algebra and multilinear algebra, a multilinear form on a vector space over a field is a map. that is separately - linear in each of its arguments. [1] More generally, one can define multilinear forms on a module over a commutative ring. The rest of this article, however, will only consider multilinear forms on ...
A moving frame on a submanifold M of G/H is a section of the pullback of the tautological bundle to M. Intrinsically a moving frame can be defined on a principal bundle P over a manifold. In this case, a moving frame is given by a G-equivariant mapping φ : P → G, thus framing the manifold by elements of the Lie group G.
In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces. Suppose that is a smooth map between smooth manifolds; then the differential of at a point , denoted , is, in some sense, the best linear approximation of near . It can be viewed as a generalization of the total derivative of ...