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Pushforward (differential), the Glossary

In differential geometry, pushforward is a linear approximation of smooth maps (formulating manifold) on tangent spaces.^[1]

29 relations: Bundle map, Commutative diagram, Derivation (differential algebra), Diffeomorphism, Differentiable manifold, Differential geometry, Einstein notation, Fiber bundle, Flow-based generative model, Function composition, Functor, Jacobian matrix and determinant, Lie algebra, Lie group, Linear approximation, Linear map, Local diffeomorphism, Manifold, Matrix (mathematics), Product rule, Pullback (differential geometry), Pullback bundle, Section (fiber bundle), Smoothness, Tangent bundle, Tangent space, Total derivative, Vector bundle, Vector field.
Generalizations of the derivative
Smooth functions

Bundle map

In mathematics, a bundle map (or bundle morphism) is a morphism in the category of fiber bundles.

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The commutative diagram used in the proof of the five lemma In mathematics, and especially in category theory, a commutative diagram is a diagram such that all directed paths in the diagram with the same start and endpoints lead to the same result.

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Derivation (differential algebra)

In mathematics, a derivation is a function on an algebra that generalizes certain features of the derivative operator.

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Diffeomorphism

In mathematics, a diffeomorphism is an isomorphism of differentiable manifolds.

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Differentiable 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.

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Differential geometry

Differential geometry is a mathematical discipline that studies the geometry of smooth shapes and smooth spaces, otherwise known as smooth manifolds.

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Einstein notation

In mathematics, especially the usage of linear algebra in mathematical physics and differential geometry, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity.

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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.

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Flow-based generative model

A flow-based generative model is a generative model used in machine learning that explicitly models a probability distribution by leveraging normalizing flow, which is a statistical method using the change-of-variable law of probabilities to transform a simple distribution into a complex one.

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Function composition

In mathematics, function composition is an operation that takes two functions and, and produces a function such that.

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Functor

In mathematics, specifically category theory, a functor is a mapping between categories.

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Jacobian matrix and determinant

In vector calculus, the Jacobian matrix of a vector-valued function of several variables is the matrix of all its first-order partial derivatives. Pushforward (differential) and Jacobian matrix and determinant are generalizations of the derivative.

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Lie algebra

In mathematics, a Lie algebra (pronounced) is a vector space \mathfrak g together with an operation called the Lie bracket, an alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g, that satisfies the Jacobi identity.

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Lie group

In mathematics, a Lie group (pronounced) is a group that is also a differentiable manifold, such that group multiplication and taking inverses are both differentiable.

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Linear approximation

In mathematics, a linear approximation is an approximation of a general function using a linear function (more precisely, an affine function).

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Linear map

In mathematics, and more specifically in linear algebra, a linear map (also called a linear mapping, linear transformation, vector space homomorphism, or in some contexts linear function) is a mapping V \to W between two vector spaces that preserves the operations of vector addition and scalar multiplication.

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Local diffeomorphism

In mathematics, more specifically differential topology, a local diffeomorphism is intuitively a map between smooth manifolds that preserves the local differentiable structure.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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Matrix (mathematics)

In mathematics, a matrix (matrices) is a rectangular array or table of numbers, symbols, or expressions, with elements or entries arranged in rows and columns, which is used to represent a mathematical object or property of such an object.

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Product rule

In calculus, the product rule (or Leibniz rule or Leibniz product rule) is a formula used to find the derivatives of products of two or more functions.

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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^*. Pushforward (differential) and pullback (differential geometry) are differential geometry.

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Pullback bundle

In mathematics, a pullback bundle or induced bundle is the fiber bundle that is induced by a map of its base-space.

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Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.

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Smoothness

In mathematical analysis, the smoothness of a function is a property measured by the number, called differentiability class, of continuous derivatives it has over its domain. Pushforward (differential) and smoothness are smooth functions.

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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.

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Tangent space

In mathematics, the tangent space of a manifold is a generalization of to curves in two-dimensional space and to surfaces in three-dimensional space in higher dimensions. Pushforward (differential) and tangent space are differential geometry.

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Total derivative

In mathematics, the total derivative of a function at a point is the best linear approximation near this point of the function with respect to its arguments.

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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.

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Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a space, most commonly Euclidean space \mathbb^n.

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References

[1] https://en.wikipedia.org/wiki/Pushforward_(differential)

Also known as Differential of a smooth map, Tangent map.