In mathematics, the push forward (or pushforward) of a smooth map F : M → N between smooth manifolds at a point p is, in some sense, the best linear approximation of F near p. It can be viewed as generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of M at p to the tangent space of N at F(p). For other meanings of mathematics or math, see mathematics (disambiguation). ... In mathematics, a smooth function is one that is infinitely differentiable, i. ... In mathematics, a manifold M is a type of space, characterized in one of two equivalent ways: near every point of the space, we have a coordinate system; or near every point, the environment is like that in Euclidean space of a given dimension. ... In mathematics, a total derivative may be either. ... In mathematics, a linear transformation (also called linear operator or linear map) is a function between two vector spaces that respects the arithmetical operations addition and scalar multiplication defined on vector spaces, or, in other words, it preserves linear combinations. Definition and first consequences Formally, if V and W are... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

The push forward of a map F is also called, by various authors, the derivative, total derivative, or differential of F.

The notion of push forward also exists in measure theory as the measure induced on Y by a measurable function , given a measure on X. In mathematics, a measure is a function that assigns a number, e. ...

Motivation

Let be a smooth map from an open subset, U, of to an open subset, V, of . Let be the coordinates in U and those in V. For any , the Jacobian of F is the matrix representation of the total derivative In mathematics, a smooth function is one that is infinitely (indefinitely) differentiable, i. ... In topology and related fields of mathematics, a set U is called open if, intuitively speaking, you can wiggle or change any point x in U by a small amount in any direction and still be inside U. In other words, if x is surrounded only by elements of U... In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of numbers to each point in an n-dimensional space. ... In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. ... In mathematics, a matrix (plural matrices) is a rectangular table of numbers or, more generally, a table consisting of abstract quantities that can be added and multiplied. ...

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We wish to generalize this to the case that F is a smooth function between any smooth manifolds M and N. On a sphere, the sum of the angles of a triangle is not equal to 180°. A sphere is not a Euclidean space, but locally the laws of the Euclidean geometry are good approximations. ...

Definition

Let be a smooth map of smooth manifolds. Given some , the push forward is a linear map

from the tangent space of M at p to the tangent space of N at F(p). The exact definition depends on the definition one uses for tangent vectors (for the various definitions see tangent space). The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ... The tangent space of a manifold is a concept which facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...

If one defines tangent vectors as equivalence classes of curves through p then the push forward is given by

Here γ is a curve in M with γ(0) = p. The push forward is just the tangent vector to the curve at 0.

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions the push forward is given by In abstract algebra, a derivation on an algebra A over a field k is a linear map D : A → A that satisfies Leibniz law: D(ab) = (Da)b + a(Db). ...

Here , therefore x is a derivative defined on M and f is a smooth real-valued function on N. By definition, the push-forward of X is in T_F(p)N and therefore itself is a derivative. In component notation the definition becomes,

therefore by identifying the following relation,

one can understand push-forward roughly as a coordinate transformation. Note that the push-forward is not guaranteed to have an inverse. Therefore in general we cannot define pullback for vectors. This article needs to be cleaned up to conform to a higher standard of quality. ...

The push forward is frequently expressed using a variety of other notations such as

Properties

One can show that push forward of a composition is the composition of push forwards (i.e., functorial behaviour), and the push forward of a local diffeomorphism is an isomorphism of tangent spaces. In mathematics, a composite function, formed by the composition of one function on another, represents the application of the former to the result of the application of the latter to the argument of the composite. ... Did somebody just say functor? In category theory, a functor is a special type of mapping between categories. ... In mathematics, a local diffeomorphism is a smooth map f : M → N between smooth manifolds such that for every point p of M there exists an open neighbourhood U of p such that f(U) is open in N and f|U : U → f(U) is a diffeomorphism. ... In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of mapping between objects, devised by Eilhard Mitscherlich, which shows a relation between two properties or operations. ...

Returning to the motivating example, it can be shown that the push forward of , in the given standard coordinates, is the matrix J whose entries are . This is the Jacobian of F. More generally, given a smooth map the push forward of F written in local coordinates will always be given by the Jacobian of F in those coordinates.

The push forward of F induces in an obvious manner a vector bundle morphism from the tangent bundle of M to the tangent bundle of N: In mathematics, a vector bundle is a geometrical construct where to every point of a topological space (or manifold, or algebraic variety) we attach a vector space in a compatible way, so that all those vector spaces, glued together, form another topological space (or manifold or variety). ... In mathematics, the tangent bundle of a differentiable manifold M, denoted by T(M) or just TM, is the disjoint union of the tangent spaces to each point of M An element of T(M) is a pair (x,v) where x ∈ M and v ∈ Tx(M), the tangent space...

Wikipedia does not have an article with this exact name. ...

Push forwards of vector fields

Although one can always push forward tangent vectors, the push forward of a vector field does not always make sense. For example, if the map F is not surjective how should one define the vector outside the range of F? Conversely, if F is not injective there may be more than one choice of the push forward of the field at a given point. Vector field given by vectors of the form (-y, x) In mathematics a vector field is a construction in vector calculus which associates a vector to every point in a Euclidean space. ...

There is one special situation where one can push forward vector fields, namely if the map F is a diffeomorphism. In this case, suppose X is a vector field on M, the push forward defines a vector field Y on N, given by Y = F _* X with In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. ...

Here, F ^{− 1}(p) maps the point p back from the manifold N to the manifold M. Then is the vector field at the point F ^{− 1}(p) on M.

Push forward in measure theory

Given two measurable spaces and , a function is called measurable if the pre-image of any measurable set in Y is measurable in X: In mathematics, a σ-algebra (or σ-field) X over a set S is a family of subsets of S which is closed under countable set operations; σ-algebras are mainly used in order to define measures on S. The concept is important in mathematical analysis and probability theory. ...

If the space is equipped with a measure , the push forward of μ by a measurable function is the measure defined by In mathematics, a measure is a function that assigns a number, e. ...

See also

• Pullback

This article needs to be cleaned up to conform to a higher standard of quality. ...

References

• John M. Lee, Introduction to Smooth Manifolds, (2003) Springer Graduate Texts in Mathematics 218.
• Jurgen Jost, Riemannian Geometry and Geometric Analysis, (2002) Springer-Verlag, Berlin ISBN 3-540-42627-2 See section 1.6.
• Ralph Abraham and Jarrold E. Marsden, Foundations of Mechanics, (1978) Benjamin-Cummings, London ISBN 0-8053-0102-X See section 1.7 and 2.3.