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Encyclopedia > Pushforward

In Mathematics is commonly defined as the study of patterns of structure, change, and space; more informally, one might say it is the study of figures and numbers. Mathematical knowledge is constantly growing, through research and application, but mathematics itself is not usually considered a natural science. One reason is that... mathematics, the push forward (or pushforward) of a In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders. A function is called C1 if it has a derivative that is a continuous function; such functions are also called continuously differentiable. A function is called Cn for n ≥ 1... smooth map F : MN between For other uses, see Manifold (disambiguation). 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... 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 In mathematics, a total derivative may mean either (i) a differential operator involving the sum of all the partial derivatives with respect to all variables in a problem, or be used compatibly (ii) to express the exterior derivative d, as applied to differential forms, and in particular as applied to... total derivative of ordinary calculus. Explicitly, it is a 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. Contents // 1 Definition and first consequences 2 Examples 3 Matrices... linear map from the In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. The elements of the tangent space are called tangent vectors at p. All the tangent spaces... tangent space of M at p to the tangent space of N at F(p).


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

Contents

Motivation

Let be a In mathematics, a smooth function is one that is infinitely differentiable, i.e., has derivatives of all finite orders. A function is called C1 if it has a derivative that is a continuous function; such functions are also called continuously differentiable. A function is called Cn for n ≥ 1... smooth map from an 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... open subset, U, of to an open subset, V, of . Let be the See Cartesian coordinate system or Coordinates (elementary mathematics) for a more elementary introduction to this topic. In mathematics as applied to geometry, physics or engineering, a coordinate system is a system for assigning a tuple of scalars to each point in an n-dimensional space. Scalars in many cases means... coordinates in U and those in V. For any , the In vector calculus, the Jacobian is shorthand for either the Jacobian matrix or its determinant, the Jacobian determinant. Also, in algebraic geometry the Jacobian of a curve means the Jacobian variety: a group structure, which can be imposed on the curve. They are all named after the mathematician Carl Gustav... Jacobian of F is the The word matrix (plural matrices) has several meanings. Generally speaking, a matrix is something that provides support or structure, especially in the sense of surrounding and/or shaping. It comes from the Latin word for womb, which itself derived from the Latin word for mother, which is mater. Various disciplines... matrix representation of the total derivative

.

We wish to generalize this to the case that F is a smooth function between any For other uses, see Manifold (disambiguation). 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... smooth manifolds M and N.


Definition

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

from the In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. The elements of the tangent space are called tangent vectors at p. All the tangent spaces... 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 In differential geometry, one can attach to every point p of a differentiable manifold a tangent space, a real vector space which intuitively contains the possible directions in which one can pass through p. The elements of the tangent space are called tangent vectors at p. All the tangent spaces... tangent space).


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

Here γ is a curve is 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 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). As a consequence, if A is unital, then D(1) = 0 since D1 = D(1ยท1) = D1 + D1... derivations acting on smooth real-valued functions the push forward is given by

Here X is a derivation on M and f is a smooth real-valued function on N. One can show that F * (X) is a indeed a derivation.


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

Properties

One can show that push forward of a 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. The functions f: X → Y and g: Y →... composition is the composition of push forwards (i.e., For functors in computer science, see the function object article. In category theory, a functor is a special type of mapping between categories. Functors can be thought of as morphisms in the category of all (small) categories. Functors were first considered in algebraic topology, where algebraic objects (like the fundamental... functorial behaviour), and the push forward of a local diffeomorphism is an In mathematics, an isomorphism (in Greek isos = equal and morphe = shape) is a kind of interesting mapping between objects. Douglas Hofstadter provides an informal definition: The word isomorphism applies when two complex structures can be mapped onto each other, in such a way that to each part of one structure... isomorphism of tangent spaces.


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 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). A typical example... vector bundle morphism from the In mathematics, the tangent bundle of a manifold is a vector bundle which as a set is the disjoint union of all the tangent spaces at every point in the manifold with natural topology and smooth structure. The tangent bundle of manifold M is usually denoted by T(M) or... tangent bundle of M to the tangent bundle of N:

Push forwards of vector fields

Although one can always push forward tangent vectors, the push forward of a 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 Euclidean space. Vector fields are often used in physics to model for example the speed and direction of a moving fluid... 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.


There is one special situation where one can push forward vector fields, namely if the map F is a In mathematics, a diffeomorphism is a kind of isomorphism of smooth manifolds. The mathematical definition is as follows. Given two differentiable manifolds M and N, a bijective map f from M to N is called a diffeomorphism if both and its inverse are smooth. Two manifolds M and N are... 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

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.


See also

  • This article discusses the pullback in differential geometry. For the pullback in category theory see pullback (category theory). In mathematics, the pullback of smooth map f : M → N between differentiable manifolds is a smooth vector bundle morphism f* : T*N → T*M, for which the following... pullback

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 3-540-4267-2 See section 1.6.
  • Ralph H. Abraham (born July 4, 1936) is an American mathematician. He has been a member of the mathematics department at the University of California, Santa Cruz since 1968. He was born in Burlington, Vermont, earned his Ph.D. from the University of Michigan in 1960, and held positions at... 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.

  Results from FactBites:
 
Diffeology - Pushforward, subduction, quotient (1682 words)
The quotient diffeology is built on the following pushforward construction.
This diffeology is called the pushforward of the diffeology of X by the map f, or simply the diffeology image.
Proof: It is a direct consequence of associativity of pushforward (see above) and that the composition of surjective maps is surjective.
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