Pushforward (differential)
In differential geometry, pushforward is a linear approximation of smooth maps 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 ordinary calculus. Explicitly, the differential is a linear map from the tangent space of  at  to the tangent space of  at , . Hence it can be used to push tangent vectors on  forward to tangent vectors on . The differential of a map  is also called, by various authors, the derivative or total derivative of . Motivation 
Let  be a smooth map from an open subset  of  to an open subset  of . For any point  in , the Jacobian of  at  (with respect to the standard coordinates) is the matrix representation of the total derivative of  at , which is a linear map

between their tangent spaces. Note the tangent spaces  are isomorphic to  and , respectively. The pushforward generalizes this construction to the case that  is a smooth function between any smooth manifolds  and . The differential of a smooth map 
Let  be a smooth map of smooth manifolds. Given  the differential of  at  is a linear map

from the tangent space of  at  to the tangent space of  at  The image  of a tangent vector  under  is sometimes called the pushforward of  by  The exact definition of this pushforward depends on the definition one uses for tangent vectors (for the various definitions see tangent space). If tangent vectors are defined as equivalence classes of the curves  for which  then the differential is given by

Here,  is a curve in  with  and  is tangent vector to the curve  at  In other words, the pushforward of the tangent vector to the curve  at  is the tangent vector to the curve  at 

Alternatively, if tangent vectors are defined as derivations acting on smooth real-valued functions, then the differential is given by

for an arbitrary function  and an arbitrary derivation  at point  (a derivation is defined as a linear map  that satisfies the Leibniz rule, see: definition of tangent space via derivations). By definition, the pushforward of  is in  and therefore itself is a derivation, . After choosing two charts around  and around   is locally determined by a smooth map  between open sets of  and , and

in the Einstein summation notation, where the partial derivatives are evaluated at the point in  corresponding to  in the given chart. Extending by linearity gives the following matrix

Thus the differential is a linear transformation, between tangent spaces, associated to the smooth map  at each point. Therefore, in some chosen local coordinates, it is represented by the Jacobian matrix of the corresponding smooth map from  to . In general, the differential need not be invertible. However, if  is a local diffeomorphism, then  is invertible, and the inverse gives the pullback of 

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

It follows from the definition that the differential of a composite is the composite of the differentials (i.e., functorial behaviour). This is the chain rule for smooth maps. Also, the differential of a local diffeomorphism is a linear isomorphism of tangent spaces. The differential on the tangent bundle 
The differential of a smooth map  induces, in an obvious manner, a bundle map (in fact a vector bundle homomorphism) from the tangent bundle of  to the tangent bundle of , denoted by , which fits into the following commutative diagram:

where  and  denote the bundle projections of the tangent bundles of  and  respectively. induces a bundle map from  to the pullback bundle φ∗TN over  via

where  and  The latter map may in turn be viewed as a section of the vector bundle  over M. The bundle map  is also denoted by  and called the tangent map. In this way,  is a functor. Pushforward of vector fields 
Given a smooth map  and a vector field X on M, it is not usually possible to identify a pushforward of X by φ with some vector field Y on N. For example, if the map φ is not surjective, there is no natural way to define such a pushforward outside of the image of φ. Also, if φ is not injective there may be more than one choice of pushforward at a given point. Nevertheless, one can make this difficulty precise, using the notion of a vector field along a map. A section of φ∗TN over M is called a vector field along φ. For example, if M is a submanifold of N and φ is the inclusion, then a vector field along φ is just a section of the tangent bundle of N along M; in particular, a vector field on M defines such a section via the inclusion of TM inside TN. This idea generalizes to arbitrary smooth maps. Suppose that X is a vector field on M, i.e., a section of TM. Then,  yields, in the above sense, the pushforward φ∗X, which is a vector field along φ, i.e., a section of φ∗TN over M.

Any vector field Y on N defines a pullback section φ∗Y of φ∗TN with . A vector field X on M and a vector field Y on N are said to be φ-related if  as vector fields along φ. In other words, for all x in M, . In some situations, given a X vector field on M, there is a unique vector field Y on N which is φ-related to X. This is true in particular when φ is a diffeomorphism. In this case, the pushforward defines a vector field Y on N, given by

A more general situation arises when φ is surjective (for example the bundle projection of a fiber bundle). Then a vector field X on M is said to be projectable if for all y in N, dφx(Xx) is independent of the choice of x in φ−1({y}). This is precisely the condition that guarantees that a pushforward of X, as a vector field on N, is well defined. Examples

Pushforward from multiplication on Lie groups 
Given a Lie group , we can use the multiplication map  to get left multiplication  and right multiplication  maps . These maps can be used to construct left or right invariant vector fields on  from its tangent space at the origin  (which is its associated Lie algebra). For example, given  we get an associated vector field  on  defined by

for every . This can be readily computed using the curves definition of pushforward maps. If we have a curve

where

we get

since  is constant with respect to . This implies we can interpret the tangent spaces  as . Pushforward for some Lie groups 
For example, if  is the Heisenberg group given by matrices

it has Lie algebra given by the set of matrices

since we can find a path  giving any real number in one of the upper matrix entries with  (i-th row and j-th column). Then, for

we have

which is equal to the original set of matrices. This is not always the case, for example, in the group

we have its Lie algebra as the set of matrices

hence for some matrix

we have

which is not the same set of matrices. See also
Pullback (differential geometry)
Flow-based generative model

References 

 
  See section 1.6. See section 1.7 and 2.3. Generalizations of the derivative
Differential geometry
Smooth functions