Wikipedia
Suppose that is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as a generalization of the total derivative of ordinary calculus. Explicitly, it is a linear map from the tangent space of M at x to the tangent space of N at φ(x). Hence it can be used to push tangent vectors on M forward to tangent vectors on N.
The differential of a map φ is also called, by various authors, the derivative or total derivative of φ, and is sometimes itself called the pushforward.
The notion of pushforward in mathematics is "dual" to the notion of pullback, and can mean a number of different, but closely related things.
- Pushforward (differential): the differential of a smooth map between manifolds, and the "pushforward" operations it defines.
- Direct image sheaf: the pushforward of a sheaf by a map.
- Pushforward (homology): the map induced in homology by a continuous map between topological spaces.
- Fiberwise integral: the direct image of a differential form or cohomology by a smooth map, defined by "integration on the fibres".
- Pushout (category theory): the categorical dual of pullback.
- Pushforward measure: measure induced on the target measure space by a measurable function.
- The transfer operator is the pushforward on the space of measurable functions; its adjoint, the pull-back, is the composition or Koopman operator.
In algebraic topology, the pushforward of a continuous function f : X → Y between two topological spaces is a homomorphism f : H(X) → H(Y) between the homology groups for n ≥ 0.
Homology is a functor which converts a topological space X into a sequence of homology groups H(X). (Often, the collection of all such groups is referred to using the notation H(X); this collection has the structure of a graded ring.) In any category, a functor must induce a corresponding morphism. The pushforward is the morphism corresponding to the homology functor.