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 Pushforward (homology), the map induced in homology by a continuous map between topological spaces Pushforward measure, measure induced on the target measure space by a measurable function Pushout (category theory), the categorical dual of pullback Direct image sheaf, the pushforward of a sheaf by a map Fiberwise integral, the direct image of a differential form or cohomology by a smooth map, defined by "integration on the fibres" Transfer operator, the pushforward on the space of measurable functions; its adjoint, the pull-back, is the composition or Koopman operator zh:推出