diffe

Diffe is a fictional term used in introductory treatments of differential geometry and related fields to denote a minimal, structured change between two nearby differentiable states. The concept is not part of standard mathematical vocabulary outside pedagogical examples, but it is used to illustrate how small perturbations propagate through a differentiable mapping.

In this framework, a diffe at a point p on a manifold M, relative to a differentiable map f: M -> N, is the linear approximation that describes how an infinitesimal input variation dx maps to an output variation df_p(dx). It captures the first-order behavior of f and is treated as invariant under smooth coordinate changes to preserve its local meaning. The diffe thus functions similarly to the classical differential, while emphasizing the interpretation as a controlled, differential change in structure.

Applications of the diffe are mainly pedagogical. It is used to contrast discrete differences with continuous derivatives, and to illustrate how linear approximations illuminate nonlinear phenomena. In exercises, it is often introduced alongside the Jacobian, gradients, and the notion of a diffeomorphism to reinforce intuition about locality, perturbation, and the propagation of small changes through mappings.

See also: differential, differential geometry, Jacobian, diffeomorphism. While related in concept, the diffe is presented here as a teaching device rather than a standard term in published mathematical literature.