differentiale

Differentiale is a mathematical term occasionally used to denote the family of infinitesimal changes associated with a function or map. While not standard in most contemporary texts, the word appears in historical literature and in discussions that emphasize the collective nature of differentials, differential maps, and related objects.

In single-variable calculus, the differential of a function f at a point x is the linear form df_x = f'(x) dx, expressing the first-order change in f corresponding to an infinitesimal change dx in the input.

More generally, for a smooth map f: M -> N between manifolds, the differential df_p is a linear map from the tangent space T_pM to T_{f(p)}N, capturing the infinitesimal behavior of f near p. In local coordinates it is represented by the Jacobian matrix.

In differential geometry and algebraic geometry, differentiale can refer to families of differential operators or to differential forms. The cotangent bundle and the space of Kähler differentials Omega^1_{A/k} are standard manifestations of such objects, encoding infinitesimal variation and derivations.

Notation typically uses d or delta for differentials rather than "Differentiale", but the term appears in some expository works to emphasize the aggregate of differential operations across a map or manifold.

See also differential, differential form, exterior derivative, Jacobian, tangent space, cotangent space.