Posttan

Posttan is a term used in theoretical mathematics to denote a post-processing operation applied to the tangent map of a differentiable function. The name reflects a step that follows the standard tangent mapping, hence “post-tangent.” It is used to study how nonlinear alterations after a tangent map affect vectors in tangent spaces.

In a typical setup, let f: M → N be a smooth map between manifolds and p ∈ M. Its differential df_p maps TpM to T_{f(p)}N. A fixed nonlinear post-processing map h: T_{f(p)}N → T_{f(p)}N defines the posttan transform of v ∈ TpM by Posttan_f(p)(v) = h(df_p(v)). Variants may allow h to depend on p or on a chosen basis, although many formulations keep h fixed across points. If h is linear, Posttan_f(p) reduces to df_p, while nonlinear h can change trajectories and stability properties of the associated dynamical system.

Examples include h(y) = Ay for a fixed linear operator A, yielding Posttan_f(p)(v) = A df_p(v), or h acting componentwise as squaring, which magnifies larger components. Applications appear in theoretical studies of robustness to post-processing of velocity fields and in educational contexts to illustrate how nonlinear post-processing interacts with linearization.

History and usage of the term vary across authors. Posttan is not a standard textbook concept, but it has appeared in advanced notes and discussions on differential geometry and dynamical systems as a way to formalize the impact of post-tangent nonlinearities on linearized models. See also tangent map, differential, Jacobian, nonlinear transformation.