Homotopy Type Theory directionally differentiable function > history (Rev #3, changes)

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< directional derivative


Given a sequentially Cauchy complete Archimedean ordered field \mathbb{R} and a real vector space VV, a scalar function f:Vf:V \to \mathbb{R} is a directionally differentiable function if for every vector v:Vv:V and line L vVL_v \subseteq V,

L v v:V a:v=awL_v \coloneqq \sum_{v:V} \Vert \sum_{a:\mathbb{R}} v = a w \Vert

with canonical equivalence m:L vm:\mathbb{R} \cong L_v and canonical inclusion i:L vVi:L_v \hookrightarrow V, the composite fimf \circ i \circ m is a differentiable function.

See also

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