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Definition

Given a calculus field$F$sequentially Cauchy complete Archimedean ordered field and a$\mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}$ F \mathbb{R} - and acalculus vector space?real vector space $V$, a scalar function $f:V\to \mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}$ f:V \to F \mathbb{R} is a directionally differentiable function if for every vector $v:V$ and line $L_v \subseteq V$,

with canonical equivalence $m:\mathrm{<span><del\; class="diffmod">\; F</del><ins\; class="diffmod">\; ℝ</ins></span>}\cong L_v$ m:F m:\mathbb{R} \cong L_v and canonical inclusion $i:L_v \hookrightarrow V$, the composite $f \circ i \circ m$ is a differentiable function.

See also

• differentiable function
• directional derivative